Nearly fibered links with genus one

TL;DR

This paper classifies nearly fibered links with genus one in the 3-sphere, characterized by specific properties of their link Floer homology, and computes their Floer homology groups.

Contribution

It provides a complete classification of nearly fibered links with genus one and computes their link Floer homology groups, extending previous knot results.

Findings

Classified all nearly fibered links with genus one in S^3.

Computed the link Floer homology groups for these links.

Connected Floer homology properties with geometric link features.

Abstract

We classify all the $n$-component links in the $3$-sphere that bound a Thurston norm minimizing Seifert surface $\Sigma$ with Euler characteristic $\chi(\Sigma)=n-2$ and that are nearly fibered, which means that their rank of the maximal (collapsed) Alexander grading $s_{\text{top}}$ of the link Floer homology group $\widehat{HFL}$ is equal to two. In other words, such a link $L$ satisfies $s_{\text{top}}=\frac{n-\chi(\Sigma)}{2}=1$, and in addition $\text{rk}\:\widehat{HFL}_{*}(L)[1]=2$ and $\text{rk}\:\widehat{HFL}_{*}(L)[s]=0$ for every $s>1$. The proof of the main theorem is inspired by the one of a similar recent result for knots by Baldwin and Sivek; and involves techniques from sutured Floer homology. Furthermore, we also compute the group $\widehat{HFL}$ for each of these links.

Tables

Table 1. Table 1 : The link Floer homology group, with collapsed Alexander grading, of every nearly fibered (multi-component) link with Seifert big genus equal to one.

${\widehat{HFL}}_d\left(L\right)\left[s\right]$	$s=-1$	$s=0$	$s=1$
$T_{2,3}\bigsqcup ○$	${𝔽}_{\left(-3\right)}\oplus {𝔽}_{\left(-2\right)}$	${𝔽}_{\left(-2\right)}\oplus {𝔽}_{\left(-1\right)}$	${𝔽}_{\left(-1\right)}\oplus {𝔽}_{\left(0\right)}$
$T_{2,-3}\bigsqcup ○$	${𝔽}_{\left(-1\right)}\oplus {𝔽}_{\left(0\right)}$	${𝔽}_{\left(0\right)}\oplus {𝔽}_{\left(1\right)}$	${𝔽}_{\left(1\right)}\oplus {𝔽}_{\left(2\right)}$
$4_1\bigsqcup ○$	${𝔽}_{\left(-2\right)}\oplus {𝔽}_{\left(-1\right)}$	${𝔽}_{\left(-1\right)}\oplus {𝔽}_{\left(0\right)}$	${𝔽}_{\left(0\right)}\oplus {𝔽}_{\left(1\right)}$
$T_{2,2}\bigsqcup ○$	${𝔽}_{\left(-3\right)}\oplus {𝔽}_{\left(-2\right)}$	${𝔽}_{\left(-2\right)}^2\oplus {𝔽}_{\left(-1\right)}^2$	${𝔽}_{\left(-1\right)}\oplus {𝔽}_{\left(0\right)}$
$T_{2,-2}\bigsqcup ○$	${𝔽}_{\left(-2\right)}\oplus {𝔽}_{\left(-1\right)}$	${𝔽}_{\left(-1\right)}^2\oplus {𝔽}_{\left(0\right)}^2$	${𝔽}_{\left(0\right)}\oplus {𝔽}_{\left(1\right)}$
$T_{2,4}^{\prime }$	${𝔽}_{\left(-2\right)}^2$	${𝔽}_{\left(-1\right)}^4$	${𝔽}_{\left(0\right)}^2$
$T_{2,-4}^{\prime }$	${𝔽}_{\left(-1\right)}^2$	${𝔽}_{\left(0\right)}^4$	${𝔽}_{\left(1\right)}^2$
$T_{2,3;2,4}^{\prime }$	${𝔽}_{\left(-3\right)}^2$	${𝔽}_{\left(-2\right)}^4\oplus {𝔽}_{\left(-1\right)}\oplus {𝔽}_{\left(0\right)}$	${𝔽}_{\left(-1\right)}^2$
$T_{2,-3;2,-4}^{\prime }$	${𝔽}_{\left(0\right)}^2$	${𝔽}_{\left(-1\right)}\oplus {𝔽}_{\left(0\right)}\oplus {𝔽}_{\left(1\right)}^4$	${𝔽}_{\left(2\right)}^2$

Equations

$$G_3(L)=\frac{n-{\chi }_3(F)}{2},$$

$${\widehat{HFL}}_{\ast }(L)[s]=\underset{s=s_1+...+s_n}{⨁}{\widehat{\text{HFL}}}_{\ast }(L)[s_1,...,s_n].$$

$$s_{\text{top}}:=\underset{s\in \mathbb{Z}}{\max }\left\{\mathrm{rk}{\widehat{HFL}}_{\ast }(L)[s]\ne 0\right\}=G_3(L),$$

$$\mathrm{rk}{\widehat{HFL}}_{\ast }(L)[s_{\text{top}}]=1.$$

$${\widehat{HFL}}_d(L^{\ast })[s]\cong {\widehat{HFL}}_{-d+1-n}(L)[-s]$$

$${\widehat{HFL}}_d(T_{2,3;2,4}^{{}^{\prime }})[s]\cong {\mathbb{F}}_{(-1)}^2[1]\oplus {\mathbb{F}}_{(-2)}^4[0]\oplus {\mathbb{F}}_{(-1)}[0]\oplus {\mathbb{F}}_{(0)}[0]\oplus {\mathbb{F}}_{(-3)}^2[-1].$$

$${\widehat{HFL}}_d(L_1\bigsqcup L_2)[s]\cong \underset{\begin{array}{c}{d}_1+d_2=d\\ s_1+s_2=s\end{array}}{⨁}{\widehat{HFL}}_{d_1}(L_1)[s_1]$$

$$\oplus \underset{\begin{array}{c}{d}_1+d_2=d+1\\ s_1+s_2=s\end{array}}{⨁}{\widehat{HFL}}_{d_1}(L_1)[s_1]\otimes {\widehat{HFL}}_{d_2}(L_2)[s_2].$$

$$2=\mathrm{rk}{\widehat{HFL}}_{\ast }(L)[1]=2\cdot \mathrm{rk}{\widehat{HFL}}_{\ast }(L_1)[G_3(L_1)]\cdot \mathrm{rk}{\widehat{HFL}}_{\ast }(L_2)[G_3(L_2)];$$

$$\mathrm{rk}{\widehat{HFL}}_{\ast }(L_i)[G_3(L_i)]=1$$

$$\{\begin{array}{rl}& g(F)=1\\ & n=k\end{array}\text{ or }\{\begin{array}{rl}& g(F)=0\\ & n-1=k\end{array}.$$

$${\lambda }_1^{\prime }\bigsqcup {\lambda }_2^{\prime }=\partial \Phi (A)=\Phi (\partial A)=\Phi ({\lambda }_1\bigsqcup {\lambda }_2)=\Phi ({\lambda }_1)\bigsqcup \Phi ({\lambda }_2)$$

$${\mathrm{\ell k}}^2(L)=|H_1(S_{0,0}^3(L);\mathbb{Z})|=|H_1(S_{0,0}^3(C_{2,4});\mathbb{Z})|=4.$$

$$\{\begin{array}{rl}& \Phi ({\mu }_1)={\mu }_1^{\prime }+p\cdot {\lambda }_1^{\prime }\\ & \Phi ({\mu }_2)={\mu }_2^{\prime }+q\cdot {\lambda }_2^{\prime }\end{array}$$

$$3=|H_1(S_{1,1}^3(L);\mathbb{Z})|=\left|H_1\left(S_{\frac{1}{p+1},\frac{1}{q+1}}^3(C_{2,4});\mathbb{Z}\right)\right|=\left|\det \left(\begin{array}{cccc}-(p+1)& 1& 0& 0\\ 1& 0& 2& 0\\ 0& 2& 0& 1\\ 0& 0& 1& -(q+1)\end{array}\right)\right|=|4(p+1)(q+1)-1|$$

$$3=|H_1(S_{-1,-1}^3(L);\mathbb{Z})|=\left|H_1\left(S_{-\frac{1}{3},-\frac{1}{3}}^3(C_{2,4});\mathbb{Z}\right)\right|=\left|\det \left(\begin{array}{cccc}3& 1& 0& 0\\ 1& 0& 2& 0\\ 0& 2& 0& 1\\ 0& 0& 1& 3\end{array}\right)\right|=35$$

$$h(T_{2,3})(k)=\{\begin{array}{rl}0& k⩾1\\ 1& k=0\\ -k& k⩽-1\end{array}\text{ and }h(T_{2,3;2,4})(k,k)=\{\begin{array}{rl}0& k⩾2\\ 1& k=0,1\\ 3& k=-1\\ -2k& k⩽-2\end{array}.$$

$${\widehat{\text{HFL}}}_d(T_{2,3;2,4})[s_1,s_2]$$

$$\oplus$$

$${\widehat{\text{HFL}}}_d(L^{\prime })[s_1,...,s_n]\cong {\widehat{\text{HFL}}}_{d-2s_i+\mathrm{\ell k}(L_i,L\setminus L_i)}(L)[s_1,...,-s_i,...,s_n].$$

$${\widehat{\text{HFL}}}_d(T_{2,3;2,4}^{\prime })[s_1,s_2]$$

$$\oplus$$

Full text

The proof of the main theorem is inspired by the one of a similar recent result for knots by Baldwin and Sivek, and involves techniques from sutured Floer homology. Furthermore, we also compute the group $\widehat{HFL}$ for each of these links.

2020 Mathematics Subject Classification: 57K10, 57K18.

1 Introduction

The Seifert genus $g_{3}$ is one of the most important invariants of knots. Its definition is standard in literature: say $\Sigma$ is a Seifert surface for a knot $K$ if it is a compact, oriented surface, embedded in $S^{3}$, such that $\partial\Sigma=K$; we call $g_{3}(K)$ the minimal genus of a connected Seifert surface for $K$. When we want to study a link $L$ with $n>1$ components this is no longer true, but there are two main possibilities:

• •

either we can consider $g_{3}(L)$, using precisely the same definition as above;

• •

or we take

[TABLE]

which we call the Seifert big genus of $L$, and we define $\chi_{3}$ as the maximal Euler characteristic of any Seifert surface for $L$ (dropping the requirement of $F$ being connected).

We recall that in $S^{3}$ we can define fibered links as the links whose complements fiber over the circle; then the following is a classical result in low dimensional topology.

Theorem 1.1** (Stallings [16]).**

The only fibered links in $S^{3}$, with Seifert big genus equal to one, are the trefoil knots $T_{2,\pm 3}$, the figure-eight knot $4_{1}$ and the Hopf links $T_{2,\pm 2}$.

We recall that the link Floer chain complex $\left(\widehat{CFL}_{*}(\mathcal{D}),\widehat{\partial}\right)$ over the field with two elements $\mathbb{F}$, introduced by Ozsváth and Szabó in [14], can be constructed from a Heegaard diagram $\mathcal{D}=(\Sigma,\alpha,\beta,\textbf{w},\textbf{z})$, where the sets of basepoints both contain $n$ elements. Under some conditions, the diagram $\mathcal{D}$ represents an $n$-component link $L$. If we ignore the information given by z then $\mathcal{D}$ is just a (multi-pointed) Heegaard diagram for $S^{3}$ and then $\left(\widehat{CFL}_{*}(\mathcal{D}),\widehat{\partial}\right)$ can be interpreted as a multi-filtered complex (the filtration is induced by the link $L$) whose total homology is $\left(\mathbb{F}_{-1}\oplus\mathbb{F}_{0}\right)^{\otimes\>n-1}$. The absolute grading $*$ is called Maslov grading, while the multi-filtration the Alexander filtration.

We write $\widehat{\textbf{HFL}}_{*}(L)[s_{1},...,s_{n}]$ for the homology of the graded object associated to $\left(\widehat{CFL}_{*}(\mathcal{D}),\widehat{\partial}\right)$. This is a multi-graded homology group, where now the Alexander filtration is turned into an absolute (multi-) grading $[s_{1},...,s_{n}]$. Such a group is an isotopy invariant of $L$ in $S^{3}$. We can define a slightly different homology group by collapsing the Alexander multi-grading as follows

[TABLE]

It is proved in [12] that

[TABLE]

while a very important result of Ghiggini [5] and Ni [11] is that a link is fibered if and only if its link Floer homology satisfies the following property:

[TABLE]

Combining Equations (1.2) and (1.3), we obtain that link Floer homology detects each of the five links mentioned in Theorem 1.1; in other words, those are the only links such that $G_{3}(L)=1$ and $\widehat{HFL}_{*}(L)[1]\cong\mathbb{F}$.

In this paper we want to take a step forward and classify all the links in $S^{3}$ such that $G_{3}(L)=1$, but this time with $\widehat{HFL}_{*}(L)[1]\cong\mathbb{F}^{2}$; we call such links nearly fibered following the terminology in [1]. The first part of this classification is the case of knots which was done recently by Baldwin and Sivek.

Theorem 1.2** (Baldwin-Sivek [1]).**

A nearly fibered knot $K$ such that $g_{3}(K)=1$ is isotopic to either the knot $5_{2}$, the knot $15_{43522}$, the $2$-framed Whitehead doubles of the trefoil knot $\emph{Wh}^{\pm}_{2}(T_{2,3})$ or a pretzel knot $P(-3,3,2m+1)$ for $m\geqslant 0$ (including the Stevedore knot $6_{1}$) up to mirroring.

Our main result extends the classification in Theorem 1.2 to links.

Theorem 1.3**.**

A nearly fibered link $L$, with $n>1$ components, such that $G_{3}(L)=1$, is isotopic to either $K\sqcup\bigcirc$, where $K$ is a trefoil knot, a figure-eight knot or a Hopf link, the link $T_{2,4}$ or the $(2,4)$-cable of $T_{2,3}$, with oppositely oriented components, up to mirroring.

The last two links are pictured in Figure 1. From now on we use the notation that $T^{{}^{\prime}}_{2,4}$ represents the torus link $T_{2,4}$ and $T^{{}^{\prime}}_{2,3;2,4}$ represents the $(2,4)$-cable of the positive trefoil knot, both with oppositely oriented components.

We can compute the link Floer homology group $\widehat{HFL}(L)$ for every link in Theorem 1.3; and then also of their mirror images using the formula

[TABLE]

for every $d,s\in\mathbb{Z}$, which is proved in [13].

Proposition 1.4**.**

We have that

[TABLE]

As far as we know this is the first time that this homology group has been computed completely. Furthermore, we obtain that $\widehat{HFL}(L)$ is different for every link in Theorems 1.2 and 1.3 and this implies that link Floer homology detects each nearly fibered link of Seifert big genus one, confirming some results of Binns and Dey in [2] and Binns and Martin in [3].

The paper is organized as follows: in Section 2 we prove Theorem 1.3, separating the case of split and non-split links. Finally, in Section 3 we present our table of genus one nearly fibered links and we explain the proof of Proposition 1.4.

Acknowledgements

AC has a CIRGET post-doctoral fellowship at the Université du Québec à Montréal. IM is supported by the Knut and Alice Wallenberg Foundation through the grant KAW 2021.0191, and by the Swedish Research Counsil through the grant number 2020-04426, as a postdoctoral researcher at Uppsala Universitet. We are both grateful to the Alfréd Rényi Institute of Mathematics in Budapest for its hospitality during the semester on "Singularities and low-dimensional topology", and for the support from the Élvonal (Frontier) grant KKP126683 (given by NKFIH). We thank Steven Sivek for a helpful conversation, and the referee for their corrections.

2 Proof of the main result

2.1 Split links

Suppose that $L$ is as in Theorem 1.3 and is a split link; in this case, we can write $L=L_{1}\sqcup L_{2}$ for some links $L_{1}$ and $L_{2}$. We recall that the link Floer homology of a disjoint union, see [14, 4], satisfies the following relation

[TABLE]

This implies that

[TABLE]

since the top-Alexander grading rank cannot be zero by definition, we have that

[TABLE]

and then $L_{i}$ is fibered for $i=1,2$, see [5, 11]. Moreover, it has to be the case that $G_{3}(L_{1})=1$ and $G_{3}(L_{2})=0$ (up to relabelling); and a big genus one fibered link is either a trefoil or a figure-eight knot or a Hopf link from Theorem 1.1, while it is a classical result that the only big genus zero fibered link is the unknot. This proves the first part of Theorem 1.3.

2.2 Non-split links

The first step in this case is the following lemma.

Lemma 2.1**.**

A non-split link $L$ such that $G_{3}(L)=1$ has at most two components.

Proof.

Say $F\hookrightarrow S^{3}$ is a surface that realizes the Seifert big genus $G_{3}$. A quick computation shows that $1=G_{3}(L)=g(F)+n-k$, where $n$ is the number of components of $L$ and $k$ the number of connected components of $F$. Hence, since $g(F)$ and $n-k$ are both non-negative, we have that either

[TABLE]

In the first case $F$ has $n$ connected components $F_{1},...,F_{n}$, each one with a knot as boundary; this implies that $F_{2},...,F_{n}$ are all diffeomorphic to disks (up to relabelling) and then $L$ has $n-1$ unlinked unknotted components. This is a contradiction, since $L$ is non-split, unless $n=1$.

In the second case $F$ has $n-1$ connected components $F_{1},...,F_{n-1}$, each one with a knot as boundary except one of them which is an annulus; this implies that $F_{2},...,F_{n-1}$ are again diffeomorphic to disks (again up to relabelling) and, in the same way as before, we obtain that $n=2$. ∎

Here, we pass to sutured Floer homology, as defined by Juhász in [9]. We recall that a balanced sutured manifold is a pair $(M,\gamma)$, where $M$ is a compact oriented 3-manifold with boundary and with no closed components, and the sutures $\gamma\subset\partial M$ are oriented $1$-manifolds, meeting each boundary component, that divide the boundary into two subsurfaces $R_{+}$ and $R_{-}$, such that $\chi(R_{+})=\chi(R_{-})$. Similarly to the link case, balanced sutured manifolds can be presented by multi-pointed Heegaard diagrams, and then sutured Floer homology is constructed in the same way and as a generalization of the hat version of Heegaard Floer homology; it assigns to $(M,\gamma)$ a finitely generated vector space $SFH(M,\gamma)$ which splits along relative $\operatorname{Spin}^{c}$ structures. The main advantage of the sutured Floer viewpoint is its neat behavior under sutured manifold decomposition.

In the case of links, we refer to a link complement with a pair of (oppositely oriented) meridional sutures on each boundary component as the sutured link complement $S^{3}(L):=\left(S^{3}\setminus\mathring{\nu(L)},\gamma_{\mu}\right)$. This way we have an isomorphism $SFH(S^{3}(L))\cong\widehat{\textbf{HFL}}(L)$ where the Alexander multi-grading is recovered by evaluations of relative $\operatorname{Spin}^{c}$ structures on particular surfaces bounded by components of $L$; in particular, for the collapsed Alexander grading we use a Seifert surface of $L$. Furthermore, looking at the sutured Seifert surface complement $S^{3}(F)=(S^{3}\setminus\mathring{\nu(F)},\gamma_{F}):=\left(S^{3}\setminus F\times(-1,1),\partial F\times\{0\}\right)$, the corresponding subspace is exactly the top Alexander grading summand $SFH(S^{3}(F))\cong\widehat{HFL}(L)[G_{3}(L)]$.

Returning to our analysis of non-split links, we have from Lemma 2.1 that $L$ bounds an annulus $F$ in $S^{3}$, and by assumption $SFH(S^{3}(F))\cong\widehat{HFL}(L)[G_{3}(L)]$ is of rank 2. Now, to understand our $(S^{3}\setminus\mathring{\nu(F)},\gamma_{F})$ we follow the argument of Baldwin and Sivek from [1, Section 5], which, in turn, relies on the properties of link complements with rank-2 sutured Floer homology [10] and some classical facts about $L$-space knots. We get, as in [1, Theorem 5.1], that up to reversing the orientation $(S^{3}\setminus\mathring{\nu(F)},\gamma_{F})$, whose boundary is a torus, is identified with $(S^{3}\setminus\mathring{\nu(C)},\gamma_{2})$, where $C$ is either the unknot $\bigcirc$ or the positive trefoil $T_{2,3}$ and the suture $\gamma_{2}$ is given by two parallel simple closed curves on $\partial\nu(F)$ oppositely oriented and with slope 2.

The idea now is to show that $L$ is actually isotopic to $C^{\prime}_{2,4}$, the $(2,4)$-cable of $C$ with oppositely oriented components, up to mirroring. This will complete the proof of Theorem 1.3.

Proposition 2.2**.**

Up to reversing the orientation, the complement $S^{3}\setminus\mathring{\nu(L)}$ is diffeomorphic to $S^{3}\setminus\mathring{\nu(C^{\prime}_{2,4})}$. Furthermore, there is such a diffeomorphism $\Phi$ for which $\Phi(\lambda_{i})=\lambda_{i}^{\prime}$ with $i=1,2$, where $\lambda_{i}$ and $\lambda_{i}^{\prime}$ are longitudes for $L$ and $C^{\prime}_{2,4}$ respectively.

Proof.

The manifold $S^{3}\setminus\mathring{\nu(L)}$ is obtained by gluing up the thickened annulus $A\times[-1,1]$ along $A\times\{-1,1\}$ to $S^{3}\setminus\mathring{\nu(F)}$. Such a gluing is determined by the attaching regions: the two parallel annuli complementary to a neighborhood of the sutures $\mathring{\nu(\gamma_{F})}$. Therefore, using again [1, Theorem 5.1] we get that $S^{3}\setminus\mathring{\nu(L)}$ is diffeomorphic to the 3-manifold given by attaching $A\times[-1,1]$ to $S^{3}\setminus\mathring{\nu(C)}$ along $\partial\nu(C)\setminus\mathring{\nu(\gamma_{2})}$, which is precisely $S^{3}\setminus\mathring{\nu(C^{\prime}_{2,4})}$. We denote this diffeomorphism by $\Phi$ and we observe that $\Phi(A)$ is an annulus whose boundary is $C_{2,4}^{\prime}$.

For the second part of the statement, we note that by the definition of longitude one has $\lambda_{1}\sqcup\lambda_{2}=\partial A$; and also $\lambda^{\prime}_{1}\sqcup\lambda^{\prime}_{2}=\partial\Phi(A)$ because $\Phi$ is orientation preserving. Hence, we can write

[TABLE]

which implies $\Phi(\lambda_{i})=\lambda_{i}^{\prime}$ for $i=1,2$ ∎

Remark 2.3**.**

It is known that the Gordon-Luecke theorem does not hold for links: there is an infinite family of links, obtained by performing annular twisting on the Whitehead link, which have diffeomorphic complements but are all non-isotopic. Proposition 2.2 is then by itself not enough to show that $L$ is isotopic to $C_{2,4}^{\prime}$. Since we are considering a link $L$ which bounds an embedded annulus in $S^{3}$, this could also be proved using the topology of the cabling construction; nonetheless, in this paper we decided to include a more direct proof.

Our strategy continues as follows: we are going to prove that the diffeomorphism $\Phi$ from Proposition 2.2 preserves the peripheral system of $L$; and since the latter is known to be a complete isotopy invariant of links [17], this is enough to prove Theorem 1.3. We then need the following two lemmas.

Lemma 2.4**.**

A link $L$ whose complement is diffeomorphic to the one of $C_{2,4}^{\prime}$, through a diffeomorphism $\Phi$ which preserves the longitudes, has linking number $\operatorname{\ell k}(L)$ equal to $\pm 2$.

Proof.

Since $\Phi$ preserves the longitudes $\lambda_{i}$, we have that $L$ and $C_{2,4}$ have diffeomorphic [math]-surgeries (with respect to the framings given by $\lambda_{i}$ and $\Phi(\lambda_{i}))$. Then one has

[TABLE]

∎

Let us denote the meridians of $L$ and $C_{2,4}^{\prime}$ by $\mu_{i}$ and $\mu_{i}^{\prime}$ respectively for $i=1,2$. Since $\Phi$ is an orientation preserving diffeomorphism we should have

[TABLE]

for some $p,q\in\mathbb{Z}$.

Lemma 2.5**.**

If the link $L$ and the diffeomorphism $\Phi$ are as previously defined then we have $p=q=0$ and then $\Phi$ also preserves meridians.

Proof.

Proposition 2.2 and Lemma 2.4 imply that the $(1,1)$-surgery on $L$ is diffeomorphic to $S^{3}_{\frac{1}{p+1},\frac{1}{q+1}}(C_{2,4})$. Therefore, after applying two slam-dunks as in Figure 2, see [6] for the precise definition, we have the following relation

[TABLE]

which implies a contradiction unless either $p=q=0$ or $p=q=-2$, but in the latter case we can repeat the same argument to obtain

[TABLE]

which is again a contradiction. ∎

Proving that $\Phi$ preserves both meridians and longitudes means exactly that the peripheral system of $L$ is invariant under $\Phi$ and this concludes the proof of Theorem 1.3, as we mentioned before.

3 Computation of the link Floer homology groups

We compute $\widehat{HFL}(L)$ for any of the links in Theorem 1.3. It can be observed from Table 1 that all such links have different homology and then their isotopy class is determined by $\widehat{HFL}$.

The homology groups of the mirror images and the split links are computed using Equations (1.4) and (2.1), starting from $\widehat{HFL}$ of the links in Theorem 1.1 which can be found in [13]. The homology of $T_{2,4}^{\prime}$ is also found in literature [14]; hence, we are only left to discuss how to compute $\widehat{HFL}(T_{2,3;2,4}^{\prime})$.

We start by computing the multi-graded link Floer homology group $\widehat{\textbf{HFL}}(T_{2,3;2,4})$, where this time the $(2,4)$-cable of the trefoil knot has components oriented in the same direction. Such a link is an $L$-space link and a cable of an $L$-space knot and then we can use the following result.

Theorem 3.1** (Gorsky-Hom [7]).**

Suppose that $K$ is an $L$-space knot in $S^{3}$; then the $(rm,rn)$-cable of $K$ is an $L$-space link for every $m,n$ coprime such that $\frac{n}{m}>2g(K)-1$ and $r\geqslant 1$.

Furthermore, the homology group $\widehat{\emph{{HFL}}}(K_{rm,rn})$ is completely determined by $m,n,r$ and the Alexander polynomial of $K$.

In our case we have $m=1,\>n=2,\>r=2$ and $K=T_{2,3}$. Since $2\geqslant 1=2g(T_{2,3})-1$ the hypothesis of the theorem are satisfied. The formulae in [7, Theorem 3] express $\widehat{\textbf{HFL}}(K_{rm,rn})$ in terms of the $h$-function of $K_{rm,rn}$, another link invariant introduced in [8] which as expected is also determined by the Alexander polynomial; moreover, in [7, Lemma 4.5] it is also explained how to compute $h(K_{rm,rn})$ from $h(K)$.

Finding the $h$-function of an $L$-space knot is very simple, given the structure of the knot Floer chain complex of this family of knots (see [15]), and in fact we can immediately write

[TABLE]

Plugging such values of $h(T_{2,3;2,4})$ into [7, Theorem 3] yields

[TABLE]

Now, in order to compute the homology of $T_{2,3;2,4}^{\prime}$ we need the following formula [13, Proposition 11.4.2], which relates the link Floer homology group of links where the orientation of one component differs: suppose that $L$ has $n$-components, and let $L^{\prime}$ be the oriented link obtained from $L$ by reversing the orientation of its $i$-th component $L_{i}$. Then

[TABLE]

Applying Equation (3.1) to $\widehat{\textbf{HFL}}(T_{2,3;2,4})$ we then obtain

[TABLE]

and it is now easy to check that, collapsing the Alexander grading as in Equation (1.1), the group $\widehat{HFL}(T_{2,3;2,4}^{\prime})$ is precisely the one appearing in Table 1.