Colored knot Floer homology: structures and examples

TL;DR

This paper introduces a new colored version of knot Floer homology by constructing it as a colimit over infinite full twists, providing explicit descriptions for L-space knots and crossing change maps, inspired by colored Khovanov homology.

Contribution

It defines a colored knot Floer homology via colimits of link Floer homology with infinite twists, extending the structure and understanding of knot invariants.

Findings

Colimit of infinite full twists forms a module over the unknot's colored knot Floer homology.

Explicit description of colored Heegaard Floer homology for L-space knots.

Construction of maps for crossing changes in colored knot Floer homology.

Abstract

Inspired by the $S^n$ colored version of Khovanov and Khovanov-Rozansky homology, we define a colored version of knot Floer homology by studying the colimit of a directed system of link Floer homology with infinite full twists. Specifically, our $n$-colored knot Floer homology of a knot $K$ is then defined as the colimit of the link Floer homology of $(n, mn)$-cables of $K$ by fixing $n$ and letting $m$ goes to infinity. We show that the colimit of the infinite full twists is a module over the colored knot Floer homology of the unknot. In addition, we give an explicit description of colored Heegaard Floer homology for L-space knots, and maps for colored knot Floer homology of crossing changes.

Equations

$$P_{\text{lx@text@underscore}}{\mathfrak{sl}}_{\text{lx@text@underscore}}N,S^n{\mathbb{C}}^N(K)=\underset{\text{lx@text@underscore}}{\lim }p\to \infty P_{\text{lx@text@underscore}}{\mathfrak{sl}}_{\text{lx@text@underscore}}N,{\mathbb{C}}^N(K_{\text{lx@text@underscore}}n,p).$$

$${\mathrm{Kh}}_{\text{lx@text@underscore}}{S}^n(K):=\underrightarrow{\lim }\left[\mathrm{Kh}(K_{\text{lx@text@underscore}}n,0)\stackrel{{\alpha }_{\text{lx@text@underscore}}n}{\to }\mathrm{Kh}(K_{\text{lx@text@underscore}}n,n)\stackrel{{\alpha }_{\text{lx@text@underscore}}n}{\to }\mathrm{Kh}(K_{\text{lx@text@underscore}}n,2n)\stackrel{{\alpha }_{\text{lx@text@underscore}}n}{\to }\cdots \right]$$

$${\mathcal{H}}_{\text{lx@text@underscore}}D(L)=\underrightarrow{\lim }\left[\mathcal{H}\mathcal{F}\mathcal{L}(L)\stackrel{{\varphi }_{\text{lx@text@underscore}}0}{\to }\mathcal{H}\mathcal{F}\mathcal{L}(L_{\text{lx@text@underscore}}1)\stackrel{{\varphi }_{\text{lx@text@underscore}}0}{\to }\mathcal{H}\mathcal{F}\mathcal{L}(L_{\text{lx@text@underscore}}2)\to \cdots \right]$$

$${\mathcal{H}\mathcal{F}\mathcal{L}}^{\mathrm{stab}}(L_{\text{lx@text@underscore}}m;\overline{\mathbf{s}}):=\mathcal{H}\mathcal{F}\mathcal{L}(L_{\text{lx@text@underscore}}m,\mathbf{s})$$

$${\mathcal{H}\mathcal{F}\mathcal{L}}^{\mathrm{stab}}(L_{\text{lx@text@underscore}}m,\overline{\mathbf{s}})\cong {\mathcal{H}\mathcal{F}\mathcal{L}}^{\mathrm{stab}}\left(L_{\text{lx@text@underscore}}m+1,\overline{\mathbf{s}}\right)$$

$$\dim {\mathcal{H}}_{\text{lx@text@underscore}}D,j(L;\overline{\mathbf{s}})\le \dim {\mathcal{H}\mathcal{F}\mathcal{L}}_{\text{lx@text@underscore}}^{\mathrm{stab}}j(L_{\text{lx@text@underscore}}m;\overline{\mathbf{s}}),m\gg 0.$$

$${\varphi }_{\text{lx@text@underscore}}0:{\mathcal{H}\mathcal{F}\mathcal{L}}^{\mathrm{stab}}\left(L_{\text{lx@text@underscore}}m;\overline{\mathbf{s}}\right)\to {\mathcal{H}\mathcal{F}\mathcal{L}}^{\mathrm{stab}}\left(L_{\text{lx@text@underscore}}m+1;\overline{\mathbf{s}}\right)$$

$${\mathcal{H}}_{\text{lx@text@underscore}}n(K)=\underrightarrow{\lim }\left[\mathcal{H}\mathcal{F}\mathcal{L}(K_{\text{lx@text@underscore}}n,0)\stackrel{{\varphi }_{\text{lx@text@underscore}}0}{\to }\mathcal{H}\mathcal{F}\mathcal{L}(K_{\text{lx@text@underscore}}n,n)\stackrel{{\varphi }_{\text{lx@text@underscore}}0}{\to }\mathcal{H}\mathcal{F}\mathcal{L}(K_{\text{lx@text@underscore}}n,2n)\to \cdots \right].$$

$$\chi ({\mathcal{H}}_{\text{lx@text@underscore}}n(K))=\underset{\text{lx@text@underscore}}{\lim }m\to \infty (t_{\text{lx@text@underscore}}1\cdots t_{\text{lx@text@underscore}}n{)}^{-m(n-1)/2}{\chi }_{\text{lx@text@underscore}}{K}_{\text{lx@text@underscore}}n,mn(t_{\text{lx@text@underscore}}1,\dots ,t_{\text{lx@text@underscore}}n)=(t_{\text{lx@text@underscore}}1\cdots t_{\text{lx@text@underscore}}n{)}^{1/2}{\chi }_{\text{lx@text@underscore}}K(t_{\text{lx@text@underscore}}1\cdots t_{\text{lx@text@underscore}}n)$$

$${\mathcal{H}}_{\text{lx@text@underscore}}D(L_{\text{lx@text@underscore}}f)=\underrightarrow{\lim }\left[\mathcal{H}\mathcal{F}\mathcal{L}(K_{\text{lx@text@underscore}}n,fn)\stackrel{{\varphi }_{\text{lx@text@underscore}}0}{\to }\mathcal{H}\mathcal{F}\mathcal{L}(K_{\text{lx@text@underscore}}n,(f+1)n)\stackrel{{\varphi }_{\text{lx@text@underscore}}0}{\to }\mathcal{H}\mathcal{F}\mathcal{L}(K_{\text{lx@text@underscore}}n,(f+2)n)\to \cdots \right].$$

$$\underrightarrow{\lim }\left[\mathcal{H}\mathcal{F}\mathcal{L}(K_{\text{lx@text@underscore}}n,r)\stackrel{{\varphi }_{\text{lx@text@underscore}}0}{\to }\mathcal{H}\mathcal{F}\mathcal{L}(K_{\text{lx@text@underscore}}n,n+r)\stackrel{{\varphi }_{\text{lx@text@underscore}}0}{\to }\mathcal{H}\mathcal{F}\mathcal{L}(K_{\text{lx@text@underscore}}n,2n+r)\to \cdots \right].$$

$${\varphi }_{\text{lx@text@underscore}}0:{\mathcal{H}\mathcal{F}\mathcal{L}}^{\mathrm{stab}}\left(K_{\text{lx@text@underscore}}n,mn+r;\tilde{\mathbf{s}}\right)\to {\mathcal{H}\mathcal{F}\mathcal{L}}^{\mathrm{stab}}\left(K_{\text{lx@text@underscore}}n,(m+1)n+r;\tilde{\mathbf{s}}\right)$$

$$U_{\text{lx@text@underscore}}i=\mathsf{A}\prod _{\text{lx@text@underscore}}j\ne i{V}_{\text{lx@text@underscore}}j,U_{\text{lx@text@underscore}}i{V}_{\text{lx@text@underscore}}i=U_{\text{lx@text@underscore}}{i}^{\prime }{V}_{\text{lx@text@underscore}}{i}^{\prime }i,i^{\prime }=1,\dots ,n.$$

$$G_{\text{lx@text@underscore}}{j}^{\mathrm{col}}:{\mathcal{H}}_{\text{lx@text@underscore}}n(K^{-})\to {\mathcal{H}}_{\text{lx@text@underscore}}n(K^{+}),F_{\text{lx@text@underscore}}{j}^{\mathrm{col}}:{\mathcal{H}}_{\text{lx@text@underscore}}n(K^{+})\to {\mathcal{H}}_{\text{lx@text@underscore}}n(K^{-})$$

$${\text{gr}}_{\text{lx@text@underscore}}\mathbf{w}(G_{\text{lx@text@underscore}}{j}^{\mathrm{col}})={\text{gr}}_{\text{lx@text@underscore}}\mathbf{w}(F_{\text{lx@text@underscore}}{j}^{\mathrm{col}})=-j^2-j,A_{\text{lx@text@underscore}}i(G_{\text{lx@text@underscore}}{j}^{\mathrm{col}})=-2j+1,A_{\text{lx@text@underscore}}i(F_{\text{lx@text@underscore}}{j}^{\mathrm{col}})=n-1.$$

$${\epsilon }_{\text{lx@text@underscore}}n(U)=\mathsf{A},{\epsilon }_{\text{lx@text@underscore}}n(V)=V_{\text{lx@text@underscore}}1\cdots V_{\text{lx@text@underscore}}n.$$

$${\mathcal{H}}_{\text{lx@text@underscore}}n(K)\simeq \mathcal{H}\mathcal{F}\mathcal{L}(K)\underset{\text{lx@text@underscore}}{⨂}\mathbb{F}[U,V]{\mathcal{A}}_{\text{lx@text@underscore}}{n}^{\mathrm{col}}$$

$$U_{\text{lx@text@underscore}}I{a}_{\text{lx@text@underscore}}k-1=V_{\text{lx@text@underscore}}\overline{I}{a}_{\text{lx@text@underscore}}k$$

$$a_{\text{lx@text@underscore}}i{a}_{\text{lx@text@underscore}}j={\mathbf{U}}^{k\ell -ij}{a}_{\text{lx@text@underscore}}k{a}_{\text{lx@text@underscore}}\ell$$

$$A(a_{\text{lx@text@underscore}}k)=\left(\frac{n-1}{2}-k,\dots ,\frac{n-1}{2}-k\right),$$

$${\mathcal{T}\mathcal{W}}_{\text{lx@text@underscore}}D(L)=\underset{\text{lx@text@underscore}}{⨁}{m=0}^{\infty }\mathcal{H}\mathcal{F}\mathcal{L}(L_{\text{lx@text@underscore}}m)$$

$${\varphi }_{\text{lx@text@underscore}}k:\mathcal{H}\mathcal{F}\mathcal{L}(L_{\text{lx@text@underscore}}m)\to \mathcal{H}\mathcal{F}\mathcal{L}(L_{\text{lx@text@underscore}}m+1).$$

$${\mathcal{T}\mathcal{W}}_{\text{lx@text@underscore}}D(O_{\text{lx@text@underscore}}n)=\underset{\text{lx@text@underscore}}{⨁}{m=0}^{\infty }\mathcal{H}\mathcal{F}\mathcal{L}(T(n,mn))$$

$$\mathbb{F}[X_{\text{lx@text@underscore}}0]=\mathcal{H}\mathcal{F}\mathcal{L}(O_{\text{lx@text@underscore}}n)$$

$$X\subset X_{\text{lx@text@underscore}}0\times {\mathbb{P}}_{\text{lx@text@underscore}}^{n-1}\mathbb{F}\subset {\mathbb{A}}_{\text{lx@text@underscore}}^{2n}\mathbb{F}\times {\mathbb{P}}_{\text{lx@text@underscore}}^{n-1}\mathbb{F}$$

$$\mathbb{F}[{\mathcal{U}}_{\text{lx@text@underscore}}0]={\mathcal{A}}_{\text{lx@text@underscore}}n[a_{\text{lx@text@underscore}}{0}^{-1}]={\mathcal{A}}_{\text{lx@text@underscore}}{n}^{\mathrm{col}}.$$

$${\mathrm{Kh}}_{\text{lx@text@underscore}}{S}^n(O_{\text{lx@text@underscore}}1)\simeq H^{\ast }\left(\mathbb{Z}[u_{\text{lx@text@underscore}}0,\dots ,u_{\text{lx@text@underscore}}n-1,{\xi }_{\text{lx@text@underscore}}0,\dots ,{\xi }_{\text{lx@text@underscore}}n-1],d_{\text{lx@text@underscore}}2\right).$$

$${\mathrm{HY}}_{\text{lx@text@underscore}}{S}^n(O)\simeq \mathbb{Q}[u_{\text{lx@text@underscore}}0,\dots ,u_{\text{lx@text@underscore}}n-1,{\xi }_{\text{lx@text@underscore}}0,\dots ,{\xi }_{\text{lx@text@underscore}}n-1,y_{\text{lx@text@underscore}}1,\dots ,y_{\text{lx@text@underscore}}n].$$

$$x_{\text{lx@text@underscore}}i=u_{\text{lx@text@underscore}}0+u_{\text{lx@text@underscore}}1y_{\text{lx@text@underscore}}i+\dots +u_{\text{lx@text@underscore}}n-1y_{\text{lx@text@underscore}}{i}^{n-1},i=1,\dots ,n.$$

$$x_{\text{lx@text@underscore}}i\mapsto U_{\text{lx@text@underscore}}i,y_{\text{lx@text@underscore}}i\mapsto V_{\text{lx@text@underscore}}i,u_{\text{lx@text@underscore}}k\mapsto (-1{)}^k{e}_{\text{lx@text@underscore}}n-1-k(V_{\text{lx@text@underscore}}1,\dots ,V_{\text{lx@text@underscore}}n)\mathsf{A}$$

Full text

AA was partly supported by NSF Grant DMS-2238103

EG was partly supported by NSF Grant DMS-2302305

BL was partly supported by NSF Grant DMS-2417229

1. Introduction

1.1. Motivation: colored invariants via cables

One of the central ideas in quantum topology is the notion of colored knot invariants [33]. Given a knot $K\subset S^{3}$, a semisimple Lie algebra $\mathfrak{g}$ and its representation $V$, one can define the Laurent polynomial $P_{\lx@text@underscore}{\mathfrak{g},V}(K)\in\mathbb{Z}[q,q^{-1}]$ which is a topological invariant of $K$. For example, for $\mathfrak{g}=\mathfrak{sl}_{\lx@text@underscore}2$ and $V=\mathbb{C}^{2}$ one gets Jones polynomial, while $\mathfrak{g}=\mathfrak{sl}_{\lx@text@underscore}2$ and $V=S^{n}\mathbb{C}^{2}\simeq\mathbb{C}^{n+1}$ corresponds to the so-called colored Jones polynomial [26]. More generally, one refers to $P_{\lx@text@underscore}{\mathfrak{g},V}(K)$ as the invariant of $K$ colored by the representation $V$.

For $\mathfrak{g}=\mathfrak{sl}_{\lx@text@underscore}N$ Rosso and Jones [34] proved that the invariants of any cable $K_{\lx@text@underscore}{n,p}$ of the knot $K$ are related to colored invariants of $K$. In particular, one has the following identity

[TABLE]

On the left hand side of (1) we get $S^{n}\mathbb{C}^{N}$-colored (or simply $S^{n}$-colored) invariant of $K$, while on the right hand side we get the limit of uncolored invariants of cables $K_{\lx@text@underscore}{n,p}$ normalized by a certain monomial in $q$ which we omit. The limit means that for all $i$ and sufficiently large $p$ the coefficient at $q^{i}$ in the right hand side stabilizes.

In recent decades, there was a lot of work [7, 10, 18, 19, 35] towards categorifying (1) in the context of Khovanov and Khovanov-Rozansky homology [20, 21, 22]. In particular, for $N=2$ the (uncolored) Jones polynomial is categorified by Khovanov homology $\mathrm{Kh}(K)$ and one can define the $S^{n}$-colored Khovanov homology as the colimit

[TABLE]

The terms on the right hand side differ by adding one full twist, and the colimit is often referred to as “infinite full twist” [35].

The key novel feature of (2) are the connecting maps $\alpha_{\lx@text@underscore}n:\mathrm{Kh}\left(K_{\lx@text@underscore}{n,mn}\right)\to\mathrm{Kh}\left(K_{\lx@text@underscore}{n,(m+1)n}\right)$ which are required for the definition of colimit. The colimit on the right hand side of (2) is a bigraded vector space which is typically infinite-dimensional, but one can prove that it is finite-dimensional in each bidegree. See [10, 18, 19, 35] and references therein for more details and an explicit description of the maps $\alpha_{\lx@text@underscore}n$, and related computations in Khovanov-Rozansky homology. See Section 1.7 for some computations of colored homology.

In this paper, we define and study some analogues of (1) and (2) in the context of Alexander polynomial and Heegaard Floer homology. In fact, the analogue of (1) is not too interesting, as the Alexander polynomials for cables of $K$ are proportional to the Alexander polynomial of $K$ evaluated at $t_{\lx@text@underscore}1\cdots t_{\lx@text@underscore}n$ (see Section 2.4 for more details). However, an analogue of (2), which we are about to define, has a rich and interesting structure.

1.2. Colored knot Floer homology

Analogously to (2), we define a colimit invariant under full twists by using link Floer homology. Let $L\subset S^{3}$ be an oriented $n$-component link and $M$ be an unknot bounding a disk $D$ that intersects every component positively at exactly one point. Let $L_{\lx@text@underscore}m$ denote the link obtained by inserting $m$ full twists in $L$. Then we can define a colimit of the following directed system of $\mathcal{H\!F\!L}(L_{\lx@text@underscore}m)$:

[TABLE]

where the connecting map $\phi_{\lx@text@underscore}0$ is a cobordism map induced by blowing down the $(-1)$-framed unknot $M$ in some $\text{Spin}^{c}$-structure (see Proposition 2.7), and we use the “full” version of link Floer homology developed in [38, 39]. In particular, $\mathcal{H\!F\!L}(L_{\lx@text@underscore}m)$ is a module over $\mathbb{F}[U_{\lx@text@underscore}1,\ldots,U_{\lx@text@underscore}n,V_{\lx@text@underscore}1,\ldots,V_{\lx@text@underscore}n]$.

In our first main result, we prove that $\mathcal{H}_{\lx@text@underscore}{D}(L)$ is well defined.

Theorem 1.1**.**

The $\mathbb{F}$-vector space $\mathcal{H}_{\lx@text@underscore}{D}(L)$ has well defined Alexander $\mathbb{Z}^{n}$-grading, and Maslov $\mathbb{Z}$-grading. The homogeneous component of each $\mathbb{Z}^{n}\oplus\mathbb{Z}$-degree is finite-dimensional.

To prove Theorem 1.1, we normalize the Alexander degrees such that they are preserved by the maps $\phi_{\lx@text@underscore}0$. More precisely, we define

[TABLE]

where $\overline{\mathbf{s}}=\mathbf{s}-(\mathbf{c}_{\lx@text@underscore}m,\ldots,\mathbf{c}_{\lx@text@underscore}m)$ and $\mathbf{c}_{\lx@text@underscore}m=m(n-1)/2$. We then prove that for any fixed normalized Alexander degree $\overline{\mathbf{s}}$ the dimension of $\mathcal{H\!F\!L}^{\mathrm{stab}}(L_{\lx@text@underscore}m;\overline{\mathbf{s}})$ stabilizes for sufficiently large $m$:

Theorem 1.2**.**

For any $\overline{\mathbf{s}}=(\overline{s}_{\lx@text@underscore}1,\overline{s}_{\lx@text@underscore}2,\cdots,\overline{s}_{\lx@text@underscore}n)$ with $\overline{s}_{\lx@text@underscore}i\geq C-m$ for all $i$ we have

[TABLE]

Here, $C$ is some constant depending on $L$ but independent of $m$.

The isomorphism in Theorem 1.2 is obtained by analyzing certain special Heegaard diagrams of $L_{\lx@text@underscore}{m}$ and $L_{\lx@text@underscore}{m+1}$, constructing an explicit bijection on generators and verifying that it is a chain map, see Theorem 3.2. Theorem 1.2 gives an upper bound for the dimension of the limit (see Lemma 2.2) for each Maslov grading $j$:

[TABLE]

In fact, we conjecture a stronger statement.

Conjecture 1.3**.**

Given a normalized Alexander degree $\overline{\mathbf{s}}$, the maps

[TABLE]

are isomorphisms for sufficiently large $m$. As a consequence, $\dim\mathcal{H}_{\lx@text@underscore}{D,j}(L;\overline{\mathbf{s}})=\dim\mathcal{H\!F\!L}^{\mathrm{stab}}_{\lx@text@underscore}j\left(L_{\lx@text@underscore}m;\overline{\mathbf{s}}\right)$ for $m\gg 0$.

Remark 1.4**.**

Based on Theorem 1.2, one might be tempted to define a “naive” version of $\mathcal{H}_{\lx@text@underscore}D(L)$ by simply computing $\mathcal{H\!F\!L}^{\mathrm{stab}}\left(L_{\lx@text@underscore}m;\overline{\mathbf{s}}\right)$ for $m\gg 0$. However, it is unclear whether the isomorphisms constructed in the proof of Theorem 1.2 have any nice topological properties.

On the other hand, the map $\phi_{\lx@text@underscore}0$ does interact well with the cobordism maps in Heegaard Floer homology which yields nice properties of the colimit (3). See, in particular, Theorems 1.13 and 1.15 below.

Remark 1.5**.**

Throughout this paper, for simplicity, we focus on the case that every component of the link intersects the disk $D$ positively in one point. However, an analogous version of Theorem 1.2 holds in general. More precisely, suppose there exists a subset $I\subset\{1,2,\cdots,n\}$ such that every component of $L_{\lx@text@underscore}I=\amalg_{\lx@text@underscore}{i\in I}L_{\lx@text@underscore}i$ (resp. $L_{\lx@text@underscore}{\bar{I}}=\amalg_{\lx@text@underscore}{i\in\bar{I}}L_{\lx@text@underscore}i$) intersects $D$ negatively (resp. positively) at one point, where $\bar{I}=\{1,2,\cdots,n\}\setminus I$. Let $|I|=l$. If we use $\phi_{\lx@text@underscore}{-l}$ to define $\mathcal{H}_{\lx@text@underscore}{D}(L)$ instead of $\phi_{\lx@text@underscore}0$ then there is a constant $C$ independent of $m$ such that for any $\bar{\mathbf{s}}$ with $\bar{s}_{\lx@text@underscore}i\leq-(C-m)$ for $i\in I$ and $\bar{s}_{\lx@text@underscore}i\geq C-m$ for $i\in\bar{I}$ we have $\mathcal{H\!F\!L}^{\mathrm{stab}}(L_{\lx@text@underscore}{m};\bar{\mathbf{s}})\cong\mathcal{H\!F\!L}^{\mathrm{stab}}(L_{\lx@text@underscore}{m+1};\bar{\mathbf{s}})$. Here, Alexander grading is renormalized such that $\bar{s}_{\lx@text@underscore}i=s_{\lx@text@underscore}i+\mathbf{c}_{\lx@text@underscore}m$ for $i\in I$ and $\bar{s}_{\lx@text@underscore}{i}=s_{\lx@text@underscore}i-\mathbf{c}_{\lx@text@underscore}m$ for $i\in\bar{I}$. This is a direct consequence of Theorem 1.2 and the symmetry of link Floer homology, that is $\mathcal{H\!F\!L}(L,\mathbf{s})\cong\mathcal{H\!F\!L}(L^{\prime};\mathbf{s}^{\prime})$ where $L^{\prime}=(-L_{\lx@text@underscore}{I})\amalg L_{\lx@text@underscore}{\bar{I}}$ and $\mathbf{s}^{\prime}$ is obtained from $\mathbf{s}$ by multiplying every $i$-th entry with $i\in I$ by $-1$. Consequently, we can define $\mathcal{H}_{\lx@text@underscore}{D}(L)$ with connecting homomorphisms $\phi_{\lx@text@underscore}{-l}$ and after proper renormalization of the Maslov grading, the limit is graded by Maslov and Alexander grading and is finite dimensional in each degree.

Given a knot $K\subset S^{3}$ and an integer $n>0$ we consider the family of $n$-component cables $K_{\lx@text@underscore}{n,mn}$ for $m\geq 0$ and their link Floer homology $\mathcal{H\!F\!L}(K_{\lx@text@underscore}{n,mn})$. All components of $K_{\lx@text@underscore}{n,mn}$ are isotopic to $K$ and their pairwise linking numbers are all equal to $m$. Applying the colimit definition to the link $K_{\lx@text@underscore}{n,mn}$, we define the $n$-colored knot Floer homology of $K$.

Definition 1.6**.**

We define the $n$-colored knot Floer homology of $K$ as the colimit of the directed system

[TABLE]

This is a special case of $\mathcal{H}_{\lx@text@underscore}D(L)$ where $L=K_{\lx@text@underscore}{n,0}$ and the disk $D$ bounds the meridian of $K$.

For $n=1$ all cables $K_{\lx@text@underscore}{1,m}$ coincide with $K$ and we get $\mathcal{H}_{\lx@text@underscore}1(K)=\mathcal{H\!F\!L}(K)$. In Lemma 6.1 we prove that Conjecture 1.3 holds whenever $K$ is an L-space knot. Assuming Conjecture 1.3, the inequality (5) becomes an equality, and we get the ranks of $\mathcal{H}_{\lx@text@underscore}n(K)$ for all (normalized) Alexander degrees, as well as its Euler characteristic.

Theorem 1.7**.**

Assuming Conjecture 1.3, the Euler characteristic of $\mathcal{H}_{\lx@text@underscore}{n}(K)$ is given by

[TABLE]

where $\chi_{\lx@text@underscore}K(t)=\frac{\Delta_{\lx@text@underscore}K(t)}{1-t^{-1}}$ is the Euler characteristic of $\mathcal{H\!F\!L}(K)$.

Remark 1.8**.**

We can choose some framing $f$ on the knot $K$ and consider the $n$-component link $L_{\lx@text@underscore}f=K_{\lx@text@underscore}{n,fn}$ whose components are pushoffs of $K$ along the framing. Adding a full twist changes $f$ to $f+1$, and we obtain the colimit

[TABLE]

It is easy to see that as a vector space this does not depend on $f$ and is isomorphic to $\mathcal{H}_{\lx@text@underscore}n(K)$. The Maslov grading does not depend on $f$, but the normalized Alexander grading is shifted by $\left(\frac{f(n-1)}{2},\ldots,\frac{f(n-1)}{2}\right)$.

Remark 1.9**.**

One can define a colimit using different choices of connecting maps $\phi_{\lx@text@underscore}k$ corresponding to different $\text{Spin}^{c}$ structures in the surgery cobordism. We expect that other $\phi_{\lx@text@underscore}k$ correspond to different “colors” of the homology. For simplicity of computation, we focus on the choice of $\phi_{\lx@text@underscore}0$ throughout the paper. By conjugation, the colimit for the choice of $\phi_{\lx@text@underscore}{n-1}$ is the same as the choice of $\phi_{\lx@text@underscore}0$ up to some regrading and interchanging $U_{\lx@text@underscore}i$ with $V_{\lx@text@underscore}i$.

Remark 1.10**.**

One can define a similar colimit when components of the cables are oriented differently, see Remark 1.5.

Instead of considering the family of $(n,mn)$ cables of $K$ with $n$ components, we can instead consider the family of $(n,mn+r)$-cables of $K$ for fixed remainder $r$. These links have $\gcd(n,r)$ connected components; in particular, for $r=1$ we get the family of knots $K_{\lx@text@underscore}{n,mn+1}$. The maps $\phi_{\lx@text@underscore}0:\mathcal{H\!F\!L}(K_{\lx@text@underscore}{n,mn+r})\to\mathcal{H\!F\!L}(K_{\lx@text@underscore}{n,(m+1)n+r})$ can be defined as above, and we can consider the directed system

[TABLE]

The following conjecture, if true, would imply that the colimit of (7) is well defined. By taking into account the Alexander grading shift of $\phi_{\lx@text@underscore}0$, one can define the normalized Alexander grading $\widetilde{\mathbf{s}}$ which generalizes $\overline{\mathbf{s}}$ and is preserved by $\phi_{\lx@text@underscore}0$.

Conjecture 1.11**.**

a) For a given normalized Alexander degree $\widetilde{\mathbf{s}}$, the dimension of $\mathcal{H\!F\!L}^{\mathrm{stab}}(K_{\lx@text@underscore}{n,mn+r},\widetilde{\mathbf{s}})$ stabilizes for sufficiently large $m$.

b) The maps

[TABLE]

are isomorphisms for sufficiently large $m$.

At the moment, we cannot generalize our proof of Theorem 1.2 to the case $r\neq 0$. However, for $r=1$ Hedden proved Conjecture 1.11(a) in [16, 17] for minus and hat versions of Heegaard Floer homology, but part (b) of the conjecture appears to be open in these cases as well.

Remark 1.12**.**

Assuming Conjecture 1.11(a), it appears that many results of the paper can be applied to (7) for arbitrary $r$, with some small modifications. We plan to investigate the properties of more general colored homology (7) in future work.

Next, we study the module structure of $\mathcal{H}_{\lx@text@underscore}{D}(L)$. Since the connecting maps $\phi_{\lx@text@underscore}0$ commute with the action of $U_{\lx@text@underscore}i,V_{\lx@text@underscore}i$, the limit is clearly a module over $\mathbb{F}[U_{\lx@text@underscore}1,\ldots,U_{\lx@text@underscore}n,V_{\lx@text@underscore}1,\ldots,V_{\lx@text@underscore}n]$. However, $\mathcal{H}_{\lx@text@underscore}{D}(L)$ has a richer module structure over a commutative algebra $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$, which is a localization of the cable algebra $\mathcal{A}_{\lx@text@underscore}n$ defined in subsection 1.5.

Theorem 1.13**.**

Consider the commutative algebra $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$ with generators $U_{\lx@text@underscore}1,\ldots,U_{\lx@text@underscore}n,V_{\lx@text@underscore}1,\ldots,V_{\lx@text@underscore}n,\mathsf{A}$ and relations

[TABLE]

Then for an arbitrary $n$-component link $L$ there is an action of $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$ on $\mathcal{H}_{\lx@text@underscore}{D}(L)$ (in particular, on $\mathcal{H}_{\lx@text@underscore}n(K)$ for any knot $K$). The generator $\mathsf{A}$ changes Alexander degree by $(-1,\ldots,-1)$ and Maslov degree by $-2$.

Problem 1.14**.**

Is $\mathcal{H}_{\lx@text@underscore}{D}(L)$ a finitely generated module over $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$?

1.3. Colored knot Floer homology and crossing changes

Let $K^{+},K^{-}$ denote two knot diagrams that differ at a single crossing, and $K^{+}$ represents the one with a positive crossing, $K^{-}$ represents the one with a negative crossing. There are cobordism maps between $\mathcal{H\!F\!L}(K^{+})$ and $\mathcal{H\!F\!L}(K^{-})$ induced by blowing down a $(-1)$-framed unknot. We study the colored knot Floer homology of $K^{+},K^{-}$ respectively, and find analogous cobordism maps between them.

Theorem 1.15**.**

For any $j\in\mathbb{Z}$, there are maps

[TABLE]

with

[TABLE]

All maps $G_{\lx@text@underscore}j^{\mathrm{col}},F_{\lx@text@underscore}j^{\mathrm{col}}$ commute with the action of the algebra $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$.

1.4. Examples

We compute a lot of examples of colored homology, starting with the unknot. In this case, $K_{\lx@text@underscore}{n,mn}=T(n,mn)$ is a torus link, and its “full” link Floer homology was computed in [5]. This computation can be used to prove the following:

Theorem 1.16**.**

If $K=O_{\lx@text@underscore}1$ is the unknot then $\mathcal{H}_{\lx@text@underscore}{n}(O_{\lx@text@underscore}1)$ is a free rank one module over $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$. In particular, it is isomorphic to $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$ as an $\mathbb{F}$-vector space.

More generally, we can compute the colored homology when $K$ is an L-space knot. Recall that an L-space is a 3-manifold with minimal possible rank of Heegaard Floer homology [29]. A knot (resp. link) $K$ is called an L-space knot (resp. L-space link) if all sufficiently large Dehn surgeries of $S^{3}$ along $K$ yield L-spaces [25]. If $K$ is an L-space knot then by [13] for $m\gg 0$ the cable $K_{\lx@text@underscore}{n,mn}$ is an L-space link, and $\mathcal{H\!F\!L}(K_{\lx@text@underscore}{n,mn})$ is determined by $\mathcal{H\!F\!L}(K)$ and, in fact, by the Alexander polynomial of $K$. This allows us to prove the following.

Theorem 1.17**.**

Consider the homomorphism $\varepsilon_{\lx@text@underscore}n:\mathbb{F}[U,V]\to\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$ defined by

[TABLE]

Assume $K$ is an L-space knot, then

[TABLE]

where we regard $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$ as a module over $\mathbb{F}[U,V]$ via the homomorphism $\varepsilon_{\lx@text@underscore}n$.

See Section 6 for more details. In particular, we give an explicit description of $\mathcal{H}_{\lx@text@underscore}{n}(K)$ by generators and relations in Theorem 6.2.

1.5. Cabled homology

In order to prove Theorem 1.13, we need to study the relation between $\mathcal{H\!F\!L}(L_{\lx@text@underscore}m)$ for different $m$ in more detail.

Definition 1.18**.**

The $n$-strand cable algebra $\mathcal{A}_{\lx@text@underscore}n$ is defined as a $\mathbb{Z}^{n}\oplus\mathbb{Z}\oplus\mathbb{Z}$ graded algebra over $\mathbb{F}[U_{\lx@text@underscore}1,\ldots,U_{\lx@text@underscore}n,V_{\lx@text@underscore}1,\ldots,V_{\lx@text@underscore}n,\mathbf{U}]$ with commuting generators $a_{\lx@text@underscore}0,\ldots,a_{\lx@text@underscore}{n-1}$ and the following relations:

• •

$U_{\lx@text@underscore}iV_{\lx@text@underscore}i=\mathbf{U},\ i=1,\ldots,n$**

• •

Linear relations

[TABLE]

where $I\subset\{1,\ldots,n\}$ is an arbitrary subset with $|I|=k$ and $\overline{I}=\{1,\ldots,n\}\setminus I$. Here, $U_{\lx@text@underscore}I=\prod_{\lx@text@underscore}{i\in I}U_{\lx@text@underscore}i$ and $V_{\lx@text@underscore}{\overline{I}}=\prod_{\lx@text@underscore}{i\notin I}V_{\lx@text@underscore}i$.

• •

Quadratic relations

[TABLE]

whenever $i+j=k+\ell$ and $i\leq k\leq\ell\leq j$.

The gradings correspond to the Alexander ($\mathbb{Z}^{n}$) and Maslov ($\mathbb{Z}$) gradings, along with an additional $\mathbb{Z}$ “twist grading”. The generator $a_{\lx@text@underscore}k$ has Alexander grading

[TABLE]

Maslov grading $\textup{gr}_{\lx@text@underscore}{{\bf{w}}}(a_{\lx@text@underscore}k)=-k^{2}-k$ and twist grading $\mathbf{tw}(a_{\lx@text@underscore}k)=1$.

Theorem 1.19**.**

Consider the direct sum

[TABLE]

with its Alexander and Maslov grading together with additional “twist” grading given by $m$. Then $\mathcal{TW}_{\lx@text@underscore}D(L)$ is a $\mathbb{Z}^{n}\oplus\mathbb{Z}\oplus\mathbb{Z}$-graded module over $\mathcal{A}_{\lx@text@underscore}n$ where the generators $a_{\lx@text@underscore}k$ act by certain cobordism maps

[TABLE]

The maps $\phi_{\lx@text@underscore}k$ are defined geometrically. Actually they are the cobordism maps induced from blowing down the $(-1)$-framed unknot corresponding to different $\text{Spin}^{c}$ structures, see Proposition 2.7. In order to prove Theorem 1.19 one needs to verify the relations (9) and (10) for them. We first verify these equations in Theorem 4.6 when $L=O_{\lx@text@underscore}n$ is the $n$-component unlink, and prove that

[TABLE]

is a free rank one module over $\mathcal{A}_{\lx@text@underscore}n.$ We then use naturality properties of link Floer homology [38] to complete the proof for general $K$ in Theorem 5.1.

Theorem 1.13 can then be deduced from Theorem 1.19 and the following result which we prove as Theorem 5.2.

Theorem 1.20**.**

The algebra $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$ is isomorphic to (graded) localization $\mathcal{A}_{\lx@text@underscore}n[a_{\lx@text@underscore}0^{-1}]$. Under this isomorphism, the generator $\mathsf{A}$ corresponds to $a_{\lx@text@underscore}1/a_{\lx@text@underscore}0$.

1.6. Algebro-geometric interpretation

Motivated by [13, 14], we can give a geometric interpretation of the algebras $\mathcal{A}_{\lx@text@underscore}n$ and $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$. It is a special case of the so-called $\mathrm{Proj}$ construction in algebraic geometry111We work over the field $\mathbb{F}$ which is not algebraically closed. For more precise statements, one needs to either consider the algebraic closure of $\mathbb{F}$ or assume that all results above have characteristic zero analogues and work over $\mathbb{C}$..

Consider the $2n$-dimensional affine space $\mathbb{A}^{2n}_{\lx@text@underscore}{\mathbb{F}}$ over the field $\mathbb{F}$ with coordinates $U_{\lx@text@underscore}1,\ldots,U_{\lx@text@underscore}n,V_{\lx@text@underscore}1,\ldots,V_{\lx@text@underscore}n.$ The equations $U_{\lx@text@underscore}iV_{\lx@text@underscore}i=U_{\lx@text@underscore}jV_{\lx@text@underscore}j$ for all $i\neq j$ define the affine algebraic variety $X_{\lx@text@underscore}0\subset\mathbb{A}^{2n}_{\lx@text@underscore}{\mathbb{F}}$ such that

[TABLE]

where $O_{\lx@text@underscore}n$ is the $n$-component unlink. Note that $\mathcal{H\!F\!L}(O_{\lx@text@underscore}n)$ is also isomorphic to the component of $\mathcal{A}_{\lx@text@underscore}n$ of twist degree 0.

To give a geometric interpretation to the whole algebra $\mathcal{A}_{\lx@text@underscore}n$, we consider an auxiliary projective space $\mathbb{P}^{n-1}_{\lx@text@underscore}{\mathbb{F}}$ with homogeneous coordinates $[a_{\lx@text@underscore}0:\ldots:a_{\lx@text@underscore}{n-1}]$. Then equations (9) and (10) define the algebraic variety

[TABLE]

By definition, the homogeneous coordinate ring of $X$ is isomorphic to $\mathcal{A}_{\lx@text@underscore}n$.

Next, consider the open chart $\{a_{\lx@text@underscore}0\neq 0\}\simeq\mathbb{A}^{n-1}_{\lx@text@underscore}{\mathbb{F}}$ in $\mathbb{P}^{n-1}_{\lx@text@underscore}{\mathbb{F}}$. By intersecting it with $X$, we obtain an open subset $\mathcal{U}_{\lx@text@underscore}0$. This is again an affine algebraic variety with the coordinate ring

[TABLE]

Finally, we can interpret Theorem 1.19 by saying that $\mathcal{TW}_{\lx@text@underscore}D(L)$ defines a quasi-coherent sheaf on $X$, and colored homology $\mathcal{H}_{\lx@text@underscore}{D}(L)$ corresponds to the restriction of this sheaf to the open chart $\mathcal{U}_{\lx@text@underscore}0.$

Note that restricting to a different chart $\{a_{\lx@text@underscore}k\neq 0\}$ would correspond to colimit (6) with connecting maps $\phi_{\lx@text@underscore}k$.

1.7. Comparison with Khovanov-Rozansky homology

As mentioned above, our definition of colored homology is inspired by the $S^{n}$-colored Khovanov and Khovanov-Rozansky homology, it would be interesting to find any relation to these. First, we recall some known computations of colored homology, starting with triply graded homology $\mathrm{HHH}$ corresponding to (colored) HOMFLY-PT polynomial.

Theorem 1.21** ([18]).**

The $S^{n}$-colored triply graded Khovanov-Rozansky homology of the unknot is a free polynomial algebra $\mathrm{HHH}_{\lx@text@underscore}{S^{n}}(O_{\lx@text@underscore}1)$ with even generators $u_{\lx@text@underscore}0,\ldots,u_{\lx@text@underscore}{n-1}$ and odd generators $\xi_{\lx@text@underscore}0,\ldots,\xi_{\lx@text@underscore}{n-1}$.

Rasmussen [32] constructed a spectral sequence from $\mathrm{HHH}$ to Khovanov homology. It is expected that it extends to colored homology, more precisely, we get the following.

Conjecture 1.22** ([15]).**

The Rasmussen spectral sequence from $\mathrm{HHH}_{\lx@text@underscore}{S^{n}}(O_{\lx@text@underscore}1)$ to $\mathrm{Kh}_{\lx@text@underscore}{S^{n}}(O_{\lx@text@underscore}1)$ has only one nontrivial differential $d_{\lx@text@underscore}2$ given by $d_{\lx@text@underscore}2(u_{\lx@text@underscore}i)=0,\ d_{\lx@text@underscore}2(\xi_{\lx@text@underscore}i)=\sum_{\lx@text@underscore}{j=0}^{i}u_{\lx@text@underscore}ju_{\lx@text@underscore}{i-j}.$ As a consequence,

[TABLE]

Beliakova, Putyra, Robert and Wagner [3] recently defined another spectral sequence from (reduced) triply graded homology to the hat-version of Heegaard Floer homology, confirming a conjecture of Dunfield, Gukov and Rasmussen [11]. It would be interesting to extend this spectral sequence to the “full” and colored versions of knot Floer homology. Following the computations in [1], we expect that $\mathcal{H\!F\!L}$ is in fact related to the so-called “$y$-ified homology” $\mathrm{HY}$ defined in [12]. We have the following result.

Theorem 1.23** ([2]).**

The $S^{n}$–colored $y$-ified homology of the unknot with rational coefficients is a free polynomial algebra:

[TABLE]

It is a module over $\mathbb{Q}[x_{\lx@text@underscore}1,\ldots,x_{\lx@text@underscore}n,y_{\lx@text@underscore}1,\ldots,y_{\lx@text@underscore}n]$ where

[TABLE]

One could hope that Theorem 1.23 holds over $\mathbb{Z}$ (or over $\mathbb{F}$) and speculate that the hypothetical spectral sequence from $\mathrm{HY}$ to $\mathcal{H\!F\!L}$ sends $x_{\lx@text@underscore}i$ to $U_{\lx@text@underscore}i$ and $y_{\lx@text@underscore}i$ to $V_{\lx@text@underscore}i$, see [1] for more details and the comparison of homology of $T(n,n)$ in both theories. Quite surprisingly, it turns out that the equations (8) are a specialization of (11), and one can hope that the colored homologies are compatible as well.

Lemma 1.24**.**

Let $e_{\lx@text@underscore}{m}(V_{\lx@text@underscore}1,\ldots,V_{\lx@text@underscore}n)$ be the elementary symmetric polynomials in $V_{\lx@text@underscore}i$. Under the homomorphism defined by

[TABLE]

the equation (11) transforms into (8).

Example 1.25**.**

For $n=3$ we get

[TABLE]

Proof of Lemma 1.24: Let $e_{\lx@text@underscore}m=e_{\lx@text@underscore}{m}(V_{\lx@text@underscore}1,\ldots,V_{\lx@text@underscore}n)$ and $\widehat{e}_{\lx@text@underscore}m=e_{\lx@text@underscore}{m}(V_{\lx@text@underscore}1,\ldots,\widehat{V_{\lx@text@underscore}i},\ldots,V_{\lx@text@underscore}n).$ It is easy to see that $e_{\lx@text@underscore}m=\widehat{e}_{\lx@text@underscore}m+V_{\lx@text@underscore}i\widehat{e}_{\lx@text@underscore}{m-1}.$ After applying the homomorphism to the right hand side of (11) we get

[TABLE]

[TABLE]

All of the terms cancel out except for

[TABLE]

which agrees with the right hand side of (8). ∎

1.8. Relation to other works

In this section, we briefly discuss the relation of our results to other work.

• (1)

The study of cables in Heegaard Floer homology has a long and rich history, starting from the pioneering works of Hedden [16, 17] and including very recent paper of Chen, Zemke and Zhou [6]. We refer to [6] for more references and context. We note, however, that most references discuss cables where the pattern has one component (so the cable of $K$ is again a knot) while we prefer working with cables which have $n$ components.

In particular, our proof of Theorem 1.1 uses Heegaard diagrams of $K_{\lx@text@underscore}{n,mn}$ which are inspired by, but slightly different from the ones used in [16] for $K_{\lx@text@underscore}{n,mn+1}$.

The paper [6] uses a powerful machinery of bordered bimodules [41] and their tensor products, it would be interesting to find a relation with our results. In particular, the answer in Theorem 1.17 above seems to have a similar flavor to the constructions in [6] but we do not know any precise connection.

• (2)

Recent works of Cooper–Deyeso [9] and Wildi [36] also define colored variants of knot Floer homology. In particular, [9, Corollary 4.6] defines it by considering a similar “infinite full twist” limit, but uses hat-version of the homology and $(n,mn+1)$ cables which have one component. It would be interesting to relate their construction to ours.

By contrast, [36] defines an analogue of “$\wedge^{n}$-colored homology” (as opposed to “$S^{n}$-colored”), so the construction is very different. Still, one might expect an interesting symmetry between $S^{n}$- and $\wedge^{n}$-colored homology, see [8].

Acknowledgments

The authors would like to thank Anna Beliakova, Daren Chen, Robert Lipshitz, Jacob Rasmussen, Arno Wildi and Ian Zemke for useful discussions.

2. Background

2.1. Colimits

We recall some useful definitions and properties of colimits that will be used throughout the paper.

Let $\{V_{\lx@text@underscore}i\ :\ i\geq 1\}$ be a collection of vector spaces, and $f_{\lx@text@underscore}i:V_{\lx@text@underscore}i\to V_{\lx@text@underscore}{i+1}$ are some linear maps (called connecting maps). We call this data a directed system and draw it as follows:

[TABLE]

Definition 2.1**.**

The colimit $\varinjlim(V_{\lx@text@underscore}{\bullet},f_{\lx@text@underscore}{\bullet})$ of the directed system $(V_{\lx@text@underscore}{\bullet},f_{\lx@text@underscore}{\bullet})$ is defined as the vector space spanned by pairs $(v,i)$ where $v\in V_{\lx@text@underscore}i$ modulo the equivalence relation $(v,i)\sim(f_{\lx@text@underscore}i(v),i+1)$.

The following easy observation will be useful.

Lemma 2.2**.**

Let $\{V_{\lx@text@underscore}i\}$ be a sequence of vector spaces together with some connecting maps $f_{\lx@text@underscore}i:V_{\lx@text@underscore}i\to V_{\lx@text@underscore}{i+1}$. Let $L=\varinjlim(V_{\lx@text@underscore}i,f_{\lx@text@underscore}i)$ be the colimit of this directed system.

If $\dim V_{\lx@text@underscore}i\leq D$ for sufficiently large $i$ then $\dim L\leq D$ (in particular, $L$ is finite-dimensional).

Proof.

Recall that $L$ is spanned by the equivalence classes of vectors $v_{\lx@text@underscore}i\in V_{\lx@text@underscore}i$ modulo relation $v_{\lx@text@underscore}i\sim f_{\lx@text@underscore}i(v_{\lx@text@underscore}i)$. We will abbreviate $f_{\lx@text@underscore}i$ to $f$ when it is clear from context, so $v_{\lx@text@underscore}i\sim f(v_{\lx@text@underscore}i)$ and $v_{\lx@text@underscore}i\sim f^{k}(v_{\lx@text@underscore}i)$ for all $k>0$.

Assume that $v_{\lx@text@underscore}{i_{\lx@text@underscore}1}\in V_{\lx@text@underscore}{i_{\lx@text@underscore}1},\ldots,v_{\lx@text@underscore}{i_{\lx@text@underscore}{D+1}}\in V_{\lx@text@underscore}{i_{\lx@text@underscore}{D+1}}$ represent equivalence classes of some vectors in $L$. Furthermore, assume that for $i>N$ we have $\dim V_{\lx@text@underscore}i\leq D$, pick $n=\max(i_{\lx@text@underscore}1,\ldots,i_{\lx@text@underscore}{D+1},N+1)$. Then the vectors

[TABLE]

all belong to $V_{\lx@text@underscore}n$ and $\dim V_{\lx@text@underscore}n\leq D$. Therefore these vectors are linearly dependent and the respective vectors in $L$ (represented by $v_{\lx@text@underscore}{i_{\lx@text@underscore}1},\ldots,v_{\lx@text@underscore}{i_{\lx@text@underscore}{D+1}}$) are linearly dependent in $L$.

We conclude that any $D+1$ vectors in $L$ are linearly dependent, so $\dim L\leq D$. ∎

Lemma 2.3**.**

Suppose that $(V_{\lx@text@underscore}{\bullet},f_{\lx@text@underscore}{\bullet})$ and $(W_{\lx@text@underscore}{\bullet},g_{\lx@text@underscore}{\bullet})$ are two directed systems, and $h_{\lx@text@underscore}i:V_{\lx@text@underscore}i\to W_{\lx@text@underscore}{i+s}$ is a collection of maps such that $g_{\lx@text@underscore}{i+s}\circ h_{\lx@text@underscore}i=h_{\lx@text@underscore}{i+1}\circ f_{\lx@text@underscore}i$. Then there is a well defined map

[TABLE]

Proof.

We define $h(v,i)=(h_{\lx@text@underscore}i(v),i+s)$ for $v\in V_{\lx@text@underscore}i$. By our assumption, $h$ preserves the equivalence relation and hence defines a map of colimits. ∎

Example 2.4**.**

For $s=1$ we require that all squares in this diagram are commutative:

[TABLE]

2.2. Heegaard Floer homology

In this subsection, we review the necessary background for link Floer homology. We assume some familiarity with the basic Heegaard Floer homology and its refinement to knots, see [28, 27, 30, 31] for details. Given a link $\mathcal{L}$ with $n$ components in a 3-manifold $Y$, one can define the “full” version of Heegaard Floer complex $\mathcal{C\!F\!L}(Y,L)$ from the Heegaard diagram by using some version of Lagrangian Floer complexes. The multi-pointed Heegaard link diagram $(\Sigma,\bm{\alpha},\bm{\beta},{\bf{w}},{\bf{z}})$ for such pair $(Y,L)$ is defined as follows:

1. (1)

$\Sigma$ is a closed oriented genus $g$ surface; 2. (2)

$\bm{\alpha}=\{\alpha_{\lx@text@underscore}1,\cdots,\alpha_{\lx@text@underscore}{g+n-1}\}$, and $\bm{\beta}=\{\beta_{\lx@text@underscore}1,\cdots,\beta_{\lx@text@underscore}{g+n-1}\}$ are collections of simple closed curves on $\Sigma$ such that $\bm{\alpha}$ and $\bm{\beta}$ each span a $g$-dimensional subspace of $H_{\lx@text@underscore}1(\Sigma,\mathbb{Z})$ and curves in $\bm{\alpha}$ (and $\bm{\beta}$) are pairwise disjoint. 3. (3)

${\bf{w}}=\{w_{\lx@text@underscore}1,\cdots,w_{\lx@text@underscore}n\}$ and ${\bf{z}}=\{z_{\lx@text@underscore}1,\cdots,z_{\lx@text@underscore}n\}$ denote sequences of basepoints on $\Sigma$ such that each component of $\Sigma\setminus\bm{\alpha}$ (respectively of $\Sigma\setminus\bm{\beta}$) contains a single point of ${\bf{w}}$ and a single point of ${\bf{z}}$.

The generators of the link Floer chain complex $\mathcal{C\!F\!L}(Y,L)$ are tuples of intersection points $\mathbf{x}\in\mathbb{T}_{\lx@text@underscore}\alpha\cap\mathbb{T}_{\lx@text@underscore}{\beta}$ of the Lagrangian tori

[TABLE]

in the symmetric product $\textup{Sym}^{g+n-1}(\Sigma)$.

We work over the coefficient field $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$ in the paper. The marked points $z_{\lx@text@underscore}i$ and $w_{\lx@text@underscore}i$ are associated with formal variables $V_{\lx@text@underscore}i$ and $U_{\lx@text@underscore}i$, and the chain complex $\mathcal{C\!F\!L}(Y,L)$ is a module over the polynomial ring $R=\mathbb{F}[U_{\lx@text@underscore}1,\cdots,U_{\lx@text@underscore}n,V_{\lx@text@underscore}1,\cdots,V_{\lx@text@underscore}n]$. The differential in $\mathcal{C\!F\!L}(Y,L)$ is obtained by counting pseudo-holomorphic disks in $\textup{Sym}^{g+n-1}(\Sigma)$ via:

[TABLE]

Here the sum is taken over all holomorphic disks from $\mathbf{x}$ to $\mathbf{y}$ in $\pi_{\lx@text@underscore}2(\mathbf{x},\mathbf{y})$ representing the homotopy classes of maps $\varphi$ from the unit disk $\mathbb{D}\subset\mathbb{C}$ to $\textup{Sym}^{g+n-1}(\Sigma)$ satisfying some boundary condition. The Maslov index of $\varphi$ is denoted by $\mu(\varphi)$. For any point $x\in\Sigma\setminus(\bm{\alpha}\cup\bm{\beta})$, $n_{\lx@text@underscore}x(\varphi)$ denotes the intersection number of the divisor $\{x\}\times\textup{Sym}^{g+n-2}(\Sigma)\subset\textup{Sym}^{g+n-1}(\Sigma)$ with $\varphi(\mathbb{D})$. The moduli space $\mathcal{M}$ consists of all pseudo-holomorphic curves representing $\varphi$ for a generic 1-parameter family of almost complex structures on $\textup{Sym}^{g+n-1}(\Sigma)$. For more details, see [28].

We denote the homology of the link Floer chain complex $\mathcal{C\!F\!L}(Y,L)$ by $\mathcal{H\!F\!L}(Y,L)$, which is an invariant of the pair $(Y,L)$. The actions of $U_{\lx@text@underscore}iV_{\lx@text@underscore}i$ on the complex $\mathcal{C\!F\!L}(L)$ are pairwise homotopic, so we introduce the following relation:

[TABLE]

and let $R_{\lx@text@underscore}{UV}$ denote the ring generated by $U_{\lx@text@underscore}1,\cdots,U_{\lx@text@underscore}n,V_{\lx@text@underscore}1,\cdots,V_{\lx@text@underscore}n$ modulo this relation. Then the action of $R$ on $\mathcal{H\!F\!L}(Y,L)$ factors through $R_{\lx@text@underscore}{UV}$. We get different versions of Heegaard Floer homology by setting extra constraints on the variables $U_{\lx@text@underscore}i,V_{\lx@text@underscore}i$. For example, $H_{\lx@text@underscore}\ast(\mathcal{C\!F\!L}(Y,L)/(V_{\lx@text@underscore}1-1,\cdots,V_{\lx@text@underscore}n-1))$ is isomorphic to $\mathit{HF}^{-}(Y)$, the minus version of Heegaard Floer homology of the 3-manifold $Y$.

In this paper, we focus on links $L=L_{\lx@text@underscore}1\cup\ldots\cup L_{\lx@text@underscore}n$ in the three-sphere and, for simplicity, we write the complex as $\mathcal{C\!F\!L}(L)$. The complex $\mathcal{C\!F\!L}(L)$ and its homology $\mathcal{H\!F\!L}(L)$ have several gradings. Firstly, there is the $\mathbb{Q}\times\mathbb{Q}$-valued Maslov bigrading, denoted by $(\textup{gr}_{\lx@text@underscore}{{\bf{w}}},\textup{gr}_{\lx@text@underscore}{{\bf{z}}})$, as well as the Alexander multi-grading $A=(A_{\lx@text@underscore}1,\ldots,A_{\lx@text@underscore}n)$ valued in the lattice

[TABLE]

where $\ell_{\lx@text@underscore}i$ is the linking number of $L_{\lx@text@underscore}i$ with the rest of the components, i.e., $\ell_{\lx@text@underscore}i=\sum_{\lx@text@underscore}{j\neq i}lk(L_{\lx@text@underscore}i,L_{\lx@text@underscore}j)$. These gradings are relatively determined by the following equations:

[TABLE]

where $\varphi\in\pi_{\lx@text@underscore}2(\mathbf{x},\mathbf{y})$ and $n_{\lx@text@underscore}{{\bf{w}}}(\varphi)=\sum_{\lx@text@underscore}{i=1}^{n}n_{\lx@text@underscore}{w_{\lx@text@underscore}i}(\varphi)$ whereas $n_{\lx@text@underscore}{{\bf{z}}}(\varphi)=\sum_{\lx@text@underscore}{i=1}^{n}n_{\lx@text@underscore}{z_{\lx@text@underscore}i}(\varphi)$. Furthermore, the actions of $U_{\lx@text@underscore}i,V_{\lx@text@underscore}i$ are homogeneous with respect to these gradings with weights

[TABLE]

where $\mathbf{e}_{\lx@text@underscore}i$ is the standard $i$-th coordinate vector in $\mathbb{R}^{n}$. The differential on $\mathcal{C\!F\!L}(L)$ preserves the Alexander grading, while dropping both $\textup{gr}_{\lx@text@underscore}{{\bf{w}}}$ and $\textup{gr}_{\lx@text@underscore}{{\bf{z}}}$ by one. In fact, these gradings satisfy the relation

[TABLE]

For any $\mathbf{s}\in\mathbb{H}_{\lx@text@underscore}L$, let $\mathcal{C\!F\!L}(L,\mathbf{s})$ denote the subcomplex of $\mathcal{C\!F\!L}(L)$ generated by all elements $x$ such that $A(x)=\mathbf{s}$. By the grading formula (15), it is straightforward to see that the product $U_{\lx@text@underscore}iV_{\lx@text@underscore}i$ preserves the Alexander gradings, and $\mathcal{C\!F\!L}(L,\mathbf{s})$ is a module over the subring $\mathbb{F}[U_{\lx@text@underscore}1V_{\lx@text@underscore}1,U_{\lx@text@underscore}2V_{\lx@text@underscore}2,\cdots,U_{\lx@text@underscore}nV_{\lx@text@underscore}n]$, and so $\mathcal{H\!F\!L}(L,\mathbf{s})$ is an $\mathbb{F}[\mathbf{U}]$-module. In particular, for any link $L$ we have

[TABLE]

and $\mathcal{H\!F\!L}(L,\mathbf{s})$ is the direct sum of one copy of $\mathbb{F}[\mathbf{U}]$ with some $\mathbf{U}$-torsion.

Recall that a rational homology sphere $M$ is an L-space if for each $\text{Spin}^{c}$-structure $\mathfrak{s}$, $HF^{-}(M,\mathfrak{s})$ is isomorphic to $\mathbb{F}[\mathbf{U}]$. A link $L$ in the three-sphere $S^{3}$ is an L-space link if the Dehn surgery $S^{3}_{\lx@text@underscore}{\mathbf{d}}(L)$ is an L-space for all $\mathbf{d}\gg 0$. Another way to characterize the L-space links is the condition that $\mathcal{H\!F\!L}(L,\mathbf{s})\cong\mathbb{F}[\mathbf{U}]$ for every lattice point $\mathbf{s}\in\mathbb{H}_{\lx@text@underscore}L$. For any link $L$ in the three sphere, we define the link invariant, known as the $h$-function.

Definition 2.5**.**

The $h$-function $h:\mathbb{H}_{\lx@text@underscore}L\to\mathbb{Z}$ of a link $L$ in the three-sphere is defined so that for a given $\mathbf{s}\in\mathbb{H}_{\lx@text@underscore}L$ the value $h(\mathbf{s})$ is $-\dfrac{1}{2}\textup{gr}_{\lx@text@underscore}{{\bf{w}}}$ for the maximal Maslov grading of non-torsion elements in $\mathcal{H\!F\!L}(L,\mathbf{s})$.

For L-space links $L$, its $h$-function can be computed from the Alexander polynomials of the link and its sublinks, see [4]. Moreover, for such links, $\mathcal{H\!F\!L}(L)$ is determined by its $h$-function (see [5]), which is determined by these Alexander polynomials.

2.3. Cobordism maps

Recall that for an $n$-component link $L\subset Y$, each component is associated with two base points $z_{\lx@text@underscore}i,w_{\lx@text@underscore}i$ in the multi-pointed Heegaard diagram. In this subsection, we review cobordisms between two pairs $(Y_{\lx@text@underscore}1,L_{\lx@text@underscore}1,{\bf{w}}_{\lx@text@underscore}1,{\bf{z}}_{\lx@text@underscore}1)$ and $(Y_{\lx@text@underscore}2,L_{\lx@text@underscore}2,{\bf{w}}_{\lx@text@underscore}2,{\bf{z}}_{\lx@text@underscore}2)$ and the induced map between $\mathcal{H\!F\!L}(Y_{\lx@text@underscore}1,L_{\lx@text@underscore}1)$ and $\mathcal{H\!F\!L}(Y_{\lx@text@underscore}2,L_{\lx@text@underscore}2)$.

A coloring of a multi-based link $(L,{\bf{w}},{\bf{z}})$ is a map $\sigma:{\bf{w}}\cup{\bf{z}}\rightarrow P$ where $P=\{p_{\lx@text@underscore}1,\cdots,p_{\lx@text@underscore}k\}$ is a finite set, considered as the set of colors. We associate a free polynomial ring with a set of colors

[TABLE]

generated by the formal variables $X_{\lx@text@underscore}{p_{\lx@text@underscore}1},X_{\lx@text@underscore}{p_{\lx@text@underscore}2},\cdots X_{\lx@text@underscore}{p_{\lx@text@underscore}k}$. A coloring $\sigma$ gives the ring $\mathcal{R}_{\lx@text@underscore}p^{-}$ the structure of an $\mathbb{F}[U_{\lx@text@underscore}{{\bf{w}}},V_{\lx@text@underscore}{{\bf{z}}}]$-module. So for a colored multi-based link $(L,{\bf{w}},{\bf{z}},\sigma)$, we define

[TABLE]

Definition 2.6**.**

[39, Definition 1.3]** A decorated link cobordism from a 3-manifold with multi-based link $(Y_{\lx@text@underscore}1,(L_{\lx@text@underscore}1,{\bf{w}}_{\lx@text@underscore}1,{\bf{z}}_{\lx@text@underscore}1))$ to another one $(Y_{\lx@text@underscore}2,(L_{\lx@text@underscore}2,{\bf{w}}_{\lx@text@underscore}2,{\bf{z}}_{\lx@text@underscore}2))$ consists of a pair $(W,\mathcal{F})$ such that

1. (1)

$W$* is a compact 4-manifold with $\partial W=-Y_{\lx@text@underscore}1\sqcup Y_{\lx@text@underscore}2$;* 2. (2)

$\mathcal{F}=(\Sigma,A)$* is an oriented, properly embedded surface $\Sigma$ in $W$, along with a properly embedded 1-manifold $A$ in $\Sigma$, called dividing arcs. Furthermore, $\Sigma\setminus A$ consists of two disjoint (possibly disconnected) subsurfaces, $\Sigma_{\lx@text@underscore}{\bf{w}},\Sigma_{\lx@text@underscore}{\bf{z}}$, such that the intersection of the closure of $\Sigma_{\lx@text@underscore}{\bf{w}}$ and $\Sigma_{\lx@text@underscore}{\bf{z}}$ is $A$;* 3. (3)

$\partial\Sigma=-L_{\lx@text@underscore}1\cup L_{\lx@text@underscore}2$; 4. (4)

Each component of $L_{\lx@text@underscore}1\setminus A$ (and $L_{\lx@text@underscore}2\setminus A)$ contains exactly one basepoint; 5. (5)

The ${\bf{w}}$ basepoints are all in $\Sigma_{\lx@text@underscore}{\bf{w}}$ and the ${\bf{z}}$ basepoints are all in $\Sigma_{\lx@text@underscore}{\bf{z}}$; 6. (6)

$\mathcal{F}$* is equipped with a coloring, i.e. a map $\sigma:C(\Sigma\setminus A)\rightarrow P$, where $C(\Sigma\setminus A)$ denotes the set of components of $\Sigma\setminus A$.*

Zemke [38, Theorem A] associated a $\text{Spin}^{c}$ functorial chain map to a decorated cobordims $(W,\mathcal{F})$ with a $\text{Spin}^{c}$ structure $\mathfrak{s}$ on $W$:

[TABLE]

where $\sigma_{\lx@text@underscore}i$ denotes the coloring on $L_{\lx@text@underscore}i$ induced by restricting $\sigma$ for $i=1,2$. The maps are $\mathcal{R}_{\lx@text@underscore}P^{-}$ equivariant, $\mathbb{Z}^{p}$-filtered, and are invariant up to $\mathcal{R}_{\lx@text@underscore}P^{-}$-equivariant, $\mathbb{Z}^{P}$-filterd chain homotopies. The Maslov grading change and Alexander multi-grading changes for the cobordism maps are given in [39, Theorem 1.4]. Here we review the grading change formula for the cobordism induced by blowing down a $(-1)$-framed unknot, which is needed to define the colored knot Floer homology.

Let $L$ be an $n$-component link in the three-sphere, and $\overline{L}$ be another link obtained from $L$ by adding a full twist on $n$ strands, that is, $\overline{L}$ is obtained from $L$ by blowing down an $(-1)$-framed unknot circling the $n$ strands of $L$ where we assume that all strands are oriented the same way. Consider the corresponding cobordism from $(S^{3},L)$ to $(S^{3},\overline{L})$ obtained by attaching a $2$-handle to $S^{3}\times I$ along the aforementioned $(-1)-$framed unknot in $S^{3}\times\{1\}$. Let $[S^{2}]$ denote the generator of the second homology class of the cobordism, and let $\phi_{\lx@text@underscore}k:\mathcal{H\!F\!L}(L)\to\mathcal{H\!F\!L}(\overline{L})$ denote the induced cobordism maps in homology in the $\text{Spin}^{c}$-structure $\mathfrak{s}_{\lx@text@underscore}k$ such that $\langle\mathfrak{s}_{\lx@text@underscore}k,[S^{2}]\rangle=2k+1$. See [1] for a detailed discussion of the properties of $\phi_{\lx@text@underscore}k$. We have

Proposition 2.7**.**

[1]** Given an n-component link $L$ in the three sphere, the cobordism maps $\phi_{\lx@text@underscore}k:\mathcal{H\!F\!L}(L)\to\mathcal{H\!F\!L}(\overline{L})$ induced from the $(-1)$-surgery on the unknot with $0\leq k\leq n-1$ satisfy the following grading properties:

[TABLE]

where $i=1,\cdots,n$.

More generally, assume that we blow down a $(-1)$-framed unknot $M$ circling the link with arbitrary orientation of the strands. Then we still get maps $\phi_{\lx@text@underscore}{k}$ with $\textup{gr}_{\lx@text@underscore}{{\bf{w}}}(\phi_{\lx@text@underscore}k)=-k^{2}-k$ and

[TABLE]

The degrees of $\phi_{\lx@text@underscore}k$ are computed using [39]. Proposition 2.7 is a special case when $\mathrm{lk}(L,M)=n$ and $\mathrm{lk}(L_{\lx@text@underscore}i,M)=1$ for all $i$.

2.4. Alexander polynomials of cable links

The link Floer homology $\mathit{HFL}^{-}(L)$ is the categorification of the multi-variable Alexander polynomials of links $L$ in the three sphere. That is,

[TABLE]

Here $\chi_{\lx@text@underscore}L(t_{\lx@text@underscore}1,\cdots,t_{\lx@text@underscore}n)$ is a normalization of Alexander polynomial $\Delta_{\lx@text@underscore}L(t_{\lx@text@underscore}1,\cdots,t_{\lx@text@underscore}n)$ of $L$ as follows.

If $L$ is a knot, then

[TABLE]

It is a polynomial in $t$ and an infinite series in $t^{-1}$. For a link $L$ with $n>1$ components, we get

[TABLE]

Given a knot $K$, recall that the multivariable Alexander polynomial of the $(n,mn)$ cable $K_{\lx@text@underscore}{n,mn}$ (with $n$ components) is given by the equation:

[TABLE]

[TABLE]

where $\mathbf{t}=t_{\lx@text@underscore}1\cdots t_{\lx@text@underscore}n$. Furthermore,

[TABLE]

so we conclude

[TABLE]

This immediately implies the following:

Lemma 2.8**.**

Let $K$ be a knot of Seifert genus $g(K)$, and $m\gg 1$, define $\mathbf{c}_{\lx@text@underscore}m=\frac{m(n-1)}{2}$. Then

[TABLE]

In particular,

[TABLE]

Furthermore, the normalized Euler characteristic $t^{-\mathbf{c}_{\lx@text@underscore}m}\chi_{\lx@text@underscore}{K_{\lx@text@underscore}{n,mn}}(t)$ stabilizes in the region

[TABLE]

and vanishes for $s\not\preceq g(K)(1,\ldots,1)$.

Here two vectors $\mathrm{v}\preceq\mathrm{w}$ means that each component $v_{\lx@text@underscore}i\leq w_{\lx@text@underscore}i$ for $i=1,\cdots,n$.

3. Link Floer homology and infinite twists

Let $L=\amalg_{\lx@text@underscore}{i=1}^{n}L_{\lx@text@underscore}i$ be an oriented $n$-component link in $S^{3}$ and $M$ be an unknot bounding a disk $D$ that intersects every $L_{\lx@text@underscore}i$ positively at exactly one point. Then, $(-1)$-surgery on $M$ will result in inserting a positive full twist in $L$. Let $L_{\lx@text@underscore}m$ denote the link obtained by performing this operation $m$ times, that is inserting $m$ full twists in $L$.

3.1. Special Heegaard diagrams

First, we consider some special Heegaard diagrams for $L_{\lx@text@underscore}m$.

Corresponding to $L$ and $D$, let $G_{\lx@text@underscore}{L,D}$ be the graph obtained by pinching $L$ along $D$ and creating a thick edge as in Figure 1. So $G_{\lx@text@underscore}{L,D}$ has two vertices and $n+1$ edges connecting them. Conversely, any two pairs $(L,D)$ associated to such a bipartite graph $G$ with one distinguished edge, differ by inserting a pure braid in a neighborhood of the disk $D$.

Let

[TABLE]

be a Heegaard diagram for $G_{\lx@text@underscore}{L,D}$ where $z$ is the basepoint corresponding to the thick edge. From $H_{\lx@text@underscore}{G}$ we construct a Heegaard diagram for $L$ as follows. First, replace $z$ with $n$ basepoints ${\bf{z}}=\{z_{\lx@text@underscore}1,\cdots,z_{\lx@text@underscore}n\}$ in the same connected component of $\Sigma\setminus\{\bm{\alpha},\bm{\beta}\}$. Let $D^{\prime}\subset\Sigma$ be a small disk containing $z_{\lx@text@underscore}1,\cdots,z_{\lx@text@underscore}n$ and disjoint from $\alpha$- and $\beta$-circles. For $1\leq i\leq n$ connect $z_{\lx@text@underscore}i$ to $w_{\lx@text@underscore}i$ with arcs $a_{\lx@text@underscore}i$ and $b_{\lx@text@underscore}i$ such that

1. (1)

$a_{\lx@text@underscore}i$ is disjoint from the $\alpha$-circles and $b_{\lx@text@underscore}i$ is disjoint from the $\beta$-circles, 2. (2)

$a_{\lx@text@underscore}1,a_{\lx@text@underscore}2,\cdots,a_{\lx@text@underscore}n$ (resp. $b_{\lx@text@underscore}1,b_{\lx@text@underscore}2,\cdots,b_{\lx@text@underscore}n$) are pairwise disjoint 3. (3)

pushing $a_{\lx@text@underscore}1,\cdots,a_{\lx@text@underscore}n$ and $b_{\lx@text@underscore}1,\cdots,b_{\lx@text@underscore}n$ in the $\alpha$- and $\beta$-handlebody, respectively, gives a link isotopic to $L$ with an isotopy that maps $D^{\prime}$ to $D$.

Note that it is easy to arrange for conditions (1) and (2), however, $a_{\lx@text@underscore}1,\cdots,a_{\lx@text@underscore}n$ and $b_{\lx@text@underscore}1,\cdots,b_{\lx@text@underscore}n$ might represent a different link $L^{\prime}$. Note that the graph associated to $(L^{\prime},D^{\prime})$ is the same as $G_{\lx@text@underscore}{L,D}$. So $L$ is obtained from $L^{\prime}$ by inserting a pure braid in a neighborhood of $D^{\prime}$, which can be arranged by applying the corresponding diffeomorphism of $D^{\prime}$ to $b_{\lx@text@underscore}1,b_{\lx@text@underscore}2,\cdots,b_{\lx@text@underscore}n$.

Next, by stabilizing the diagram $H_{\lx@text@underscore}G$ as in Figure 2, we arrange for each $a_{\lx@text@underscore}i$ to be disjoint from $b_{\lx@text@underscore}j$ for $j\neq i$, while $a_{\lx@text@underscore}i\cap b_{\lx@text@underscore}i=\{z_{\lx@text@underscore}i,w_{\lx@text@underscore}i\}$. We keep using the same notation for the stabilized diagram.

Then, add $\alpha$- and $\beta$-circles $\alpha_{\lx@text@underscore}{g+1},\cdots,\alpha_{\lx@text@underscore}{g+n-1}$ and $\beta_{\lx@text@underscore}{g+1},\cdots,\beta_{\lx@text@underscore}{g+n-1}$ such that

• •

$\alpha_{\lx@text@underscore}{g+i}$ and $\beta_{\lx@text@underscore}{g+i}$ bound disks containing $a_{\lx@text@underscore}i$ and $b_{\lx@text@underscore}i$, respectively,

• •

For any $1\leq i,j\leq n-1$, if $i\neq j$ then $\alpha_{\lx@text@underscore}{g+i}\cap\beta_{\lx@text@underscore}{g+j}=\emptyset$. Otherwise, $\alpha_{\lx@text@underscore}{g+i}$ intersects $\beta_{\lx@text@underscore}{g+i}$ in exactly $4$ points, as vertices of two bigons, one containing $z_{\lx@text@underscore}i$ and another containing $w_{\lx@text@underscore}i$.

See Figure 3 for a local picture with $n=3$. Denote the final diagram of $L$ by $H$, that is

[TABLE]

A Heegaard diagram $H_{\lx@text@underscore}m$ for $L_{\lx@text@underscore}m$ is obtained from $H$ by twisting $\beta_{\lx@text@underscore}{g+1},\beta_{\lx@text@underscore}{g+2},\cdots,\beta_{\lx@text@underscore}{g+n-1}$ around $z$-basepoints $m$ times as in Figure 4. We write

[TABLE]

Every generator $\mathbf{x}$ of the link Floer chain complex $\mathcal{C\!F\!L}(H_{\lx@text@underscore}{m})$ is a $(g+n-1)$-tuple of intersection points and can be denoted as $\mathbf{x}=\{x_{\lx@text@underscore}1,x_{\lx@text@underscore}2,\cdots,x_{\lx@text@underscore}{g+n-1}\}$ where $x_{\lx@text@underscore}i\in\alpha_{\lx@text@underscore}i$. Denote the set of generators for $\mathcal{C\!F\!L}(H_{\lx@text@underscore}m)$ by $\mathcal{G}(H_{\lx@text@underscore}m)$.

For every distinct $1\leq i,j\leq n-1$, $\beta^{m}_{\lx@text@underscore}{g+j}$ intersects $\alpha_{\lx@text@underscore}{g+i}$ in $4m$ points, while $\alpha_{\lx@text@underscore}{g+i}\cap\beta^{m}_{\lx@text@underscore}{g+i}$ consists of $4m+4$ points. We label these intersection points (similar to [23]) as follows. First, we label the intersection points of $\alpha_{\lx@text@underscore}{g+i}\cap\beta_{\lx@text@underscore}{g+i}$ on the boundary of the bigons containing $w_{\lx@text@underscore}i$ and $z_{\lx@text@underscore}i$ basepoints by $E_{\lx@text@underscore}i,E_{\lx@text@underscore}i^{\prime}$ and $I_{\lx@text@underscore}i,I_{\lx@text@underscore}i^{\prime}$, respectively, as in Figure 5. The $\ell$-th winding block of intersection points is the $2(n-1)\times 2(n-1)$ grid of intersection points that appears when we wind $\beta^{\ell-1}_{\lx@text@underscore}{g+1},\cdots,\beta^{\ell-1}_{\lx@text@underscore}{g+n-1}$ one more time about the ${\bf{z}}$ basepoints to obtain $\beta^{\ell}_{\lx@text@underscore}{g+1},\cdots,\beta^{\ell}_{\lx@text@underscore}{g+n-1}$, respectively. Then, we denote the four intersection points $\alpha_{\lx@text@underscore}{g+i}\cap\beta_{\lx@text@underscore}{g+j}$ in the $\ell$-th winding block by $G_{\lx@text@underscore}{ij,\ell}$, $G_{\lx@text@underscore}{ij,\ell}^{\prime\prime}$, $G_{\lx@text@underscore}{ij,\ell}^{r}$ and $G_{\lx@text@underscore}{ij,\ell}^{l}$ as in Figure 5.

Lemma 3.1**.**

For any $0\leq\ell\leq m$, let $H^{\ell}_{\lx@text@underscore}{m+1}$ be the Heegaard diagram obtained from $H_{\lx@text@underscore}{m+1}$ by moving the basepoints ${\bf{z}}$ to the $(\ell+1)-$th winding block as in Figure 6, and denote them by ${\bf{z}}^{\ell}=\{z^{\ell}_{\lx@text@underscore}1,z^{\ell}_{\lx@text@underscore}2,\cdots,z^{\ell}_{\lx@text@underscore}n\}$. Then, $H^{\ell}_{\lx@text@underscore}{m+1}$ is a Heegaard diagram for $L_{\lx@text@underscore}{\ell}$.

Proof.

It is straightforward to check that $H_{\lx@text@underscore}{\ell}$ can be obtained from $H^{\ell}_{\lx@text@underscore}{m+1}$ by isotopy on the curves $\alpha_{\lx@text@underscore}{g+1},\cdots,\alpha_{\lx@text@underscore}{g+n-1}$ and $\beta_{\lx@text@underscore}{g+1},\cdots,\beta_{\lx@text@underscore}{g+n-1}$. For instance, the local picture for $\ell=m-1$ (and $n=3$) is depicted in Figure 7. In this figure all intersection points except for the four that are highlighted green can be removed by isotopy and the result is $H_{\lx@text@underscore}{m-1}$. ∎

Obviously, $\mathcal{C\!F\!L}(H_{\lx@text@underscore}{m+1}^{\ell})$ and $\mathcal{C\!F\!L}(H_{\lx@text@underscore}{m+1})$ have the same set of generators, that is $\mathcal{G}(H_{\lx@text@underscore}{m+1}^{\ell})=\mathcal{G}(H_{\lx@text@underscore}{m+1})$. For any $\mathbf{x}\in\mathcal{G}(H_{\lx@text@underscore}{m+1})$ we denote the Alexander grading of $\mathbf{x}$ as a generator of $\mathcal{C\!F\!L}(H_{\lx@text@underscore}{m+1}^{\ell})$ by $A^{\ell}(\mathbf{x})=(A_{\lx@text@underscore}1^{\ell}(\mathbf{x}),\cdots,A_{\lx@text@underscore}{n}^{\ell}(\mathbf{x}))$.

Let $\mathcal{G}^{o}_{\lx@text@underscore}{\ell}(H_{\lx@text@underscore}{m+1})\subset\mathcal{G}(H_{\lx@text@underscore}{m+1})$ denote the set of intersection points $\mathbf{x}$ that do not contain any points in the $i$-th winding block for $i\geq\ell+1$ and $\mathbf{x}\cap\left(\cup_{\lx@text@underscore}{j=1}^{n-1}\{I_{\lx@text@underscore}j,I_{\lx@text@underscore}j^{\prime}\}\right)=\emptyset$. Let $\mathcal{G}_{\lx@text@underscore}{\ell}^{\iota}(H_{\lx@text@underscore}{m+1})=\mathcal{G}(H_{\lx@text@underscore}{m+1})\setminus\mathcal{G}^{o}_{\lx@text@underscore}\ell(H_{\lx@text@underscore}{m+1})$.

3.2. Stabilization of homology

In this subsection, we prove the main result of Section 3 modulo some technical lemmas whose proofs are postponed to the next subsection.

Theorem 3.2**.**

Let $C$ be the constant from Lemma 3.5 below. For any $\overline{\mathbf{s}}=(\overline{s}_{\lx@text@underscore}1,\overline{s}_{\lx@text@underscore}2,\cdots,\overline{s}_{\lx@text@underscore}n)$ with $\overline{s}_{\lx@text@underscore}i\geq C-m$ for all $i$ we have

[TABLE]

In particular, for fixed $\overline{\mathbf{s}}$ and $m\gg 0$ the homology $\mathcal{H\!F\!L}^{\mathrm{stab}}(L_{\lx@text@underscore}m,\overline{s})$ stabilizes.

Proof.

Let $s_{\lx@text@underscore}i=\overline{s}_{\lx@text@underscore}i+\frac{(m+1)(n-1)}{2}$, $\widetilde{s}_{\lx@text@underscore}i=\overline{s}_{\lx@text@underscore}i+\frac{m(n-1)}{2}$ and set $\mathbf{s}=(s_{\lx@text@underscore}1,s_{\lx@text@underscore}2,\cdots,s_{\lx@text@underscore}n)$ and $\widetilde{\mathbf{s}}=(\widetilde{s}_{\lx@text@underscore}1,\widetilde{s}_{\lx@text@underscore}2,\cdots,\widetilde{s}_{\lx@text@underscore}n)$. Note that $s_{\lx@text@underscore}i-\widetilde{s}_{\lx@text@underscore}i=\frac{n-1}{2}$. As before, consider Heegaard diagrams $H_{\lx@text@underscore}{m+1}$ and $\widetilde{H}_{\lx@text@underscore}m:=H^{m}_{\lx@text@underscore}{m+1}$ for $L_{\lx@text@underscore}{m+1}$ and $L_{\lx@text@underscore}m$, respectively. For simplicity, we denote the Alexander grading of $\mathcal{C\!F\!L}(\widetilde{H}_{\lx@text@underscore}m)$ by $\widetilde{A}=(\widetilde{A}_{\lx@text@underscore}1,\cdots,\widetilde{A}_{\lx@text@underscore}m)$ i.e. $\widetilde{A}_{\lx@text@underscore}i=A^{m}_{\lx@text@underscore}i$.

The chain complex $\mathcal{C\!F\!L}(H_{\lx@text@underscore}{m+1},\mathbf{s})$ is generated by $p_{\lx@text@underscore}{\mathbf{x}}\mathbf{x}$ where the monomial $p_{\lx@text@underscore}{\mathbf{x}}$ is equal to

[TABLE]

Here, $I_{\lx@text@underscore}{\mathbf{s}}(\mathbf{x})=\{1\leq i\leq n\ |\ A_{\lx@text@underscore}i(\mathbf{x})>s_{\lx@text@underscore}i\}$ and $J_{\lx@text@underscore}{\mathbf{s}}(\mathbf{x})=\{1,2,\cdots,n\}\setminus I_{\lx@text@underscore}{\mathbf{s}}$. Similarly, $\mathcal{C\!F\!L}(\widetilde{H}_{\lx@text@underscore}{m},\tilde{\mathbf{s}})$ is generated by $\widetilde{p}_{\lx@text@underscore}{\mathbf{x}}\mathbf{x}$ where

[TABLE]

for $I_{\lx@text@underscore}{\widetilde{\mathbf{s}}}(\mathbf{x})=\{1\leq i\leq n\ |\ \widetilde{A}_{\lx@text@underscore}i(\mathbf{x})>\widetilde{s}_{\lx@text@underscore}i\}$ and $J_{\lx@text@underscore}{\widetilde{\mathbf{s}}}(\mathbf{x})=\{1,2,\cdots,n\}\setminus I_{\lx@text@underscore}{\widetilde{\mathbf{s}}}$.

We define an isomorphism $F$ from $\mathcal{C\!F\!L}(\widetilde{H}_{\lx@text@underscore}{m},\widetilde{\mathbf{s}})$ to $\mathcal{C\!F\!L}(H_{\lx@text@underscore}{m+1},\mathbf{s})$ by setting $F(\widetilde{p}_{\lx@text@underscore}\mathbf{x}\mathbf{x})=p_{\lx@text@underscore}\mathbf{x}\mathbf{x}$. We need to check that $F$ is a chain map. Denote the differential on $\mathcal{C\!F\!L}(\widetilde{H}_{\lx@text@underscore}{m},\widetilde{\mathbf{s}})$ and $\mathcal{C\!F\!L}(H_{\lx@text@underscore}{m+1},\mathbf{s})$ by $\widetilde{\partial}$ and $\partial$, respectively. Assume $\varphi\in\pi_{\lx@text@underscore}2(\mathbf{x},\mathbf{x}^{\prime})$ be a Maslov index one disk contributing nontrivially to $\partial\mathbf{x}$ and to $\widetilde{\partial}\mathbf{x}$. Then, the contributions of $\varphi$ to $\partial(p_{\lx@text@underscore}{\mathbf{x}}\mathbf{x})$ and $\widetilde{\partial}(\widetilde{p}_{\lx@text@underscore}{\mathbf{x}}\mathbf{x})$ are equal to $\mathbf{U}^{n(\varphi)}(p_{\lx@text@underscore}{\mathbf{x}^{\prime}}\mathbf{x}^{\prime})$ and $\mathbf{U}^{\widetilde{n}(\varphi)}(\widetilde{p}_{\lx@text@underscore}{\mathbf{x}^{\prime}}\mathbf{x}^{\prime})$, respectively, where

[TABLE]

in $R_{\lx@text@underscore}{UV}$. Here, $\widetilde{z}_{\lx@text@underscore}i=z_{\lx@text@underscore}i^{m}$. We need to prove $n(\varphi)=\widetilde{n}(\varphi)$, this would imply that $F$ is indeed a chain map.

Note that by Lemmas 3.4 and 3.3 for any $\mathbf{x}\in\mathcal{G}^{o}_{\lx@text@underscore}{m}(H_{\lx@text@underscore}{m+1})$ we have that $A_{\lx@text@underscore}i(\mathbf{x})-\widetilde{A}_{\lx@text@underscore}i(\mathbf{x})=\frac{n-1}{2}$ and so $p_{\lx@text@underscore}{\mathbf{x}}=\widetilde{p}_{\lx@text@underscore}{\mathbf{x}}$. Now we consider several cases:

1. If both $\mathbf{x},\mathbf{x}^{\prime}\in\mathcal{G}^{o}_{\lx@text@underscore}{m}(H_{\lx@text@underscore}{m+1})$ then $p_{\lx@text@underscore}{\mathbf{x}}=\widetilde{p}_{\lx@text@underscore}{\mathbf{x}}$, $p_{\lx@text@underscore}{\mathbf{x}^{\prime}}=\widetilde{p}_{\lx@text@underscore}{\mathbf{x}^{\prime}}$ and $n_{\lx@text@underscore}{z_{\lx@text@underscore}i}(\varphi)=n_{\lx@text@underscore}{\widetilde{z}_{\lx@text@underscore}i}(\varphi)$. Thus, $n(\varphi)=\widetilde{n}(\varphi)$.

2. Suppose $\mathbf{x}\in\mathcal{G}^{o}_{\lx@text@underscore}{m}(H_{\lx@text@underscore}{m+1})$ while $\mathbf{x}^{\prime}\in\mathcal{G}^{\iota}_{\lx@text@underscore}{m}(H_{\lx@text@underscore}{m+1})$. Then, by Lemma 3.5

[TABLE]

and $\widetilde{p}_{\lx@text@underscore}{\mathbf{x}^{\prime}}=\prod_{\lx@text@underscore}{i=1}^{n}V_{\lx@text@underscore}i^{\widetilde{s}_{\lx@text@underscore}i-\widetilde{A}_{\lx@text@underscore}i(\mathbf{x}^{\prime})}$. Similarly, $p_{\lx@text@underscore}{\mathbf{x}^{\prime}}=\prod_{\lx@text@underscore}{i=1}^{n}V_{\lx@text@underscore}i^{s_{\lx@text@underscore}i-A_{\lx@text@underscore}i(\mathbf{x}^{\prime})}$. Now we can compare the total $U$-degrees in both sides of (18), let $n_{\lx@text@underscore}{U_{\lx@text@underscore}i}(p_{\lx@text@underscore}{\mathbf{x}})$ denote the exponent of $U_{\lx@text@underscore}i$ in $p_{\lx@text@underscore}{\mathbf{x}}$. We have

[TABLE]

and so $p_{\lx@text@underscore}{\mathbf{x}}=\widetilde{p}_{\lx@text@underscore}{\mathbf{x}}$ implies that $n(\varphi)=\widetilde{n}(\varphi)$.

3. If $\mathbf{x}\in\mathcal{G}^{\iota}_{\lx@text@underscore}{m}(H_{\lx@text@underscore}{m+1})$ and $\mathbf{x}^{\prime}\in\mathcal{G}^{o}_{\lx@text@underscore}{m}(H_{\lx@text@underscore}{m+1})$, then an analogous argument implies that

[TABLE]

and so $p_{\lx@text@underscore}{\mathbf{x}^{\prime}}=\widetilde{p}_{\lx@text@underscore}{\mathbf{x}^{\prime}}$ implies that $n(\varphi)=\widetilde{n}(\varphi)$.

4. Finally, assume $\mathbf{x},\mathbf{x}^{\prime}\in\mathcal{G}_{\lx@text@underscore}{m+1}^{\iota}(H_{\lx@text@underscore}{m+1})$. In this case, $n(\varphi)=\sum_{\lx@text@underscore}{i=1}^{n}n_{\lx@text@underscore}{w_{\lx@text@underscore}i}(\varphi)=\widetilde{n}(\varphi)$ and so we are done.

∎

3.3. Estimates for the Alexander gradings

In the following lemmas we compare the Alexander degrees of the generators for different choices of ${\bf{z}}$ basepoints.

Lemma 3.3**.**

For any $\mathbf{x},\mathbf{x}^{\prime}\in\mathcal{G}^{o}_{\lx@text@underscore}{\ell}(H_{\lx@text@underscore}{m+1})$ we have $A(\mathbf{x})-A(\mathbf{x}^{\prime})=A^{i}(\mathbf{x})-A^{i}(\mathbf{x}^{\prime})$ for all $\ell\leq i\leq m$.

Proof.

By Equation 14, we have $A(\mathbf{x})-A(\mathbf{x}^{\prime})=A^{i}(\mathbf{x})-A^{i}(\mathbf{x}^{\prime})$ if and only if for any $1\leq j\leq n$ and any $\varphi\in\pi_{\lx@text@underscore}2(\mathbf{x},\mathbf{x}^{\prime})$ we have $n_{\lx@text@underscore}{z_{\lx@text@underscore}j}(\varphi)=n_{\lx@text@underscore}{z_{\lx@text@underscore}j^{i}}(\varphi)$. On the other hand, as discussed in the proof of Lemma 3.1 the Heegaard diagram $H_{\lx@text@underscore}{\ell}$ is obtained from $H_{\lx@text@underscore}{m+1}^{\ell}$ by certain isotopies that do not move the intersection points in $\mathcal{G}^{o}_{\lx@text@underscore}{\ell}(H_{\lx@text@underscore}{m+1})$. So, every $\mathbf{x}\in\mathcal{G}^{o}_{\lx@text@underscore}{\ell}(H_{\lx@text@underscore}{m+1})$ has a canonical corresponding intersection point in $\mathcal{G}(H_{\lx@text@underscore}{\ell})$. Moreover, every Whitney disk $\varphi\in\pi_{\lx@text@underscore}2(\mathbf{x},\mathbf{x}^{\prime})$ comes from extending a disk between corresponding intersection points in $\mathcal{G}(H_{\lx@text@underscore}{\ell})$. In fact, any such disk can be uniquely extended to a disk from $\mathbf{x}$ to $\mathbf{x}^{\prime}$, and will have the same coefficient at $z_{\lx@text@underscore}{j^{i}}$ for all $\ell\leq i\leq m$. For instance, see Figure 8 for the local coefficient of such a disk when $n=3$ at $z_{\lx@text@underscore}1,z_{\lx@text@underscore}2,z_{\lx@text@underscore}3$ and the $(m+1)$-th winding block. So, $n_{\lx@text@underscore}{z_{\lx@text@underscore}j}=n_{\lx@text@underscore}{z_{\lx@text@underscore}j^{m}}$. The general case is similar.

∎

Lemma 3.4**.**

For any intersection point of the form $\mathbf{x}=(\mathbf{x}^{\prime},\{E_{\lx@text@underscore}1,E_{\lx@text@underscore}2,\cdots,E_{\lx@text@underscore}{n-1}\})$ in $\mathcal{G}(H_{\lx@text@underscore}{m+1})$ we have

[TABLE]

for all $0\leq\ell\leq m$ where $A^{m+1}(\mathbf{x}):=A(\mathbf{x})$.

Proof.

Let $H_{\lx@text@underscore}G$ denote the underlying Heegaard diagram for $G_{\lx@text@underscore}{L,D}$. Following the discussions in [39, Section 5.4] if this property holds, then it will still hold if we change $H_{\lx@text@underscore}G$ (and so $H_{\lx@text@underscore}{m+1}$) by a Heegaard move. Therefore, it is enough to prove it for one Heegaard diagram $H_{\lx@text@underscore}G$.

First, we prove this for the special case where $L$ is the unlink and so $L_{\lx@text@underscore}{\ell}=T(n,\ell n)$. In this case, the corresponding graph is trivial (i.e. embeds in $S^{2}$), and thus it has a genus zero Heegaard diagram with no $\alpha$- and $\beta$-circles, that is $H_{\lx@text@underscore}G=(S^{2},z,w_{\lx@text@underscore}1,\cdots,w_{\lx@text@underscore}n)$. Starting with this special diagram, the corresponding diagram for $T(n,(m+1)n)$ has genus zero and is of the form depicted in Figure 4. In [23], Licata shows that for the special case of $\ell=1$ i.e. $T(n,n)$, the Alexander grading $A^{1}(E_{\lx@text@underscore}1,\cdots,E_{\lx@text@underscore}{n-1})=\left(\frac{n-1}{2},\frac{n-1}{2},\cdots,\frac{n-1}{2}\right)$. In general, a similar computation of relative Alexander gradings shows that $A^{\ell}_{\lx@text@underscore}i(E_{\lx@text@underscore}1,\cdots,E_{\lx@text@underscore}{n-1})\geq A^{\ell}_{\lx@text@underscore}i(\mathbf{x})$ for any other intersection point $\mathbf{x}$, so considering $\widehat{\mathit{HFL}}(T(n,\ell n))$ and the Alexander polynomial of $T(n,\ell n)$ we have $\{E_{\lx@text@underscore}1,\cdots,E_{\lx@text@underscore}{n-1}\}$ is the generator of $\widehat{\mathit{HFL}}(T(n,\ell n),\mathbf{s})$ for $\mathbf{s}=\left(\frac{\ell(n-1)}{2},\frac{\ell(n-1)}{2},\cdots,\frac{\ell(n-1)}{2}\right)$. So, $A^{\ell}(E_{\lx@text@underscore}1,E_{\lx@text@underscore}2,\cdots,E_{\lx@text@underscore}{n-1})=\left(\frac{\ell(n-1)}{2},\frac{\ell(n-1)}{2},\cdots,\frac{\ell(n-1)}{2}\right)$, and the claim holds for the unlink.

Let $O_{\lx@text@underscore}n\subset S^{3}$ be the $n$-component unlink in $S^{3}$ and $D^{\prime}$ be a small disk whose interior intersects each component of $O_{\lx@text@underscore}n$ is exactly one point with positive sign. There is a framed link $\mathbb{S}$ in the complement of $O_{\lx@text@underscore}n$ and disjoint from $D^{\prime}$ such that surgery on $S^{3}$ along $\mathbb{S}$ is diffeomorphic to $S^{3}$ and this diffeomorphism maps $(O_{\lx@text@underscore}n,D^{\prime})$ to $(L,D)$. Attaching $2$-handles along $\mathbb{S}$ gives a cobordisms from $O_{\lx@text@underscore}n$ to $L$ and also $G_{\lx@text@underscore}{O_{\lx@text@underscore}n,D^{\prime}}$ to $G_{\lx@text@underscore}{L,D}$. So, we may consider a Heegaard triple $\mathcal{T}_{\lx@text@underscore}G=(\Sigma,\bm{\alpha},\bm{\beta},\bm{\gamma},z,{\bf{w}})$ corresponding to the cobordism between graphs that can be modified to a triple for the cobordism between links. More precisely, $H_{\lx@text@underscore}{\bm{\alpha}\bm{\beta}}=(\Sigma,\bm{\alpha},\bm{\beta},z,{\bf{w}})$, $H_{\lx@text@underscore}{\bm{\alpha}\bm{\gamma}}=(\Sigma,\bm{\alpha},\bm{\gamma},z,{\bf{w}})$ and $H_{\lx@text@underscore}{\bm{\beta}\bm{\gamma}}=(\Sigma,\bm{\beta},\bm{\gamma},z,{\bf{w}})$ are Heegaard diagrams for $G_{\lx@text@underscore}{O_{\lx@text@underscore}n,D^{\prime}}$, $G_{\lx@text@underscore}{L,D}$ and the trivial graph in some connected sum of $S^{1}\times S^{2}$s. Moreover, ${\bf{w}}$ and $z$ are in the same connected component of $\Sigma\setminus(\bm{\beta}\cup\bm{\gamma})$, and a Heegaard triple $\mathcal{T}$ for the link cobordism from $O_{\lx@text@underscore}n$ to $L$ is obtained from $\mathcal{T}_{\lx@text@underscore}G$ by replacing $z$ with $z_{\lx@text@underscore}1,\cdots,z_{\lx@text@underscore}n$ and adding $\alpha_{\lx@text@underscore}{g+1},\cdots,\alpha_{\lx@text@underscore}{g+n-1}$, $\beta_{\lx@text@underscore}{g+1},\cdots,\beta_{\lx@text@underscore}{g+n-1}$ and $\gamma_{\lx@text@underscore}{g+1},\cdots,\gamma_{\lx@text@underscore}{g+n-1}$ similar to Figure 4. Note that each $\gamma_{\lx@text@underscore}{g+i}$ is a small Hamiltonian isotope of $\beta_{\lx@text@underscore}{g+i}$ and intersects it in exactly two points.

Note that twisting $\beta_{\lx@text@underscore}{g+1},\cdots,\beta_{\lx@text@underscore}{g+n-1}$ (and correspondingly $\gamma_{\lx@text@underscore}{g+1},\cdots,\gamma_{\lx@text@underscore}{g+n-1}$) around ${\bf{z}}$ basepoints $(m+1)$-times gives a Heegaard triple $\mathcal{T}_{\lx@text@underscore}{m+1}$ for the corresponding cobordism from $T(n,(m+1)n)$ to $L_{\lx@text@underscore}{m+1}$. Moreover, for any $1\leq\ell\leq m$ moving the basepoints ${\bf{z}}$ to the $(\ell+1)-$th winding block will result a Heegaard triple $\mathcal{T}_{\lx@text@underscore}{m+1}^{\ell}$ for the corresponding cobordism from $T(n,\ell n)$ to $L_{\lx@text@underscore}{\ell}$.

Assume $\varphi^{\prime}\in\pi_{\lx@text@underscore}2(\mathbf{y}^{\prime},\theta^{\prime},\mathbf{x}^{\prime})$ is a triangle in $\mathcal{T}_{\lx@text@underscore}{G}$ connecting intersection points $\mathbf{y}^{\prime}\in\mathcal{G}(H_{\lx@text@underscore}{\bm{\alpha}\bm{\beta}})$, $\theta^{\prime}\in\mathcal{G}(H_{\lx@text@underscore}{\bm{\beta}\bm{\gamma}})$ and $\mathbf{x}^{\prime}\in\mathcal{G}(H_{\lx@text@underscore}{\bm{\alpha}\bm{\gamma}})$. Let $\mathbf{x}=(\mathbf{x}^{\prime},\{E_{\lx@text@underscore}1,E_{\lx@text@underscore}2,\cdots,E_{\lx@text@underscore}{n-1}\})$, $\theta=(\theta^{\prime},\{\theta_{\lx@text@underscore}1^{+},\cdots,\theta_{\lx@text@underscore}{n-1}^{+}\})$ and $\mathbf{y}=(\mathbf{y}^{\prime},\{\bar{E}_{\lx@text@underscore}1,\bar{E}_{\lx@text@underscore}2,\cdots,\bar{E}_{\lx@text@underscore}{n-1}\})$ where $\theta_{\lx@text@underscore}j^{+},E_{\lx@text@underscore}j$ and $\bar{E}_{\lx@text@underscore}j$ are depicted in Figure 9. Corresponding to $\varphi^{\prime}$ there is a triangle $\varphi\in\pi_{\lx@text@underscore}2(\mathbf{y},\theta,\mathbf{x})$ obtained by adding small triangles on $\Sigma$ connecting $E_{\lx@text@underscore}j,\theta_{\lx@text@underscore}j^{+}$ and $\bar{E}_{\lx@text@underscore}j$ as in Figure 9.

Then, by [39, Formula 5.2] we have

[TABLE]

Here, $F_{\lx@text@underscore}i^{\ell}$ denotes the component of $F^{\ell}$ with boundary on the $i$-th components of $T(n,\ell n)$ and $L_{\lx@text@underscore}{\ell}$ i.e. the components containing $z_{\lx@text@underscore}i$ and $w_{\lx@text@underscore}i$. Moreover, $\hat{F}^{\ell}=\amalg_{\lx@text@underscore}i\hat{F}^{\ell}_{\lx@text@underscore}i$ is the result of capping $F^{\ell}$ with a Seifert surface for $T(n,\ell n)$ in $S^{3}$ and a Seifert surface for $L_{\lx@text@underscore}{\ell}$ in $S^{3}(\mathbb{S})$. Note that $A_{\lx@text@underscore}i^{\ell}(\theta)=A_{\lx@text@underscore}i^{\ell+1}(\theta)$ and $n_{\lx@text@underscore}{z_{\lx@text@underscore}i^{\ell}}(\varphi)=n_{\lx@text@underscore}{z_{\lx@text@underscore}i^{\ell+1}}(\varphi)$. Therefore,

[TABLE]

Note that changing $\ell$ will change the surface $\hat{F}_{\lx@text@underscore}{i}^{\ell}$ but it will not change its homology class i.e. $[\hat{F}_{\lx@text@underscore}i^{\ell}]$. That is because $[\hat{F}_{\lx@text@underscore}i^{\ell}]$ depends on the linking number of the $i$-th components of $T(n,\ell n)$ with $\mathbb{S}$ which does not change by changing $\ell$ as the disk $D$ is disjoint from $\mathbb{S}$. Consequently, both

[TABLE]

vanish. Thus, $A_{\lx@text@underscore}i^{\ell+1}(\mathbf{x})-A_{\lx@text@underscore}i^{\ell}(\mathbf{x})=A_{\lx@text@underscore}i^{\ell+1}(\mathbf{y})-A_{\lx@text@underscore}i^{\ell}(\mathbf{y})$. For such generators $\mathbf{y}$, we have shown the following

[TABLE]

for all $1\leq i\leq n$. Thus, the claim holds for $L$.

∎

Lemma 3.5**.**

There exists a constant $C$ independent of $m$ and $\ell$ such that for any $\mathbf{x}\in\mathcal{G}_{\lx@text@underscore}{\ell}^{\iota}(H_{\lx@text@underscore}{m+1})$

[TABLE]

for all $i$.

Proof.

Suppose $\mathbf{x}\in\mathcal{G}^{\iota}_{\lx@text@underscore}\ell(H_{\lx@text@underscore}{m+1})$. Let $\mathbf{x}^{1}\in\mathcal{G}^{o}_{\lx@text@underscore}1(H_{\lx@text@underscore}{m+1})$ be the intersection point obtained from $\mathbf{x}$ by replacing every $I_{\lx@text@underscore}i$, $I_{\lx@text@underscore}i^{\prime}$ and $G^{*}_{\lx@text@underscore}{ij,\ell}$ with $\ell>1$ in $\mathbf{x}$ by $G_{\lx@text@underscore}{ii,1}^{l}$, $G^{r}_{\lx@text@underscore}{ii,1}$ and $G^{*}_{\lx@text@underscore}{ij,1}$, respectively. Then, there is a disk $\varphi\in\pi_{\lx@text@underscore}2(\mathbf{x}^{1},\mathbf{x})$ whose domain is a union/concatenation of disk domains similar to the one in Figure 10, such that for all $1\leq i\leq n$

[TABLE]

Thus, $A_{\lx@text@underscore}i^{\ell}(\mathbf{x})\leq A_{\lx@text@underscore}i^{\ell}(\mathbf{x}^{1})-(\ell-1)$.

Consider an intersection point of the form $\mathbf{y}=(\mathbf{y}^{\prime},\{E_{\lx@text@underscore}1,E_{\lx@text@underscore}2,\cdots,E_{\lx@text@underscore}{n-1}\})$. By Lemma 3.3 we have

[TABLE]

Note that $A_{\lx@text@underscore}i^{1}(\mathbf{x}^{1})-A_{\lx@text@underscore}i^{1}(\mathbf{y})$ is equal to the $i$-th relative Alexander grading of the corresponding points of $\mathbf{x}^{1}$ and $\mathbf{y}$ in the Heegaard diagram $H_{\lx@text@underscore}1$ for $L_{\lx@text@underscore}1$. So there is a constant $C^{\prime}$ independent of $m$ such that

[TABLE]

for all $\mathbf{x}$. Consequently, $A_{\lx@text@underscore}i^{\ell}(\mathbf{x})\leq A_{\lx@text@underscore}i^{\ell}(\mathbf{y})+C^{\prime}-(\ell-1)$. Then, by Lemma 3.4, $A_{\lx@text@underscore}i^{\ell}(\mathbf{y})=A_{\lx@text@underscore}i^{1}(\mathbf{y})+\frac{(\ell-1)(n-1)}{2}$ and so the claim holds by setting

[TABLE]

∎

4. Colored knot Floer homology of the unknot

In this section, we compute the colored knot Floer homology of the unknot.

4.1. Homology of $T(n,mn)$

The first step is computing the link Floer homology of the torus link $T(n,nm)$. Since $m$ is fixed in this subsection, we will denote $\mathbf{c}_{\lx@text@underscore}m=\frac{m(n-1)}{2}$ by $\mathbf{c}$ to simplify notations.

Theorem 4.1**.**

The homology $\mathcal{H\!F\!L}(T(n,mn))$ as a module over $R_{\lx@text@underscore}{UV}$ is generated by $Y_{\lx@text@underscore}0,\ldots,Y_{\lx@text@underscore}{m(n-1)}$ with Alexander and Maslov degrees

[TABLE]

Furthermore, the relations between $Y_{\lx@text@underscore}i$ are spanned by $U_{\lx@text@underscore}IY_{\lx@text@underscore}i=V_{\lx@text@underscore}{\overline{I}}Y_{\lx@text@underscore}{i+1}$ for all subsets $I\subset\{1,2,\cdots,n\}$ with $q+1$ elements. Here, $\overline{I}=\{1,2,\cdots,n\}\setminus I$, $U_{\lx@text@underscore}I=\prod_{\lx@text@underscore}{j\in I}U_{\lx@text@underscore}i$ and $V_{\lx@text@underscore}{\overline{I}}=\prod_{\lx@text@underscore}{j\in\overline{I}}V_{\lx@text@underscore}j$.

Proof.

The proof is similar to the proof of [5, Theorem 7.3]. By [5, Lemma 7.1], to write down the generators we need to compute the $h$-function for $T(n,mn)$. Since the Alexander polynomial for $T(n,mn)$ is symmetric in $t_{\lx@text@underscore}1,\ldots,t_{\lx@text@underscore}n$, the $h$-function is symmetric in $s_{\lx@text@underscore}1,s_{\lx@text@underscore}2,\cdots,s_{\lx@text@underscore}n$ so we consider the case $s_{\lx@text@underscore}1\leq s_{\lx@text@underscore}2\leq\cdots\leq s_{\lx@text@underscore}n$. By [13, Theorem 4.3] for $s_{\lx@text@underscore}1\leq s_{\lx@text@underscore}2\leq\cdots\leq s_{\lx@text@underscore}n$ we get

[TABLE]

where

[TABLE]

In particular, for $\mathbf{c}\leq s_{\lx@text@underscore}1\leq\ldots\leq s_{\lx@text@underscore}n$ we get $h(s_{\lx@text@underscore}1,\ldots,s_{\lx@text@underscore}n)=0$. Suppose that $s=\mathbf{c}-i$ where $i=mq+r\geq 0$ then (assuming $i\leq m(n-1)=2\mathbf{c})$ we get

[TABLE]

so

[TABLE]

We claim that

[TABLE]

If $r<m-1$ then going from $s$ to $s-1$ changes $i$ to $i+1$ and $r$ to $r+1$ while preserving $q$, so (20) is clear. If $r=m-1$ then $i=qm+(m-1),i+1=(q+1)m+0$ and

[TABLE]

hence (20) holds as well.

Furthermore, one can check that

[TABLE]

Let $Y_{\lx@text@underscore}i$ be the generator of the $\mathbb{F}[\mathbf{U}]$-tower in Alexander grading $(\mathbf{c}-i,\ldots,\mathbf{c}-i)$. It will have Maslov grading

[TABLE]

and by [5, Lemma 7.1], $Y_{\lx@text@underscore}0,Y_{\lx@text@underscore}1,\cdots,Y_{\lx@text@underscore}{m(n-1)}$ generate $\mathcal{H\!F\!L}(T(n,mn))$.

By [5, Lemma 7.1], the relations are spanned by the relations between $Y_{\lx@text@underscore}a$ and $Y_{\lx@text@underscore}b$ for any $0\leq a\leq b\leq(n-1)m$ which are the ones spanned by $(U_{\lx@text@underscore}iV_{\lx@text@underscore}i+U_{\lx@text@underscore}jV_{\lx@text@underscore}j)Y_{\lx@text@underscore}a$ and $(U_{\lx@text@underscore}iV_{\lx@text@underscore}i+U_{\lx@text@underscore}jV_{\lx@text@underscore}j)Y_{\lx@text@underscore}b$ as well as sums

[TABLE]

ranging over sequences of nonnegative integer $I=(I_{\lx@text@underscore}1,\cdots,I_{\lx@text@underscore}n)$ and $J=(J_{\lx@text@underscore}1,\cdots,J_{\lx@text@underscore}n)$ such that

[TABLE]

where $a=q_{\lx@text@underscore}am+r_{\lx@text@underscore}a$ and $b=q_{\lx@text@underscore}bm+r_{\lx@text@underscore}b$. Here, $U^{I}=\prod_{\lx@text@underscore}{k}U_{\lx@text@underscore}k^{I_{\lx@text@underscore}k}$ and $V^{J}=\prod_{\lx@text@underscore}kV_{\lx@text@underscore}k^{J_{\lx@text@underscore}k}$.

If $a=i$ and $b=i+1$, then $(q_{\lx@text@underscore}b+1)(q_{\lx@text@underscore}bm+r_{\lx@text@underscore}b)/2-(q_{\lx@text@underscore}a+1)(q_{\lx@text@underscore}am+r_{\lx@text@underscore}a)/2=q+1$ and so these relations are exactly the ones described in the statement.

Suppose $b>a+1$. We show that the relations between $Y_{\lx@text@underscore}a$ and $Y_{\lx@text@underscore}b$ are in the span of the relations between consecutive $Y_{\lx@text@underscore}i$ and $Y_{\lx@text@underscore}{i+1}$. We prove this claim by induction on $b$. First, we show that there is a tuple $I^{\prime}=(I^{\prime}_{\lx@text@underscore}1,\cdots,I^{\prime}_{\lx@text@underscore}{n})$ such that for all $1\leq i\leq n$, $0\leq I^{\prime}_{\lx@text@underscore}i\leq I_{\lx@text@underscore}i$, and

[TABLE]

For $k\in\mathbb{N}$, we write

[TABLE]

Then

[TABLE]

and so by Equation (20) $\sum_{\lx@text@underscore}{k=1}^{b-a}a_{\lx@text@underscore}k=\sum_{\lx@text@underscore}{i=a}^{b-1}(q_{\lx@text@underscore}i+1)$. Consequently, the number of entries $I_{\lx@text@underscore}i$ that are equal to $b-a$ is at most $q_{\lx@text@underscore}{b-1}+1$. So, such $I^{\prime}$ exists if and only if $a_{\lx@text@underscore}1+a_{\lx@text@underscore}2\geq q_{\lx@text@underscore}{b-1}+1$. Suppose to the contrary that $a_{\lx@text@underscore}1+a_{\lx@text@underscore}2<q_{\lx@text@underscore}{b-1}+1$. Then $a_{\lx@text@underscore}2<(q_{\lx@text@underscore}{b-1}+1)/2$ and therefore

[TABLE]

On the other hand, $q_{\lx@text@underscore}{b-1}+1\leq q_{\lx@text@underscore}{b+a-1}+1\leq(q_{\lx@text@underscore}{a+i}+1)+(q_{\lx@text@underscore}{b-i-1}+1)$ implies that

[TABLE]

which is a contradiction.

Therefore such a tuple $I^{\prime}$ exists. By induction, for $J^{\prime}=(b-a-1,\cdots,b-a-1)-(I-I^{\prime})$, the relation

[TABLE]

is in the span of the claimed relations between consecutive generators. Then, this relation implies that

[TABLE]

If $|I^{\prime}|_{\lx@text@underscore}{L^{\infty}}=1$, the relations between $Y_{\lx@text@underscore}{b-1}$ and $Y_{\lx@text@underscore}b$ imply that

[TABLE]

Otherwise, for any index $i$ with $I^{\prime}_{\lx@text@underscore}i=2$, we observe that $(I-I^{\prime})_{\lx@text@underscore}i<b-a-1$, hence $J^{\prime}_{\lx@text@underscore}i>0$. Then $U^{I^{\prime}}V^{J^{\prime}}$ contains a factor of $U_{\lx@text@underscore}iV_{\lx@text@underscore}i$. Since, $|I^{\prime}_{\lx@text@underscore}i|_{\lx@text@underscore}{L_{\lx@text@underscore}1}=q_{\lx@text@underscore}{b-1}+1\leq n$ there is an index $j$ with $I^{\prime}_{\lx@text@underscore}j=0$. We will trade the $U_{\lx@text@underscore}iV_{\lx@text@underscore}i$ factor with $U_{\lx@text@underscore}jV_{\lx@text@underscore}j$. By repeating this process enough times, we obtain another tuple $I^{\prime\prime}$ from $I^{\prime}$ such that $|I^{\prime\prime}|_{\lx@text@underscore}{L^{1}}=|I^{\prime}|_{\lx@text@underscore}{L^{1}}$ and $|I^{\prime\prime}|_{\lx@text@underscore}{L^{\infty}}=1$. That is

[TABLE]

∎

Lemma 4.2**.**

Let $\mathbf{s}=(s_{\lx@text@underscore}1,s_{\lx@text@underscore}2,\cdots,s_{\lx@text@underscore}{n})$ and $\min\{s_{\lx@text@underscore}1,s_{\lx@text@underscore}2,\cdots,s_{\lx@text@underscore}n\}\geq\mathbf{c}-m$. The $\mathbb{F}[\mathbf{U}]$-tower $\mathcal{H\!F\!L}(T_{\lx@text@underscore}{n,mn},\mathbf{s})$ is generated by

• (a)

$V^{s_{\lx@text@underscore}1-\mathbf{c}}_{\lx@text@underscore}1\cdots V^{s_{\lx@text@underscore}n-\mathbf{c}}_{\lx@text@underscore}nY_{\lx@text@underscore}0$* if $\min(s_{\lx@text@underscore}1,\ldots,s_{\lx@text@underscore}n)\geq\mathbf{c}$,*

• (b)

$V^{s_{\lx@text@underscore}1-\mathbf{c}+k}_{\lx@text@underscore}1\cdots V^{s_{\lx@text@underscore}n-\mathbf{c}+k}_{\lx@text@underscore}nY_{\lx@text@underscore}k$* if $\min(s_{\lx@text@underscore}1,\ldots,s_{\lx@text@underscore}n)=\mathbf{c}-k$ for $0\leq k\leq m$.*

Proof.

By symmetry assume $s_{\lx@text@underscore}1\leq\ldots\leq s_{\lx@text@underscore}n$, and so $\min(s_{\lx@text@underscore}1,\ldots,s_{\lx@text@underscore}n)=s_{\lx@text@underscore}1$.

(a) If $s_{\lx@text@underscore}1\geq\mathbf{c}$, then $s_{\lx@text@underscore}i-\mathbf{c}+(i-1)m\geq 0$ for all $i\geq 1$ and so by Equation (19) $h(s_{\lx@text@underscore}1,s_{\lx@text@underscore}2,\cdots,s_{\lx@text@underscore}n)=0$. Therefore, the generator of $\mathcal{H\!F\!L}(T_{\lx@text@underscore}{n,mn},\mathbf{s})$ has Maslov grading equal to [math]. On the other hand,

[TABLE]

so $V^{s_{\lx@text@underscore}1-\mathbf{c}}_{\lx@text@underscore}1\cdots V^{s_{\lx@text@underscore}n-\mathbf{c}}_{\lx@text@underscore}nY_{\lx@text@underscore}0$ generates $\mathcal{H\!F\!L}(T_{\lx@text@underscore}{n,mn},\mathbf{s})$.

(b) If $\mathbf{c}-m\leq s_{\lx@text@underscore}1\leq\mathbf{c}$, then $s_{\lx@text@underscore}1-\mathbf{c}\leq 0$, while $s_{\lx@text@underscore}i-\mathbf{c}+(i-1)m\geq 0$ for $i>1$. Therefore, by Equation (19) $h(s_{\lx@text@underscore}1,\ldots,s_{\lx@text@underscore}n)=\mathbf{c}-s_{\lx@text@underscore}1=k$ and so the generator of $\mathcal{H\!F\!L}(T_{\lx@text@underscore}{n,mn},\mathbf{s})$ has Maslov grading $-2k$. On the other hand, for $k\leq m$ we get

[TABLE]

so $V^{s_{\lx@text@underscore}1-\mathbf{c}+k}_{\lx@text@underscore}1\cdots V^{s_{\lx@text@underscore}n-\mathbf{c}+k}_{\lx@text@underscore}nY_{\lx@text@underscore}k$ generates $\mathcal{H\!F\!L}(T_{\lx@text@underscore}{n,mn},\mathbf{s})$. ∎

Lemma 4.3**.**

For any $\overline{\mathbf{s}}=(\overline{s}_{\lx@text@underscore}1,\overline{s}_{\lx@text@underscore}2,\cdots,\overline{s}_{\lx@text@underscore}n)$ with $\min(\overline{s}_{\lx@text@underscore}1,\ldots,\overline{s}_{\lx@text@underscore}n)\geq-m$, the map

[TABLE]

is an isomorphism.

Proof.

The map $\phi_{\lx@text@underscore}0$ has Alexander degree $\left(\frac{n-1}{2},\ldots,\frac{n-1}{2}\right)$ and Maslov degree 0. Since $T(n,mn)$ and $T(n,(m+1)n)$ are both L-space links and $\phi_{\lx@text@underscore}0$ is injective, it is sufficient to check that the Maslov degrees of the generators of the two $\mathbb{F}[\mathbf{U}]$-towers $\mathcal{H\!F\!L}^{\mathrm{stab}}\left(T(n,mn),\overline{\mathbf{s}}\right)$ and $\mathcal{H\!F\!L}^{\mathrm{stab}}\left(T(n,(m+1)n),\overline{\mathbf{s}}\right)$ match.

If $\min(\overline{s}_{\lx@text@underscore}1,\ldots,\overline{s}_{\lx@text@underscore}n)\geq 0$, then by part (a) of Lemma 4.2 the generators of $\mathcal{H\!F\!L}^{\mathrm{stab}}\left(T(n,mn),\overline{\mathbf{s}}\right)$ and $\mathcal{H\!F\!L}^{\mathrm{stab}}\left(T(n,(m+1)n),\overline{\mathbf{s}}\right)$ have Maslov degree [math]. If $\min(\overline{s}_{\lx@text@underscore}1,\ldots,\overline{s}_{\lx@text@underscore}n)\geq-k$ for some $0<k\leq m$, by part (b) of Lemma 4.2, the generators of $\mathcal{H\!F\!L}^{\mathrm{stab}}\left(T(n,mn),\overline{\mathbf{s}}\right)$ and $\mathcal{H\!F\!L}^{\mathrm{stab}}\left(T(n,(m+1)n),\overline{\mathbf{s}}\right)$ have Maslov degree $-2k$. So $\phi_{\lx@text@underscore}0$ is an isomorphism in both cases.

∎

Corollary 4.4**.**

For any $\overline{\mathbf{s}}=(\overline{s}_{\lx@text@underscore}1,\overline{s}_{\lx@text@underscore}2,\cdots,\overline{s}_{\lx@text@underscore}n)$ with $\min(\overline{s}_{\lx@text@underscore}1,\ldots,\overline{s}_{\lx@text@underscore}n)\geq-m$

[TABLE]

Proof.

It follows from the definition of the colimit: suppose that $Y\in\mathcal{H\!F\!L}^{\mathrm{stab}}(T(n,m^{\prime}n),\overline{\mathbf{s}})$ for some $m^{\prime}$. Choose $m$ such that $m\geq m^{\prime}$ and $\min(\overline{s}_{\lx@text@underscore}1,\ldots,\overline{s}_{\lx@text@underscore}n)\geq-m$. Then, $Y$ is equivalent to $\left(\phi_{\lx@text@underscore}0\right)^{m-m^{\prime}}(Y)$ in the colimit which is in $\mathcal{H\!F\!L}^{\mathrm{stab}}(T(n,mn),\overline{\mathbf{s}})$. ∎

4.2. Cable algebra and cobordism maps

Next, we explicitly describe the cobordism maps $\phi_{\lx@text@underscore}k$ corresponding to adding full twists for $T(n,mn)$. In order to do that, we consider the direct sum of homology for all $n$-strand cables of the unknot with nonnegative integer slopes:

[TABLE]

where $O$ is the unknot and $O_{\lx@text@underscore}n$ is the $n$-component unlink.

Each summand is $\mathbb{Z}^{n}\oplus\mathbb{Z}$ graded by Alexander multi-grading ($\mathbb{Z}^{n}$) and Maslov grading ($\mathbb{Z}$). Moreover, $\mathrm{Cab}_{\lx@text@underscore}n(O)$ has an additional $\mathbb{Z}$ grading given by the cabling slope $m$, we call it twist grading and denote it by $\mathbf{tw}$. Note that cobordism maps $\phi_{\lx@text@underscore}k$, corresponding to adding a full twist, act naturally on $\mathrm{Cab}_{\lx@text@underscore}n(O)$ with twist grading $1$ i.e. they map $\mathcal{H\!F\!L}(T(n,mn))$ to $\mathcal{H\!F\!L}(T(n,(m+1)n))$.

Definition 4.5**.**

The $n$-strand cable algebra $\mathcal{A}_{\lx@text@underscore}n$ is defined as a $\mathbb{Z}^{n}\oplus\mathbb{Z}\oplus\mathbb{Z}$ graded algebra over $R_{\lx@text@underscore}{UV}$ with commuting generators $a_{\lx@text@underscore}0,\ldots,a_{\lx@text@underscore}{n-1}$ and the following relations:

• •

Linear relations

[TABLE]

*where $I\subset\{1,\ldots,n\}$ is an arbitrary subset with $|I|=k$ and $\overline{I}=\{1,\ldots,n\}\setminus I.$ *

• •

Quadratic relations

[TABLE]

whenever $i+j=k+\ell$ and $i\leq k\leq\ell\leq j$.

As above, the gradings correspond to the Alexander ($\mathbb{Z}^{n}$) and Maslov ($\mathbb{Z}$) gradings, along with an additional $\mathbb{Z}$ twist grading. The generator $a_{\lx@text@underscore}k$ has Alexander grading

[TABLE]

Maslov grading $\textup{gr}_{\lx@text@underscore}{{\bf{w}}}(a_{\lx@text@underscore}k)=-k^{2}-k$ and twist grading $\mathbf{tw}(a_{\lx@text@underscore}k)=1$.

It is easy to check that the relations are homogeneous with respect to all three gradings.

Theorem 4.6**.**

The homology $\mathrm{Cab}_{\lx@text@underscore}n(O)$ is a $\mathbb{Z}^{n}\oplus\mathbb{Z}\oplus\mathbb{Z}$ graded module over the algebra $\mathcal{A}_{\lx@text@underscore}n$ where the generators $a_{\lx@text@underscore}k$ act by cobordism maps $\phi_{\lx@text@underscore}k$. Furthermore, $\mathrm{Cab}_{\lx@text@underscore}n(O)$ is a free $\mathcal{A}_{\lx@text@underscore}n$-module generated by a single class $1\in\mathcal{H\!F\!L}(T(n,0))$.

Proof.

We proceed in several steps.

Step 1: We show that there is a well-defined action of algebra $\mathcal{A}_{\lx@text@underscore}n$ on $\mathrm{Cab}_{\lx@text@underscore}n(O)$, where $a_{\lx@text@underscore}k$ acts by the cobordism map $\phi_{\lx@text@underscore}k$. First, we prove that the actions of $\phi_{\lx@text@underscore}{k}$ and $\phi_{\lx@text@underscore}{k^{\prime}}$ on $\mathrm{Cab}_{\lx@text@underscore}n(O)$ commute. By [13] all links $T(n,nm)$ are L-space links, so $\mathcal{H\!F\!L}(T(n,mn))$ is free of rank 1 over $\mathbb{F}[\mathbf{U}]$ in each Alexander degree. The maps $\phi_{\lx@text@underscore}{k}$ and $\phi_{\lx@text@underscore}{k^{\prime}}$ correspond to negative definite cobordisms, so they are both injective [1], and hence determined by their Alexander and Maslov degrees. Now we observe that both compositions $\phi_{\lx@text@underscore}{k}\circ\phi_{\lx@text@underscore}{k^{\prime}}$ and $\phi_{\lx@text@underscore}{k^{\prime}}\circ\phi_{\lx@text@underscore}{k}$ have the same Alexander and Maslov degrees by Proposition 2.7, and hence coincide.

Next, we check that these cobordism maps satisfy relations (22) and (23). Note that $a_{\lx@text@underscore}k$ has the same Alexander, Maslov and twist grading as $\phi_{\lx@text@underscore}k$, thus relations (22) and (23) being homogeneous, implies that the cobordism maps satisfy these relations as well and the action is well-defined.

Step 2: We prove that $\mathrm{Cab}_{\lx@text@underscore}n(O)$ is generated by a single class $1\in\mathcal{H\!F\!L}(T(n,0))$ under the action of $\mathcal{A}_{\lx@text@underscore}n$. We follow Theorem 4.1 and claim that the generator $Y_{\lx@text@underscore}i\in\mathcal{H\!F\!L}(T(n,mn))$ can be identified with the monomial $a_{\lx@text@underscore}{q}^{m-r}a_{\lx@text@underscore}{q+1}^{r}$ (where $i=mq+r$) applied to $1$. By the above, the element $a_{\lx@text@underscore}{q}^{m-r}a_{\lx@text@underscore}{q+1}^{r}(1)$ is nonzero, and it is sufficient to verify that it has the same Alexander and Maslov degree as $Y_{\lx@text@underscore}i$. Indeed, the Alexander grading equals

[TABLE]

and the Maslov grading equals

[TABLE]

To sum up, we have a surjective map of $\mathcal{A}_{\lx@text@underscore}n$ modules from $\mathcal{A}_{\lx@text@underscore}n$ to $\mathrm{Cab}_{\lx@text@underscore}n(O)$.

Step 3: Finally, we need to prove that this map is an isomorphism and $\mathrm{Cab}_{\lx@text@underscore}n(O)\simeq\mathcal{A}_{\lx@text@underscore}n$. For this, we need to find the generators of $\mathcal{A}_{\lx@text@underscore}n$ over $R_{\lx@text@underscore}{UV}$, relations between them and compare these with Theorem 4.1.

Define $\widetilde{Y_{\lx@text@underscore}i}=a_{\lx@text@underscore}{q}^{m-r}a_{\lx@text@underscore}{q+1}^{r}\in\mathcal{A}_{\lx@text@underscore}n$ where $i=mq+r$. Let us prove that $\widetilde{Y_{\lx@text@underscore}i}$ generate $\mathcal{A}_{\lx@text@underscore}n$ over $R_{\lx@text@underscore}{UV}$. Indeed, consider a monomial $M=a_{\lx@text@underscore}{i_{\lx@text@underscore}1}\cdots a_{\lx@text@underscore}{i_{\lx@text@underscore}m}$ with $i_{\lx@text@underscore}1\leq i_{\lx@text@underscore}2\leq\cdots\leq i_{\lx@text@underscore}m$. If $i_{\lx@text@underscore}m-i_{\lx@text@underscore}1\leq 1$ then $M=\widetilde{Y}_{\lx@text@underscore}{i_{\lx@text@underscore}1+\ldots+i_{\lx@text@underscore}m}$ by the above. Otherwise we can use (23) to write

[TABLE]

since $(i_{\lx@text@underscore}1+1)(i_{\lx@text@underscore}m-1)-i_{\lx@text@underscore}1i_{\lx@text@underscore}m=i_{\lx@text@underscore}m-i_{\lx@text@underscore}1-1$. This decreases the power of $a_{\lx@text@underscore}{i_{\lx@text@underscore}1}$ and we can proceed by induction.

Finally, we need to verify that the relations from Theorem 4.1 hold for $\widetilde{Y_{\lx@text@underscore}i}$. This is clear since the relations between $\widetilde{Y}_{\lx@text@underscore}i=a_{\lx@text@underscore}{q}^{m-r}a_{\lx@text@underscore}{q+1}^{r}$ and $\widetilde{Y}_{\lx@text@underscore}{i+1}=a_{\lx@text@underscore}{q}^{m-r-1}a_{\lx@text@underscore}{q+1}^{r+1}$ agree with the relations between $a_{\lx@text@underscore}{q}$ and $a_{\lx@text@underscore}{q+1}$, up to multiplication by $a_{\lx@text@underscore}{q}^{m-r-1}a_{\lx@text@underscore}{q+1}^{r}$. ∎

Example 4.7**.**

The homology of $T(2,2m)$ is generated by $a_{\lx@text@underscore}0^{m},a_{\lx@text@underscore}0^{m-1}a_{\lx@text@underscore}1,\ldots,a_{\lx@text@underscore}1^{m}$ modulo relations $U_{\lx@text@underscore}1a_{\lx@text@underscore}0=V_{\lx@text@underscore}2a_{\lx@text@underscore}1$ and $U_{\lx@text@underscore}2a_{\lx@text@underscore}0=V_{\lx@text@underscore}1a_{\lx@text@underscore}1$, times a degree $(m-1)$ monomial.

Example 4.8**.**

The homology of $T(3,6)$ is generated by six degree 2 monomials in $a_{\lx@text@underscore}0,a_{\lx@text@underscore}1,a_{\lx@text@underscore}2$, but in fact we have a relation $a_{\lx@text@underscore}0a_{\lx@text@underscore}2=\mathbf{U}a_{\lx@text@underscore}1^{2}$ since $a_{\lx@text@underscore}0a_{\lx@text@underscore}2$ and $a_{\lx@text@underscore}1^{2}$ have the same Alexander degree. So $\mathcal{H\!F\!L}(T(3,6))$ is generated by $a_{\lx@text@underscore}0^{2},a_{\lx@text@underscore}0a_{\lx@text@underscore}1,a_{\lx@text@underscore}1^{2},a_{\lx@text@underscore}1a_{\lx@text@underscore}2,a_{\lx@text@underscore}2^{2}$.

Example 4.9**.**

Similarly, the homology of $T(3,3m)$ is generated by $2m+1$ monomials

[TABLE]

4.3. Colored knot Floer homology of the unknot

Finally, we are ready to compute the colored homology $\mathcal{H}_{\lx@text@underscore}{n}(O)$. Recall that this invariant is defined as the limit of the directed system of $\mathcal{H\!F\!L}(T(n,mn))$ along the directed system of maps $\phi_{\lx@text@underscore}{0}$. We define the algebraic analogue of this as follows:

Definition 4.10**.**

Let the algebra $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$ be the (graded) localization of $\mathcal{A}_{\lx@text@underscore}n$ in $a_{\lx@text@underscore}0$:

[TABLE]

Note that since $\phi_{\lx@text@underscore}0:\mathrm{Cab}_{\lx@text@underscore}n(O)\to\mathrm{Cab}_{\lx@text@underscore}n(O)$ is injective, by Theorem 4.6 the multiplication by $a_{\lx@text@underscore}0$ in $\mathcal{A}_{\lx@text@underscore}n$ is injective as well. Therefore $a_{\lx@text@underscore}0$ is not a zero divisor in $\mathcal{A}_{\lx@text@underscore}n$, and $\mathcal{A}_{\lx@text@underscore}n[a_{\lx@text@underscore}0^{-1}]$ embeds into $\mathrm{Frac}(\mathcal{A}_{\lx@text@underscore}n).$

Lemma 4.11**.**

The colored homology of the unknot $\mathcal{H}_{\lx@text@underscore}{n}(O)$ is a free rank 1 module over $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$.

Proof.

This is an easy consequence of Theorem 4.6. We identify an element $Y\in\mathcal{A}_{\lx@text@underscore}n$ such that $\mathbf{tw}(Y)=m$ with a class $Y(1)\in\mathcal{H\!F\!L}(T(n,mn))$. By Theorem 4.6 this yields an isomorphism $\mathcal{H\!F\!L}(T(n,mn))\simeq\mathrm{span}\left\{Y\in\mathcal{A}_{\lx@text@underscore}n\mid\mathbf{tw}(Y)=m\right\}$.

The colored homology $\mathcal{H}_{\lx@text@underscore}{n}(O)$ is spanned by classes $Y(1)\in\mathcal{H\!F\!L}(T(n,mn))$ for all $m$ modulo relations $Y(1)\sim\phi_{\lx@text@underscore}0(Y(1))$, equivalently, by the elements $Y\in\mathcal{A}_{\lx@text@underscore}n$ modulo relations $Y\sim a_{\lx@text@underscore}0Y$. Given such an element $Y$, we can associate to it the fraction $\frac{Y}{a_{\lx@text@underscore}0^{m}}\in\mathrm{Frac}(\mathcal{A}_{\lx@text@underscore}n)$. This agrees with our equivalence relation since

[TABLE]

and the result follows. ∎

Next, we would like to have some concrete description of $\mathcal{A}_{\lx@text@underscore}n[a_{\lx@text@underscore}0^{-1}]$ and $\mathcal{H}_{\lx@text@underscore}n(O)$.

Theorem 4.12**.**

The colored homology of unknot is isomorphic to

[TABLE]

where $\mathsf{A}=a_{\lx@text@underscore}1/a_{\lx@text@underscore}0$. The generator $\mathsf{A}$ has Maslov grading $\textup{gr}_{\lx@text@underscore}{{\bf{w}}}=-2$ and renormalized Alexander grading $(-1,\ldots,-1)$.

Proof.

The first isomorphism follows from the Lemma 4.11, so we focus on $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}=\mathcal{A}_{\lx@text@underscore}n[a_{\lx@text@underscore}0^{-1}]$. The algebra $\mathcal{A}_{\lx@text@underscore}n$ is generated by $a_{\lx@text@underscore}0,\ldots,a_{\lx@text@underscore}{n-1}$, so $\mathcal{A}_{\lx@text@underscore}n[a_{\lx@text@underscore}0^{-1}]$ is generated by $a_{\lx@text@underscore}k/a_{\lx@text@underscore}0$. We claim that

[TABLE]

This is equivalent to the identity $a_{\lx@text@underscore}ka_{\lx@text@underscore}0^{k-1}=\mathbf{U}^{\frac{k(k-1)}{2}}a_{\lx@text@underscore}1^{k}$ which can be proved by repeated application of (23) and induction in $k$:

[TABLE]

Therefore $\mathcal{A}_{\lx@text@underscore}n[a_{\lx@text@underscore}0^{-1}]$ is generated by the powers of $\mathsf{A}$ over $R_{\lx@text@underscore}{UV}$.

The relations (22) for $k=1$ have the form $U_{\lx@text@underscore}ia_{\lx@text@underscore}0=V_{\lx@text@underscore}{\overline{i}}a_{\lx@text@underscore}1$ and imply $U_{\lx@text@underscore}i=V_{\lx@text@underscore}{\overline{i}}\mathsf{A}$. Let us check that these relations in turn imply all other relations in $\mathcal{A}_{\lx@text@underscore}n[a_{\lx@text@underscore}0^{-1}]$. First note that

[TABLE]

does not depend on $i$, and we do not need to add this as an additional relation.

Let $|I|=k>1$. We claim that

[TABLE]

Indeed, if $I=\{i_{\lx@text@underscore}1,\ldots,i_{\lx@text@underscore}k\}$ then

[TABLE]

hence

[TABLE]

Now (26) implies (after multiplication by $\mathbf{U}^{\frac{(k-1)(k-2)}{2}}\mathsf{A}^{k-1}$):

[TABLE]

By applying (25) this can be rewritten as $U_{\lx@text@underscore}I\frac{a_{\lx@text@underscore}{k-1}}{a_{\lx@text@underscore}0}=V_{\lx@text@underscore}{\overline{I}}\frac{a_{\lx@text@underscore}k}{a_{\lx@text@underscore}0}$, and after multiplication by $a_{\lx@text@underscore}0$ we get (22).

To prove (23), we divide both sides by $a_{\lx@text@underscore}0^{2}$ and get

[TABLE]

and this is immediate from (25).

∎

Remark 4.13**.**

We can also directly check the isomorphism (24) using Lemma 4.2 and Corollary 4.4. By Lemma 4.2, we have two cases. If $\min(\overline{s}_{\lx@text@underscore}1,\ldots,\overline{s}_{\lx@text@underscore}n)\geq 0$ then $\mathcal{H\!F\!L}^{\mathrm{stab}}(T(n,mn),\overline{s}_{\lx@text@underscore}1,\ldots,\overline{s}_{\lx@text@underscore}n)$ is generated by

[TABLE]

If $-m\leq\min(\overline{s}_{\lx@text@underscore}1,\ldots,\overline{s}_{\lx@text@underscore}n)\leq 0$ then $\mathcal{H\!F\!L}^{\mathrm{stab}}(T(n,mn),\overline{s}_{\lx@text@underscore}1,\ldots,\overline{s}_{\lx@text@underscore}n)$ is generated by

[TABLE]

Therefore the limiting homology is indeed isomorphic to

[TABLE]

as a graded vector space.

5. Module structures of Colored knot Floer homology

In this section, we use the algebra $\mathcal{A}_{\lx@text@underscore}n$ and its localization to study the module structure for cabled and colored knot Floer homology.

5.1. Action of cobordism maps: general case

Recall that for an $n$-component link $L$ in $S^{3}$ such that every component intersects a disk $D$ positively exactly once,

[TABLE]

As before, this is a $\mathbb{Z}^{n}\oplus\mathbb{Z}\oplus\mathbb{Z}$ graded vector space with Alexander, Maslov and twist gradings.

Theorem 5.1**.**

The space $\mathcal{TW}_{\lx@text@underscore}D(L)$ is a triply graded module over the algebra $\mathcal{A}_{\lx@text@underscore}n$.

Proof.

We have cobordism maps $\phi_{\lx@text@underscore}k:\mathcal{H\!F\!L}(L_{\lx@text@underscore}m)\to\mathcal{H\!F\!L}(L_{\lx@text@underscore}{m+1})$. We construct the action of the algebra $\mathcal{A}_{\lx@text@underscore}n$ such that the generators $a_{\lx@text@underscore}k$ act on $\mathcal{TW}_{\lx@text@underscore}D(L)$ via the maps $\phi_{\lx@text@underscore}k$. For this to make sense, we need to verify that the maps $\phi_{\lx@text@underscore}k$ pairwise commute and satisfy the relations (22) and (23). We closely follow the proof of [1, Proposition 3.13].

We would like to compare the cobordism maps from $\mathcal{H\!F\!L}(L_{\lx@text@underscore}m)$ to $\mathcal{H\!F\!L}(L_{\lx@text@underscore}{m+1})$ and the similar maps from $\mathcal{H\!F\!L}(T(n,mn))$ to $\mathcal{H\!F\!L}(T(n,(m+1)n))$. To distinguish them, we denote them respectively by $\phi_{\lx@text@underscore}k^{L}$ and $\phi_{\lx@text@underscore}k^{O}$.

Let us denote by $L^{\prime}_{\lx@text@underscore}{m}$ the link $L_{\lx@text@underscore}{m}$ with an extra pair of basepoints $w^{\prime}_{\lx@text@underscore}i,z^{\prime}_{\lx@text@underscore}i$ per component. We extend the coloring $\sigma$ on $L_{\lx@text@underscore}{m}$ to $L^{\prime}_{\lx@text@underscore}{m}$ so that its codomain is $P^{\prime}=P\sqcup\{p^{\prime}_{\lx@text@underscore}1,\cdots,p^{\prime}_{\lx@text@underscore}n\}$ and $\sigma^{\prime}(w^{\prime}_{\lx@text@underscore}i)=p^{\prime}_{\lx@text@underscore}i,\sigma^{\prime}(z^{\prime}_{\lx@text@underscore}i)=\sigma(z_{\lx@text@underscore}i)$. Then $\sigma^{\prime}$ restricts to a coloring of $L_{\lx@text@underscore}{m}$ with codomain $P^{\prime}$, and for simplicity, we still denote it by $\sigma^{\prime}$. The ring isomorphism $\mathcal{R}_{\lx@text@underscore}P^{-}\cong\mathbb{F}[U_{\lx@text@underscore}1,\cdots,U_{\lx@text@underscore}n,V_{\lx@text@underscore}1,\cdots V_{\lx@text@underscore}n]$ extends to an isomorphism

[TABLE]

by sending formal variables $X_{\lx@text@underscore}{p^{\prime}_{\lx@text@underscore}i}$ to $U^{\prime}_{\lx@text@underscore}i$. Then, we have the following isomorphisms

[TABLE]

and

[TABLE]

Let $\mathcal{S}_{\lx@text@underscore}m$ be the quasi-stabilization cobordism from $L_{\lx@text@underscore}m$ to $L^{\prime}_{\lx@text@underscore}{m}$. Then, the composition of the induced cobordism map from $\mathcal{H\!F\!L}(L_{\lx@text@underscore}m)$ to $\mathcal{H\!F\!L}(L^{\prime}_{\lx@text@underscore}m)$ with the latter isomorphism is identity.

Observe that $L_{\lx@text@underscore}m$ can be obtained as band connected sum of $L$ and $T(n,mn)$ where we add one band per link component. Consider the following cobordisms:

• •

$\mathcal{C}_{\lx@text@underscore}{m}$ is the decorated cobordism from $L_{\lx@text@underscore}m$ to $L_{\lx@text@underscore}{m+1}$ obtained from the $(-1)$-surgery on an unknot bounding a disk intersecting each component of $L_{\lx@text@underscore}m$ positively at exactly one point,

• •

$\mathcal{C}^{\prime}_{\lx@text@underscore}m$ is the induced decorated cobordism from $L^{\prime}_{\lx@text@underscore}{m}$ to $L^{\prime}_{\lx@text@underscore}{m+1}$ similar to $\mathcal{C}_{\lx@text@underscore}m$,

• •

$\mathcal{B}_{\lx@text@underscore}m$ from $L_{\lx@text@underscore}{m}$ to $L_{\lx@text@underscore}{m}\sqcup O_{\lx@text@underscore}n$ corresponding to birth of an $n$-component unlink,

• •

$\mathcal{C}_{\lx@text@underscore}{b,m}$ from $L_{\lx@text@underscore}{m}\sqcup O_{\lx@text@underscore}n$ to $L^{\prime}_{\lx@text@underscore}{m}$ is given by (decorated) band attachment. More generally, band attachment gives a cobordism $\mathcal{C}_{\lx@text@underscore}{b,m,j}$ from $L_{\lx@text@underscore}{m}\sqcup T(n,jm)$ to $L^{\prime}_{\lx@text@underscore}{m+j}$,

• •

$\mathcal{C}_{\lx@text@underscore}{O_{\lx@text@underscore}n,m}$ is the cobordism from $L_{\lx@text@underscore}{m}\sqcup O_{\lx@text@underscore}n$ to $L_{\lx@text@underscore}{m}\sqcup T(n,n)$ given by the 2-handle attachment along the (-1)-framed unknot for inserting a full twist in $O_{\lx@text@underscore}n$.

Define

[TABLE]

Under the aforementioned isomorphism $\mathcal{H\!F\!L}(L_{\lx@text@underscore}{m+1}^{{}^{\prime}\sigma^{\prime}})\cong\mathcal{H\!F\!L}(L_{\lx@text@underscore}{m+1})$, the homomorphism induced by the cobordism map $F_{\lx@text@underscore}{\tilde{\mathcal{C}}_{\lx@text@underscore}m,\mathfrak{s}_{\lx@text@underscore}k}$ from $\mathcal{H\!F\!L}(L_{\lx@text@underscore}{m})$ to $\mathcal{H\!F\!L}(L_{\lx@text@underscore}{m+1})$ is equal to $\phi^{L}_{\lx@text@underscore}k$. Note that $\mathcal{S}_{\lx@text@underscore}m$ can be decomposed as the cobordism $\mathcal{B}_{\lx@text@underscore}m$ containing $n$ births from $L_{\lx@text@underscore}{m}$ to $L_{\lx@text@underscore}{m}\sqcup O_{\lx@text@underscore}n$ followed by $n$ band attachments cobordism $\mathcal{C}_{\lx@text@underscore}{b,m}$.

By arguing as in [1, Proposition 3.13] (see also Proposition 3.8 in [1]) we can isotope the attaching circle of the 2-handle in $\tilde{\mathcal{C}}_{\lx@text@underscore}m$ so that we can change the order of 2-handle attachment and band attachments, which shows that the composition

[TABLE]

is isotopic to the composition

[TABLE]

Therefore by naturality ([38]) we get

[TABLE]

Here, $\phi_{\lx@text@underscore}k^{O}:\mathcal{H\!F\!L}(O_{\lx@text@underscore}{n})\rightarrow\mathcal{H\!F\!L}(T(n,n))$ denotes the full twist cobordism map. Since all the maps are $R_{\lx@text@underscore}{UV}$-linear and $\phi_{\lx@text@underscore}k^{O}$ satisfy linear relations (22) by Theorem 4.6, the maps $\phi_{\lx@text@underscore}k^{L}$ satisfy (22) as well.

Next, we need to study the compositions of two maps $\phi_{\lx@text@underscore}{\ell}^{L}:\mathcal{H\!F\!L}(L_{\lx@text@underscore}{m})\rightarrow\mathcal{H\!F\!L}(L_{\lx@text@underscore}{m+1})$ and $\phi_{\lx@text@underscore}{k}^{L}:\mathcal{H\!F\!L}(L_{\lx@text@underscore}{m+1})\rightarrow\mathcal{H\!F\!L}(L_{\lx@text@underscore}{m+2})$

[TABLE]

By isotopying the attaching circle of the second $2$-handle, changing the order of band attachments and the second 2-handle attachment, and using the naturality of link Floer homology we have

[TABLE]

Consequently, since (23) and commutation relations hold for the unlink by Theorem 4.6, they hold for $L$ as well. ∎

5.2. Colored homology: general case

Next, we turn to the limit $\mathcal{H}_{\lx@text@underscore}D(L)$ for a link $L$ with the decorated disk $D$.

Theorem 5.2**.**

The colored homology $\mathcal{H}_{\lx@text@underscore}{D}(L)$ is a module over the algebra $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$.

Proof.

This is a consequence of Theorem 5.1. Consider an element $b=\frac{a}{a_{\lx@text@underscore}0^{t}}\in\mathcal{A}_{\lx@text@underscore}n[a_{\lx@text@underscore}0^{-1}]$ with $\mathbf{tw}(a)=t$, and a representative in an equivalence class $Y\in\mathcal{H\!F\!L}(L_{\lx@text@underscore}{m})$. We define $b(Y):=[a(Y)]$ where the action of $a$ is given by Theorem 5.1.

We need to check that this is well-defined under the equivalence relations for $b$ and $Y$. First, we can write

[TABLE]

Then

[TABLE]

Second, we have $Y\sim\phi_{\lx@text@underscore}0(Y)$ and

[TABLE]

Note that in both cases we need to use the fact that the actions of $a$ and $\phi_{\lx@text@underscore}0$ commute. This follows from the fact (proven in Theorem 5.1) that the actions of $\phi_{\lx@text@underscore}k$ and $\phi_{\lx@text@underscore}0$ commute. In fact, by Lemma 2.3 this is enough to ensure the action of $a$ on the colimit, as in the second case. ∎

5.3. Braid group action

The homology of cables has a natural braid group action by cobordisms that swap different components. We summarize its properties in the following proposition.

Proposition 5.3**.**

The braid group $\mathrm{Br}_{\lx@text@underscore}n$ acts on $\mathcal{H\!F\!L}(L_{\lx@text@underscore}{m})$ for all $m$. Given a braid $\beta\in\mathrm{Br}_{\lx@text@underscore}n$ and the corresponding permutation $w_{\lx@text@underscore}{\beta}\in S_{\lx@text@underscore}n$ we have

[TABLE]

Furthermore, the action of $\mathrm{Br}_{\lx@text@underscore}n$ commutes with the maps $\phi_{\lx@text@underscore}k:\mathcal{H\!F\!L}(L_{\lx@text@underscore}{m})\to\mathcal{H\!F\!L}(L_{\lx@text@underscore}{m+1})$:

[TABLE]

Proof.

All statements follow from the naturality of cobordism maps in link Floer homology [38]. For the last equation, note that $\phi_{\lx@text@underscore}k$ corresponds to blowing down the meridian which commutes (up to isotopy) with permuting the components of the link.

∎

Corollary 5.4**.**

There is a natural $S_{\lx@text@underscore}n$ action on the algebra $\mathcal{A}_{\lx@text@underscore}n$ which permutes $U_{\lx@text@underscore}i,V_{\lx@text@underscore}i$ and fixes $\mathbf{U}$ and $a_{\lx@text@underscore}0,\ldots,a_{\lx@text@underscore}{n-1}$. Furthermore, the action of $\mathrm{Br}_{\lx@text@underscore}n$ on $\mathcal{TW}_{\lx@text@underscore}D(L)$ and the action of $S_{\lx@text@underscore}n$ on $\mathcal{A}_{\lx@text@underscore}n$ are compatible via the action of $\mathcal{A}_{\lx@text@underscore}n$ on $\mathcal{TW}_{\lx@text@underscore}D(L)$ from Theorem 5.1.

Note that the relations (22) and (23) are clearly invariant under the $S_{\lx@text@underscore}n$ action. We can apply these results to colored homology.

Proposition 5.5**.**

There is a natural action of $\mathrm{Br}_{\lx@text@underscore}n$ on the colored homology $\mathcal{H}_{\lx@text@underscore}{D}(L)$. It is compatible with the action of $S_{\lx@text@underscore}n$ on $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}=\mathcal{A}_{\lx@text@underscore}n[a_{\lx@text@underscore}0^{-1}]$ which permutes $U_{\lx@text@underscore}i,V_{\lx@text@underscore}i$ and fixes $\mathsf{A}$.

Proof.

This follows from the fact that braid group action fixes $a_{\lx@text@underscore}0$, and Lemma 2.3. ∎

6. Examples: L-space knots

As in (4), we define $\mathbf{c}_{\lx@text@underscore}m=m(n-1)/2$ and $s_{\lx@text@underscore}i=\overline{s}_{\lx@text@underscore}i+\mathbf{c}_{\lx@text@underscore}m$. Similarly, we define $h^{\mathrm{stab}}_{\lx@text@underscore}{n,mn}(\overline{\mathbf{s}})=h_{\lx@text@underscore}{n,mn}(\mathbf{s})$. If $K_{\lx@text@underscore}{n,mn}$ is an L-space link then

[TABLE]

6.1. Colored homology: description

We study cables $K_{\lx@text@underscore}{n,mn}$ with $n$ components. Each of these components is isotopic to $K_{\lx@text@underscore}{1,m}=K$. In this section, we assume that $K$ is an L-space knot and $m$ is large, so that $K_{\lx@text@underscore}{n,mn}$ is an $L$-space link by [13]. First, we can describe the colored homology as a graded $\mathbb{F}[\mathbf{U}]$-module, one Alexander grading at a time.

Lemma 6.1**.**

Let us fix the renormalized Alexander grading $\overline{\mathbf{s}}=(\overline{s}_{\lx@text@underscore}1,\ldots,\overline{s}_{\lx@text@underscore}n)$. Then for $m$ large enough we have

[TABLE]

where $\min(\overline{\mathbf{s}})=\min\{\overline{s}_{\lx@text@underscore}1,\cdots,\overline{s}_{\lx@text@underscore}n\}$, and the map $\phi_{\lx@text@underscore}0:\mathcal{H\!F\!L}^{\mathrm{stab}}(K_{\lx@text@underscore}{n,mn},\overline{\mathbf{s}})\to\mathcal{H\!F\!L}^{\mathrm{stab}}(K_{\lx@text@underscore}{n,(m+1)n},\overline{\mathbf{s}})$ is an isomorphism. In particular, for $m$ large enough we get $\mathcal{H\!F\!L}^{\mathrm{stab}}(K_{\lx@text@underscore}{n,mn},\overline{\mathbf{s}})\simeq\mathcal{H}_{\lx@text@underscore}n(K,\overline{\mathbf{s}}).$

Proof.

As in Theorem 4.6, we argue that the map $\phi_{\lx@text@underscore}0:\mathcal{H\!F\!L}^{\mathrm{stab}}(K_{\lx@text@underscore}{n,mn})\to\mathcal{H\!F\!L}^{\mathrm{stab}}(K_{\lx@text@underscore}{n,(m+1)n})$ is nonzero, has (stable) Alexander degree zero and its action on the tower $\mathcal{H\!F\!L}^{\mathrm{stab}}(K_{\lx@text@underscore}{n,mn},\overline{\mathbf{s}})\simeq\mathbb{F}[\mathbf{U}]$ is determined by its Maslov degree.

Note that the $h$-function is symmetric, and by [13, Theorem 4.3] we can write it for $K_{\lx@text@underscore}{n,mn}$ as

[TABLE]

where $s_{\lx@text@underscore}1\leq s_{\lx@text@underscore}2\leq\ldots\leq s_{\lx@text@underscore}n$ and $h_{\lx@text@underscore}K$ denotes the $h$-function of $K$. So,

[TABLE]

We can choose $m$ large enough so that $\overline{s}_{\lx@text@underscore}i>g(K)-m$ for all $i$, then $h_{\lx@text@underscore}K(\overline{s_{\lx@text@underscore}k}+(k-1)m)=0$ for $k>1$. We conclude that for large $m$

[TABLE]

and the result follows. ∎

Next, we describe the module structure of this homology over $\mathbb{F}[U_{\lx@text@underscore}1,\ldots,U_{\lx@text@underscore}n,V_{\lx@text@underscore}1,\ldots,V_{\lx@text@underscore}n]$. First, we need to introduce some notations. Following [29] we have

[TABLE]

where $S=\{\sigma_{\lx@text@underscore}1,\sigma_{\lx@text@underscore}2,\ldots\}$ is some bounded above subset of $\mathbb{Z}$ and we assume $\sigma_{\lx@text@underscore}1>\sigma_{\lx@text@underscore}2>\ldots$ where $\sigma_{\lx@text@underscore}i=1-i$ for $i\geq g+1$. We can describe $\mathcal{H\!F\!L}(K)$ by generators and relations:

• •

The generators are $z_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i}$ for $\sigma_{\lx@text@underscore}i\in S$, with Alexander degree $\sigma_{\lx@text@underscore}i$ and homological degree $-2h_{\lx@text@underscore}K(\sigma_{\lx@text@underscore}i)=-2(i-1)$.

• •

The relations are

[TABLE]

Indeed, both $Uz_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i}$ and $V^{\sigma_{\lx@text@underscore}i-\sigma_{\lx@text@underscore}{i+1}-1}z_{\lx@text@underscore}{\sigma_{\lx@text@underscore}{i+1}}$ have Alexander degree $\sigma_{\lx@text@underscore}i-1$ and homological degree $-2i=-2(i-1)-2$. Note that when $\sigma_{\lx@text@underscore}{i+1}=\sigma_{\lx@text@underscore}i-1$ we have $Uz_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i}=z_{\lx@text@underscore}{\sigma_{\lx@text@underscore}{i+1}}$, so we can in principle eliminate $z_{\lx@text@underscore}{\sigma_{\lx@text@underscore}{i+1}}$. In particular, for $i\geq g+1$ we can write $z_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i}=U^{i-g-1}z_{\lx@text@underscore}{-g}$.

Theorem 6.2**.**

The colored homology $\mathcal{H}_{\lx@text@underscore}{n}(K)$ has generators $\widetilde{z}_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i}$ of Alexander degree

[TABLE]

and Maslov degree $-2(i-1)$. The relations are given by

[TABLE]

where $\mathbf{e}=\sum_{\lx@text@underscore}{j=1}^{n}\mathbf{e}_{\lx@text@underscore}j$, and $\mathbf{e}_{\lx@text@underscore}j$ is the unit vector with $j$-th entry equal one. Here,

[TABLE]

Proof.

Let $z(\overline{\mathbf{s}})$ denote the generator of the $\mathbb{F}[\mathbf{U}]$ tower in $\mathcal{H}_{\lx@text@underscore}{n}(K)$ of renormalized Alexander degree $\overline{\mathbf{s}}$. By Lemma 6.1 the Maslov degree of $z(\overline{\mathbf{s}})$ equals $-2h_{\lx@text@underscore}K(\min(\overline{\mathbf{s}}))$.

We can describe the action of $U_{\lx@text@underscore}i$ and $V_{\lx@text@underscore}i$ on generators $z(\overline{\mathbf{s}})$ as follows:

Case 1: $\min(\overline{\mathbf{s}})=\min(\overline{\mathbf{s}}+\mathbf{e}_{\lx@text@underscore}i)=\sigma$. In this case

[TABLE]

Note that this implies

[TABLE]

and the diagonal elements generate $\mathcal{H}_{\lx@text@underscore}{n}(K)$ under the action of $V_{\lx@text@underscore}i$.

Case 2: $\min(\overline{\mathbf{s}})=\sigma,\min(\overline{\mathbf{s}}+\mathbf{e}_{\lx@text@underscore}i)=\sigma+1$. In this case

[TABLE]

By combining (29) and (30) we get

[TABLE]

In particular, $V_{\lx@text@underscore}1\cdots V_{\lx@text@underscore}nz(\sigma,\ldots,\sigma)=z(\sigma+1,\ldots,\sigma+1)$ whenever $h_{\lx@text@underscore}K(\sigma)=h_{\lx@text@underscore}K(\sigma+1)$, so the homology is generated by

[TABLE]

The relations (28) follow directly from grading computation and by a similar argument as Theorem 4.1, the relations (28) generate all relations among the generators.

∎

Remark 6.3**.**

We get an infinite chain of generators $\widetilde{z}_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i}$ for $i\geq g+1$, these have Alexander degrees $A(\widetilde{z}_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i})=(1-i,\ldots,1-i)$ and Maslov degrees $\textup{gr}_{\lx@text@underscore}{{\bf{w}}}(\widetilde{z}_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i})=-2(i-1)$. The relations between these generators are nontrivial and given by

[TABLE]

In particular, $\mathcal{H}_{\lx@text@underscore}{n}(K)$ is infinitely generated over $\mathbb{F}[U_{\lx@text@underscore}1,\ldots,U_{\lx@text@underscore}n,V_{\lx@text@underscore}1,\ldots,V_{\lx@text@underscore}n]$.

6.2. Colored homology: module structure

In this subsection we determine the module structure of $\mathcal{H}_{\lx@text@underscore}{n}(K)$ over the cabled algebra $\mathcal{A}_{\lx@text@underscore}n[a_{\lx@text@underscore}0^{-1}]$ assuming that $K$ is an L-space knot.

Lemma 6.4**.**

In the notations of Theorem 6.2 the action of $\mathsf{A}=a_{\lx@text@underscore}1/a_{\lx@text@underscore}0$ is given by

[TABLE]

Proof.

The action of $\phi_{\lx@text@underscore}1:\mathcal{H\!F\!L}(K_{\lx@text@underscore}{n,mn})\to\mathcal{H\!F\!L}(K_{\lx@text@underscore}{n,(m+1)n})$ is nontrivial and sends an $\mathbb{F}[\mathbf{U}]$ tower inside another $\mathbb{F}[\mathbf{U}]$ tower. Hence by construction in Theorem 5.2 the action of $\mathsf{A}=a_{\lx@text@underscore}1/a_{\lx@text@underscore}0$ on colored homology is nontrivial as well, and is determined by its shift of Alexander and Maslov degrees.

The element $\mathsf{A}\widetilde{z}_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i}$ has (renormalized) Alexander degree $(\sigma_{\lx@text@underscore}i-1,\ldots,\sigma_{\lx@text@underscore}i-1)$ and Maslov degree $-2i$ which agrees with the Alexander and Maslov degrees of $(V_{\lx@text@underscore}1\cdots V_{\lx@text@underscore}n)^{\sigma_{\lx@text@underscore}i-\sigma_{\lx@text@underscore}{i+1}-1}\widetilde{z}_{\lx@text@underscore}{\sigma_{\lx@text@underscore}{i+1}}.$ ∎

Remark 6.5**.**

For $i\geq g+1$ we have $\mathsf{A}\widetilde{z}_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i}=\widetilde{z}_{\lx@text@underscore}{\sigma_{\lx@text@underscore}{i+1}},$ so we can compactly write

[TABLE]

In particular, $\mathcal{H}_{\lx@text@underscore}{n}(K)$ is finitely generated over $\mathbb{F}[U_{\lx@text@underscore}1,\ldots,U_{\lx@text@underscore}n,V_{\lx@text@underscore}1,\ldots,V_{\lx@text@underscore}n,\mathsf{A}]$, in contrast with Remark 6.3.

Theorem 6.6**.**

Assume $K$ is an L-space knot, then

[TABLE]

where we regard $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$ as a $\mathbb{F}[U,V]$-module via the homomorphism $\varepsilon_{\lx@text@underscore}n:\mathbb{F}[U,V]\to\mathbb{F}[U_{\lx@text@underscore}1,\ldots,U_{\lx@text@underscore}n,V_{\lx@text@underscore}1,\ldots,V_{\lx@text@underscore}n,\mathsf{A}]$ defined by

[TABLE]

Proof.

We would like to compare the right hand side (as an $R_{\lx@text@underscore}{UV}$-module) with Theorem 6.2.

First, we claim that the elements $z_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i}\otimes 1$ generate the right hand side over $R_{\lx@text@underscore}{UV}$, and can be identified with $\widetilde{z}_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i}$ in the left hand side. Indeed, $z_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i}$ generate $\mathcal{H\!F\!L}(K)$ over $\mathbb{F}[U,V]$ and by Theorem 4.12, $\mathsf{A}^{k},k\geq 0$ generate $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$ over $R_{\lx@text@underscore}{UV}$. Furthermore,

[TABLE]

so the powers of $\mathsf{A}$ can be eliminated.

Next, we need to check that the relations on the right hand side match the ones on the left hand side. By (31) and Lemma 6.4 the relation (27) in $\mathcal{H\!F\!L}(K)$ translates to

[TABLE]

which holds in $\mathcal{H}_{\lx@text@underscore}{n}(K)$. Next, on the right hand side we have the relation

[TABLE]

By again applying (31) this is equivalent to

[TABLE]

which coincides with (28). ∎

Remark 6.7**.**

If $K$ is a plumbed L-space knot, the sufficiently large cables $K_{\lx@text@underscore}{n,mn}$ are plumbed L-space links and by the main result of [5] the chain complexes $\mathcal{C\!F\!L}(K_{\lx@text@underscore}{n,mn})$ are homotopy equivalent to the resolutions of $\mathcal{H\!F\!L}(K_{\lx@text@underscore}{n,mn})$. It would be interesting to lift Theorem 6.6 to the level of chain complexes by considering the minimal resolution of colored homology defined by (28).

6.3. Example: $(3,4)$ torus knot

Suppose that $K=T(3,4)$, then $\Delta_{\lx@text@underscore}{K}(t)=t^{3}-t^{2}+1-t^{-2}+t^{-3}$ and $\chi_{\lx@text@underscore}K(t)=t^{3}+1+t^{-1}+t^{-3}+t^{-4}+\ldots$, so that $S=\{3,0,-1,-3,-4,\ldots\}$. The $h$-function of $T(3,4)$ is given by the following table (where $\sigma_{\lx@text@underscore}i\in S$ are marked in bold):

[TABLE]

The homology $\mathcal{H\!F\!L}(K)$ has generators $z_{\lx@text@underscore}{3},z_{\lx@text@underscore}0,z_{\lx@text@underscore}{-1},z_{\lx@text@underscore}{-3},z_{\lx@text@underscore}{-4},z_{\lx@text@underscore}{-5},\ldots$ and relations

[TABLE]

Now let us proceed to the $(2,2m)$ cables of $T(3,4)$ for sufficiently large $m$. Let $m=6$, then the $h$-function (in normalized Alexander grading) of the $(2,12)$ cable of $T(3,4)$ is given by

[TABLE]

and shown in the following table:

$\overline{s}_{\lx@text@underscore}1$543210-1-2-3-4-5$\overline{s}_{\lx@text@underscore}2$543210-1-2-3-4-50001112334500011123345000111233451111112334511111123345111111233452222222334533333333345333333333454444444445655555555566

The stable range is given by $\overline{s}_{\lx@text@underscore}1,\overline{s}_{\lx@text@underscore}2\geq g(K)-m=3-6=-3,$ and the stable generators $\widetilde{z}_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i}$ are marked by circles.

For $m=7$ the $h$-function is very similar, except for

[TABLE]

and there is another stable generator $\widetilde{z}_{\lx@text@underscore}{-4}$ with $\textup{gr}_{\lx@text@underscore}{{\bf{w}}}(\widetilde{z}_{\lx@text@underscore}{-4})=-8$.

For $m\geq 8$ we get

[TABLE]

and there are stable generators $\widetilde{z}_{\lx@text@underscore}{-4}$, $\widetilde{z}_{\lx@text@underscore}{-5}$ with with $\textup{gr}_{\lx@text@underscore}{{\bf{w}}}(\widetilde{z}_{\lx@text@underscore}{-4})=-8,\textup{gr}_{\lx@text@underscore}{{\bf{w}}}(\widetilde{z}_{\lx@text@underscore}{-5})=-10$. The stable relations between $\widetilde{z}_{\lx@text@underscore}{\sigma_{\lx@text@underscore}i}$ can be written in two ways:

1. Following [5] and Theorem 6.2, we can compute for $m\geq 6$:

[TABLE]

[TABLE]

[TABLE]

For $m\geq 7$ we get additional relations

[TABLE]

for $m\geq 8$ we get additional relations

[TABLE]

and so on.

2. Following Lemma 6.4, we can instead describe the action of $\mathsf{A}$. For $m\geq 6$ we get

[TABLE]

The last two relations make sense for $m\geq 7$ and $m\geq 8$ respectively. Then all of the above relations are determined by (33) and $U_{\lx@text@underscore}1=\mathsf{A}V_{\lx@text@underscore}2,\ U_{\lx@text@underscore}2=\mathsf{A}V_{\lx@text@underscore}1.$ Finally, note that (33) can be obtained from (32) by changing $z_{\lx@text@underscore}i$ to $\widetilde{z}_{\lx@text@underscore}i$, $U$ to $\mathsf{A}$ and $V$ to $V_{\lx@text@underscore}1V_{\lx@text@underscore}2$, in agreement with Theorem 6.6.

7. Application: crossing change maps

Suppose two knot diagrams $K^{+}$ and $K^{-}$ differ at a single crossing which is positive in $K^{+}$ and negative in $K^{-}$. We would like to find a relation between their colored homologies, but first we relate the homology of cables.

The blackboard framings for both $K^{+}$ and $K^{-}$ correspond to the writhe of their diagrams. Note that the writhe of $K^{+}$ is 2 more than the writhe of $K^{-}$. The $n$-component cables of $K^{+}$ and $K^{-}$ are related by $n^{2}$ crossing changes. More precisely, changing $n^{2}$ negative crossings in $K^{-}_{\lx@text@underscore}{n,mn}$ to positive results in $K^{+}_{\lx@text@underscore}{n,(m+2)n}$ since the linking number between each pair of components increases by 2.

Lemma 7.1**.**

For $j\in\mathbb{Z}$ there is a map

[TABLE]

of Maslov degree $-j^{2}-j$ and Alexander degree $A_{\lx@text@underscore}i(G_{\lx@text@underscore}{j,m})=-2j-1+2n,\ i=1,\ldots,n$. Furthermore, the maps $G_{\lx@text@underscore}{j,m}$ commute with the connecting maps $\phi_{\lx@text@underscore}k$:

[TABLE]

Proof.

Consider the following braids on $2n$ strands: $\mathrm{FT}_{\lx@text@underscore}{[1,n]}$ is the full twist on the left $n$ strands, $\mathrm{FT}_{\lx@text@underscore}{[n+1,2n]}$ is the full twist on the right $n$ strands, and $\mathrm{FT}_{\lx@text@underscore}{[1,2n]}$ is the full twist on all strands. Furthermore, consider the “positive cabled crossing” braid $\beta_{\lx@text@underscore}{n,n}$ which is the positive braid lift of the permutation $[n+1,\ldots,2n,1,\ldots,n]$:

$\beta_{\lx@text@underscore}{n,n}=$

It is easy to see that $\mathrm{FT}_{\lx@text@underscore}{[1,2n]}=\mathrm{FT}_{\lx@text@underscore}{[1,n]}\mathrm{FT}_{\lx@text@underscore}{[n+1,2n]}\beta_{\lx@text@underscore}{n,n}^{2}$, and so $\mathrm{FT}_{\lx@text@underscore}{[1,2n]}\beta_{\lx@text@underscore}{n,n}^{-1}=\mathrm{FT}_{\lx@text@underscore}{[1,n]}\mathrm{FT}_{\lx@text@underscore}{[n+1,2n]}\beta_{\lx@text@underscore}{n,n}$, that is:

-1$$=$$\mathrm{FT}$$=$$\mathrm{FT}$$\mathrm{FT}

This means that adding a full twist to $K^{-}_{\lx@text@underscore}{n,mn}$ results in $K^{+}_{\lx@text@underscore}{n,(m+2)n}$ with two additional full twists which is isotopic to $K^{+}_{\lx@text@underscore}{n,(m+4)n}$. Now the construction of $G_{\lx@text@underscore}{j,m}$ follows from [1, Proposition 3.10], this also gives the Maslov grading change.

For the Alexander grading we use equation (16). Let $M$ be the $(-1)$-framed unknot. Since $\mathrm{lk}(K^{-}_{\lx@text@underscore}{n,mn},M)=2n$ and $\mathrm{lk}(L_{\lx@text@underscore}i,M)=2$ for each component $L_{\lx@text@underscore}i$, we get

[TABLE]

The cobordism defining $\phi_{\lx@text@underscore}k$ corresponds to blowing down a meridian of $K^{-}_{\lx@text@underscore}{n,mn}$ (resp. $K^{+}_{\lx@text@underscore}{n,(m+4)n}$) which can be done away from the crossing, and therefore the two cobordisms commute and induced maps in Heegaard Floer homology commute. ∎

Lemma 7.2**.**

For $j\in\mathbb{Z}$ there is a map

[TABLE]

of Maslov degree $-j^{2}-j$ and Alexander degree $A_{\lx@text@underscore}i=0$. Furthermore, the maps $F_{\lx@text@underscore}{j,m}$ commute with the connecting maps $\phi_{\lx@text@underscore}k$:

[TABLE]

Proof.

The proof is similar to the proof of Lemma 7.1, where we now use the following diagram (compare with [1, Proposition 3.1]):

$-1$

Blowing down the circle changes the positive cabled crossing in $K^{+}_{\lx@text@underscore}{n,mn}$ to the negative one (resulting in $K^{-}_{\lx@text@underscore}{n,(m-2)n}$), introduces a positive full twist on one set of $n$ strands and a negative full twist on the second set of $n$ strands. Since $K$ is connected, we can slide the full twists along $K$ and cancel with each other. As a result, we are left with $K^{-}_{\lx@text@underscore}{n,(m-2)n}$.

For the Alexander degree, we denote the $(-1)$-framed unknot by $M$, and use (16) again. We have $\mathrm{lk}(L_{\lx@text@underscore}i,M)=0$ for all component $L_{\lx@text@underscore}i$ of $K^{+}_{\lx@text@underscore}{n,mn}$, so the Alexander grading does not change. ∎

Theorem 7.3**.**

a) The maps $G_{\lx@text@underscore}{j,m}$ induce maps

[TABLE]

of twist degree $4$ which commute with the action of the cable algebra $\mathcal{A}_{\lx@text@underscore}n$.

b) We obtain maps

[TABLE]

which commute with the action of the algebra $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$.

c) The maps $F_{\lx@text@underscore}{j,m}$ induce maps

[TABLE]

of twist degree $-2$ which commute with the action of the cable algebra $\mathcal{A}_{\lx@text@underscore}n$.

d) We obtain maps

[TABLE]

which commute with the action of the algebra $\mathcal{A}_{\lx@text@underscore}n^{\mathrm{col}}$.

Proof.

Part (a) is a restatement of Lemma 7.1, and (b) follows from (a) after regrading

[TABLE]

Similarly, (c) follows from Lemma 7.2 and (d) follows from (c) after regrading

[TABLE]

∎

Remark 7.4**.**

The cables $K^{+}_{\lx@text@underscore}{n,(m+2)n}$ and $K^{-}_{\lx@text@underscore}{n,mn}$ are related by a sequence of $n^{2}$ crossing changes, and we can instead consider compositions of local cobordisms maps at these crossings as in [1, Section 5].

Let $Z\subset\{1,\ldots,n\}\times\{1,\ldots,n\}$ be an arbitrary subset. Then there exists a map

[TABLE]

of Maslov degree zero and Alexander degree

[TABLE]

It is easy to see that, as before, $\Psi_{\lx@text@underscore}{Z,m}$ commutes with the connecting maps $\phi_{\lx@text@underscore}k$, and hence defines a map $\mathcal{H}_{\lx@text@underscore}n(K^{+})\to\mathcal{H}_{\lx@text@underscore}n(K^{-})$.