The Dihedral Genus of a Knot

TL;DR

This paper explores the relationship between Fox p-colored knots, dihedral branched covers, and the four-genus of knots, providing classification results and conditions linking knot signatures to four-manifold invariants.

Contribution

It introduces a new connection between dihedral covers, knot signatures, and the four-genus, including a classification of signatures of certain dihedral cover manifolds.

Findings

Identifies conditions under which the four-genus is realized by a minimal genus surface.

Classifies signatures of dihedral cover 4-manifolds arising from p-colored knots.

Constructs infinite families of knots with prescribed properties.

Abstract

Let $K\subset S^3$ be a Fox $p$-colored knot and assume $K$ bounds a locally flat surface $S\subset B^4$ over which the given $p$-coloring extends. This coloring of $S$ induces a dihedral branched cover $X\to S^4$. Its branching set is a closed surface embedded in $S^4$ locally flatly away from one singularity whose link is $K$. When $S$ is homotopy ribbon and $X$ a definite four-manifold, a condition relating the signature of $X$ and the Murasugi signature of $K$ guarantees that $S$ in fact realizes the four-genus of $K$. We exhibit an infinite family of knots $K_m$ with this property, each with a {Fox 3-}colored surface of minimal genus $m$. As a consequence, we classify the signatures of manifolds $X$ which arise as dihedral covers of $S^4$ in the above sense.

Figures

Equations

$${\Xi }_p(K,\rho )=-\sigma (X)$$

$${\Xi }_p(K,\rho )=-\sigma (X^{\prime },\partial X^{\prime })$$

$$f_m:\#(2m+1){\overline{\mathbb{C}P}}^2\to S^4.$$

$${\mathfrak{g}}_p(K,\rho )\ge \frac{|{\Xi }_p(K,\rho )|-\text{rk }{H}_1(M;\mathbb{Z})}{p-1}-\frac{1}{2}$$

$$|\sigma (K)|=\frac{2|{\Xi }_p(K,\rho )|}{p-1}-1$$

$$|{\Xi }_p(K,\rho )|-\text{rk }{H}_1(M;\mathbb{Z})$$

$${\mathfrak{g}}_p(K)\ge \frac{{\text{min}}_p(K)}{p-1}-\frac{1}{2}.$$

$$g_4(K_m)={\mathfrak{g}}_3(K_m)=\frac{|{\Xi }_3(K_m,{\rho }_m)|}{2}-\frac{1}{2}=m.$$

$${\Xi }_p(K,\rho )=\frac{p^2-1}{6p}{L}_V(\beta ,\beta )+\sigma (W(K,\beta ))+\sum _{i=1}^{p-1}{\sigma }_{{\zeta }^i}(\beta ).$$

$${\Xi }_p(K)=-\frac{p-1}{4}e(F)-\sigma (X^{\prime },\partial X^{\prime }),$$

$${\mathfrak{g}}_p(K,\rho )\ge \frac{|{\Xi }_p(K,\rho )|-\text{rk }{H}_1(M)}{p-1}-\frac{1}{2},$$

$${\pi }_1(c(M)){\ast }_{{\pi }_1(M)}{\pi }_1(X^{\prime })\simeq {\pi }_1(X^{\prime })/\text{im }{\mathsf{i}}_{\ast }({\pi }_1(M)).$$

$$\chi (X)=\text{rk }{H}_4(X)-\text{rk }{H}_3(X)+\text{rk }{H}_2(X)+1.$$

$$H_n(X)=H_n(X^{\prime }/\partial X^{\prime })={\tilde{H}}_n(X^{\prime },\partial X^{\prime }).$$

$$\text{rk }{H}_3(X)=\text{rk }{H}_3(X^{\prime },\partial X^{\prime })=\text{rk }{H}^1(X^{\prime })=\text{rk }{H}_1(X^{\prime }),$$

$$\text{rk }{H}_4(X)=\text{rk }{H}_4(X^{\prime },\partial X^{\prime })=\text{rk }{H}^0(X^{\prime })=\text{rk }{H}_0(X^{\prime })=1,$$

$$\chi (X)=2+\text{rk }{H}_2(X)-\text{rk }{H}_1(X^{\prime }).$$

$$\chi (X)=2p-\frac{p-1}{2}\chi (F)-\frac{p-1}{2}.$$

$$|{\Xi }_p(K,\rho )|=\sigma (X,\partial X^{\prime })\le \text{rk }\text{im }{\mathfrak{i}}_{\ast }\le \text{rk }{H}_2(X^{\prime },\partial X^{\prime })=\text{rk }{H}_2(X).$$

$$\chi (F)=2-2g(F)$$

$$\text{rk }{H}_2(X)=\chi (X)-2+\text{rk }{H}_1(X^{\prime })$$

$$|{\Xi }_p(K,\rho )|\le 2p-\frac{p-1}{2}(2-2g(F))-\frac{p-1}{2}-2+\text{rk }{H}_1(X^{\prime }).$$

$$g(F)\ge \frac{|{\Xi }_p(K,\rho )|-\text{rk }{H}_1(X^{\prime })}{p-1}-\frac{1}{2}.$$

$$g(F)\ge \frac{|{\Xi }_p(K,\rho )|-\text{rk }{H}_1(M)}{p-1}-\frac{1}{2}.$$

$${\mathfrak{g}}_p(K,\rho )=\frac{|{\Xi }_p(K,\rho )|}{p-1}-\frac{1}{2}.$$

$$|{\Xi }_p(K,\rho )|=|\sigma (X)|=\chi (X)-2.$$

$$\chi (X)=2p-\frac{p-1}{2}(2-2g(F))-\frac{p-1}{2}.$$

$$g(F)=\frac{|{\Xi }_p(K,\rho )|}{p-1}-\frac{1}{2}.$$

$$g(F^{\prime })\ge {\mathfrak{g}}_p(K,\rho )\ge \frac{|{\Xi }_p(K,\rho )|}{p-1}-\frac{1}{2}.$$

$$g_4(K_m)={\mathfrak{g}}_3(K_m)=\frac{|{\Xi }_3(K_m,{\rho }_m)|}{2}-\frac{1}{2}=m$$

Full text

This work was supported by NSF-DMS grant 1821212 to P. Cahn and 1821257 to A. Kjuchukova

1. Introduction

The Slice-Ribbon Conjecture of Fox [7] asks whether every smoothly slice knot in $S^{3}$ bounds a ribbon disk in the four-ball. The analogous question can be asked in the topological category, namely: does every topologically slice knot bound a locally flat homotopy ribbon disk in $B^{4}$? Recall that a properly embedded surface with boundary $F^{\prime}\subset B^{4}$ is homotopy ribbon if the fundamental group of its complement is generated by meridians of $\partial F^{\prime}$ in $S^{3}$. Ribbon disks are easily seen to be homotopy ribbon whereas homotopy ribbon disks need not be smooth.

For knots of higher genus, the generalized topological Slice-Ribbon Conjecture asks whether the topological four-genus of a knot is always realized by a homotopy ribbon surface in $B^{4}$. When a knot $K$ admits Fox $p$-colorings, we approach this problem by studying locally flat, oriented surfaces $F^{\prime}\subset B^{4}$ with $\partial F^{\prime}=K$ over which some $p$-coloring of $K$ extends, in the sense defined in Section 2.1. The minimal genus of such a surface, when one exists, we call the $p$-dihedral genus of $K$.

When $K$ is slice and $p$ square-free, it is classically known that the colored surface $F^{\prime}$ for $K$ can always be chosen to be a disk. This is essentially a consequence of [6, Lemma 3]; a detailed explanation can be found in [9, Lemma 9]. Put differently, $p$-dihedral genus and classical four-genus coincide for slice knots. Furthermore, the topological Slice-Ribbon Conjecture is true for $p$-colorable slice knots if and only if the minimal $p$-dihedral genus for these knots can always be realized by homotopy ribbon surfaces. With this in mind, given a square-free integer $p$ and a $p$-colorable knot $K$, we ask:

Question 1. Is the (topological) four-genus of $K$ equal to its (topological) $p$-dihedral genus?

Question 2. Is the $p$-dihedral genus of $K$ realized by a homotopy ribbon surface?

When both of these questions are answered in the affirmative for a knot $K$ with respect to some integer $p$, it follows that the topological four-genus and homotopy ribbon genus of $K$ are equal; that is, the generalized topological Slice-Ribbon Conjecture holds for $K$. If $K$ is not slice, requiring that it satisfies Questions 1 and 2 is a priori a stronger condition than satisfying the generalized Slice-Ribbon Conjecture; however, the advantage of this point of view is that dihedral genus can be studied using dihedral branched covers.

Specifically, our approach is the following. Start with a branched cover of $f^{\prime}:X^{\prime}\to B^{4}$ branched along a locally flat properly embedded surface $F^{\prime}$ with $\partial F^{\prime}=K$; that is, $F^{\prime}$ is a properly embedded topological submanifold of $B^{4}$. We now construct a new cover $f:X\to S^{4}$ by taking the cones of $\partial X^{\prime}$, $S^{3}$ and the map $f$. The branching set of $f$ is a surface $F$ embedded in $S^{4}$ locally flatly except for one singular point whose link is $K$. Depending on the knot $K$ and the map $f$, it may be the case that this construction yields a total space $X$ that is again a manifold. In general, $X$ has one singular (non-manifold) point $\mathfrak{z}$, the pre-image of the singularity on $F$. The link of $\mathfrak{z}$ is the cover $f_{|}$ of $S^{3}$ branched along $K$. We will consider the signature of $X$ whether it is a manifold or has an isolated singularity. In the latter case, by the signature of $X$ we mean the Novikov signature of the manifold with boundary obtained by deleting an open neighborhood of $\mathfrak{z}$ in $X$.

When $f:X\to S^{4}$ in the above construction is a $p$-fold irregular dihedral cover (Definition 1), an invariant of ($p$-colored) knots, $\Xi_{p}$, is extracted from this construction. This invariant is our main tool. In a general setting, $\Xi_{p}$ can be thought of as a defect term in the formula for the signature of a branched cover, resulting from the fact that the branching set is not locally flat. Put differently, the presence of a cone singularity $K$ on the branching set causes the signature of the cover to deviate from the smooth case by a term denoted $\Xi_{p}(K,\rho)$. This term depends only on the isotopy class of the knot $K$ and its Fox $p$-coloring $\rho$, but not on the locally flat part of the branching set.

Given a dihedral cover $f:X\to S^{4}$ whose branching set is orientable with one singularity, we in fact have

[TABLE]

by [13, Theorem 1.4], when $X$ is a manifold, and

[TABLE]

when $X$ has a singularity [9, Theorem 7]. In the latter formula, $\partial X^{\prime}$ is the dihedral cover of $K$ induced by $f_{|}$, and $\sigma(\cdot,\cdot)$ denotes the Novikov signature of a manifold with boundary. Of course, the first formula for $\Xi_{p}(K,\rho)$ in terms of $X$ is a special case of the second, since the signature of a manifold is unchanged by deleting an open neighborhood of a point.

Unless explicitly stated otherwise, we will only consider orientable branching sets. Thus, we take the above signature equation to be the definition of $\Xi_{p}(K,\rho)$. In Equation 5 we recall an explicit formula [13] for $\Xi_{p}$ which does not rely on constructing the cover $X$. We also note that $\Xi_{p}(K,\rho)$ can be computed algorithmically from a colored diagram of $K$ [4]. We often suppress notation and write $\Xi_{p}(K)$ when the choice of coloring is clear, or when a knot admits a unique $p$-coloring (up to permuting the colors). Thus, for a two-bridge knot $K$, we will write simply $\Xi_{p}(K)$. The main result of this paper, Theorem 1, obtains a certain genus bound for $K$ from $\Xi_{p}(K)$.

As implied by the above, the signature defect $\Xi_{p}(K)$ is defined for a knot $K$ which arises as the only singularity on the branching set (not necessarily orientable) of an irregular dihedral cover [9]. A knot $K$ is called $p$-admissible over $S^{4}$, or simply $p$-admissible, if there exists a $p$-fold dihedral cover $f:X\to S^{4}$ whose branching set is embedded and locally flat except for one singularity whose link is $K$. If, in addition, the covering space $X$ is a topological manifold, $K$ is called strongly $p$-admissible. The distinguishing property of strongly $p$-admissible knots111Like the invariant $\Xi_{p}$, the notion of (strong) $p$-admissibility of a knot may depend on the choice of coloring. We do not dwell on this presently since all examples in this paper are two-bridge knots and their colorings are unique. is that their dihedral covers are $S^{3}$. Admissibility of knots is studied in [kjorr2017admissible].

In Section  2, we put side by side the relevant notions of knot four-genus, recall several definitions, and state our main results: Theorems 1,  2 and  3. In Theorem  1, we give a lower bound on the homotopy-ribbon $p$-dihedral genus of a colored knot $K$ in terms of the invariant $\Xi_{p}(K)$. We also give a sufficient condition for when this bound is sharp.

In Theorem 2 and Theorem 3, we construct, for any integer $m\geq 0$, infinite families of knots for which the 3-dihedral genus and the topological four-genus are both equal to $m$. The basis of this construction are the knots $K_{m}$ pictured in Figure 1. The various four-genera of these knots are computed with the help of Theorem 1. In particular, for these knots, the lower bound on genus obtained via branched covers is exact and the generalized topological Slice-Ribbon Conjecture is seen to hold. The proofs of Theorems 1, 2 and 3 are given in Section  4.

The technique we apply is the following. Given a strongly $p$-admissible knot $K$, one can evaluate $\Xi_{p}(K)$ by realizing $K$ as the only singularity on the branch surface of a dihedral cover of $S^{4}$. Each of the knots $K_{m}$ arises as the only singularity on the branching set of a 3-fold dihedral cover

[TABLE]

The branching set of $f_{m}$ is the boundary union of the cone on $K_{m}$ with the surface $F_{m}^{\prime}$ realizing the four-genus of $K_{m}$. We construct these covering maps explicitly using singular triplane diagrams, a technique introduced in [3]. Equivalently, we construct a family of covers $\#(2m+1)\mathbb{C}P^{2}\rightarrow S^{4}$, again with oriented, connected branching sets, with the mirror images of the knots $K_{m}$ as singularities. This construction appears in Section  3. As a corollary of this construction, we realize all odd integers as values of $\Xi_{3}$. In Theorem  5, we prove that the range of values of $\Xi_{3}$ on strongly admissible knots is precisely the set of odd integers.

We work in the topological category, except where explicitly stated otherwise. Throughout, $F$ denotes a closed, connected, oriented surface, and $F^{\prime}$ a connected, oriented surface with boundary. $D_{p}$ denotes the dihedral group of order $2p$, and $p$ is always assumed odd.

Acknowledgments. The idea of dihedral genus first appeared in discussion with Kent Orr [kjorr2017admissible]. The examples used in Theorem 2 were inspired by joint work with Ryan Blair [1].

2. Dihedral four-genus and the main theorems

2.1. Some old and new notions of knot genus

We study the interplay between the following notions of four-genus for a Fox $p$-colorable knot $K\subset S^{3}$. Classically, the smooth (resp. topological) four-genus is the minimum genus of a smooth (resp. locally flat) embedded orientable surface in $B^{4}$ with boundary $K$. The smooth (resp. topological) $p$-dihedral genus of a $p$-colored knot $K$ is, informally, the minimum genus of such a surface $F^{\prime}$ in $B^{4}$ over which the $p$-coloring of $K$ extends. Precisely, a given $p$-coloring $\rho$ of $K$ extends over $F^{\prime}$ if there exists a homomorphism $\bar{\rho}$ which makes the following diagram commute (where $i_{\ast}$ the map induced by inclusion):

{\pi_{1}(S^{3}-K)}$${\pi_{1}(B^{4}-F^{\prime})}$${D_{p}}$$\scriptstyle{i_{*}}$$\scriptstyle{\rho}$$\scriptstyle{\bar{\rho}}

The $p$-dihedral genus above is defined for a knot $K$ with a fixed coloring $\rho$, hence we denote it $g_{p}(K,\rho)$ in the topological case. We define the $p$-dihedral genus of a $p$-colorable knot $K$ to be the minimum $p$-dihedral genus of $K$ over all $p$-colorings $\rho$ of $K$, and denote this by $g_{p}(K)$ in the topological case. Note that not every $p$-colored knot $K$ admits a surface $F^{\prime}$ as above. In [kjorr2017admissible], we determine a necessary and sufficient condition for the existence of a connected oriented surface that fits into this diagram. When there is no surface over which a given coloring $\rho$ of $K$ extends, we define $g_{p}(K,\rho)$ to be infinite, and similarly for the refined notions of dihedral genus defined below.

The ribbon genus of $K$ is the minimum genus of a smooth embedded orientable surface $F^{\prime}$ in $B^{4}$ with boundary $K$, such that $F^{\prime}$ has only local minima and saddles with respect to the radial height function on $B^{4}$. The smooth (topological) homotopy ribbon genus of a knot $K$ is the minimum genus of a smooth (locally flat) embedded orientable surface $F^{\prime}$ in $B^{4}$ with boundary $K$ such that $i_{\ast}:\pi_{1}(S^{3}-K)\twoheadrightarrow\pi_{1}(B^{4}-F^{\prime})$, that is, inclusion of the boundary into the surface complement induces a surjection on fundamental groups. Finally, given a $p$-colorable or $p$-colored knot, its ribbon $p$-dihedral genus or smooth (topological) homotopy ribbon $p$-dihedral genus are defined in the obvious way. Observe that all notions of dihedral genus refer to surfaces embedded in the four-ball, even though “four” is not among the multitude of qualifiers we inevitably use.

As a straight-forward consequence of the definitions, the following inequalities hold among the smooth four-genera of a knot:

four-genushom. ribbon genusribbon genus{p\text{-dihedral genus}}$${p\text{-dihedral hom. ribbon genus}}$${p\text{-dihedral ribbon genus}}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}

Excluding the last column, the inequalities make sense and hold in the topological category too.

2.2. The Main Theorems

Denote by $g_{4}(K)$ the topological 4-genus of a knot $K$, and by $\mathfrak{g}_{{p}}(K,\rho)$ the topological homotopy-ribbon $p$-dihedral genus of a knot $K$ with coloring $\rho$. Again, the minimum such genus over all colorings $\rho$ of $K$ is $\mathfrak{g}_{{p}}(K)$. Let $\sigma(K)$ be the (Murasugi) signature of the knot $K$. We relate $\mathfrak{g}_{{p}}(K,\rho)$, $\Xi_{p}(K,\rho)$ and $\sigma(K)$. Here, $\Xi_{p}(K,\rho)$ denotes the invariant discussed in the Introduction; it is reviewed in more detail in this section and, in particular, we recall that it can be computed using Equation 5.

Theorem 1**.**

(A) Let $K$ be a $p$-admissible knot with $p$-coloring $\rho$ and denote by $M$ the irregular dihedral cover of $K$ determined by $\rho$. The following inequality holds:

[TABLE]

(B) Let $K$ be a $p$-admissible knot and $F^{\prime}\subset B^{4}$ a locally flat homotopy ribbon oriented surface for $K$ over which a given $p$-coloring $\rho$ of $K$ extends. Denote by $c(K)$ the cone on $K$, seen as embedded in $D^{4}=c(S^{3})$. If the associated singular dihedral cover of $S^{4}$ branched along $F^{\prime}\cup_{K}c(K)$ is a definite manifold, then the inequality (3) is sharp. In particular, $F^{\prime}$ realizes the dihedral genus $\mathfrak{g}_{{p}}(K,\rho)$ of $K$. If, in addition, the equality

[TABLE]

holds, then the topological four-genus and the topological homotopy ribbon $p$-dihedral genus of $K$ coincide and equal $\frac{|\sigma(K)|}{2}$, so the generalized topological Slice-Ribbon Conjecture holds for $K$.

Remark. If $K$ has multiple $p$-colorings, denote by $\text{min}_{p}(K)$ the minimum value of

[TABLE]

over all such colorings of $K$. Theorem 1 implies:

[TABLE]

Theorem 2**.**

For every integer $m\geq 0$, there exists a knot $K_{m}$ and corresponding 3-coloring $\rho_{m}$, such that:

[TABLE]

That is, the inequality (4) is sharp for these knots and computes their $3$-dihedral genus as well as their topological four-genus. The generalized Slice-Ribbon Conjecture holds for these knots.

Theorem 3**.**

For any integer $m\geq 0$, there exist infinite families of knots whose 3-dihedral genus and topological four-genus are both equal to $m$.

2.3. Singular dihedral covers of $S^{4}$ and the invariant $\Xi_{p}$

In this section, we revisit the definition of a singular branched cover, and dihedral covers in particular. We also review the context in which the invariant $\Xi_{p}$ arises, as well as a couple of techniques for its calculation.

Definition 1**.**

Let $Y$ be a manifold and $B\subset Y$ a codimension-two submanifold with the property that there exists a surjection $\varphi:\pi_{1}(Y-B)\twoheadrightarrow D_{p}$. Denote by $\mathring{X}$ the covering space of $Y-B$ corresponding to the conjugacy class of subgroups $\varphi^{-1}(\mathbb{Z}/2\mathbb{Z})$ in $\pi_{1}(Y-B)$, where $\mathbb{Z}/2\mathbb{Z}\subset D_{p}$ is any reflection subgroup. The completion of $\mathring{X}$ to a branched cover $f:X\to Y$ is called the irregular dihedral $p$-fold cover of $Y$ branched along $B$.

The manifolds whose irregular dihedral covers we will consider are $S^{3}$, $B^{4}$ and $S^{4}$. The $\Xi_{p}$ invariant was originally defined in the more general context of a dihedral cover of an arbitrary four-manifold $Y$ with a singularly embedded branching set [13].

Recall the following construction from the introduction. Let $F^{\prime}$ be a surface with connected boundary $K$, properly embedded in $B^{4}$ and locally flat. Given a branched cover of manifolds with boundary $f^{\prime}:X^{\prime}\to B^{4}$, one constructs a singular branched cover of $S^{4}$ by coning off $\partial X^{\prime}$, $\partial B^{4}$ and the map $f^{\prime}$. The resulting covering map, $f:X\to S^{4}$, has total space $X:=X^{\prime}\cup_{\partial X^{\prime}}c(\partial X^{\prime})$, where $c(\partial X^{\prime})$ denotes the cone on $\partial X^{\prime}$. The branching set is a closed surface $F:=F^{\prime}\cup_{K}c(K)$ embedded in $S^{4}$ with a singularity (the cone point) whose link is $K$. The space $X$ obtained in this way is a manifold if and only if $\partial X^{\prime}\cong S^{3}$.

Denote by $\sigma(X^{\prime},\partial X^{\prime})$ the Novikov signature of the manifold with boundary given as a cover of $B^{4}$ branched along $F^{\prime}$. When $\partial X^{\prime}=S^{3}$, denote by $\sigma(X)$ the signature of the manifold $X$. In this case, we have $\sigma(X)=\sigma(X^{\prime},\partial X^{\prime})$.

Given $f^{\prime}:X^{\prime}\to B^{4}$ as before with $f^{\prime}$ an (irregular) dihedral covering map, we always assume that the associated homomorphism $\rho:\pi_{1}(S^{3}-K)\to D_{p}$ is surjective or, equivalently, that $\partial X^{\prime}$ is connected. In this case, assuming $F^{\prime}$ is orientable, $\Xi_{p}(K,\rho)=-\sigma(X^{\prime},\partial X^{\prime})$ by [9, Theorem 7]. In particular, when $X$ is a manifold, this equation reduces to the earlier result $\Xi_{p}(K,\rho)=-\sigma(X)$ [13, Theorem 1.4].

Below, we recall two formulas for $\Xi_{p}(K,\rho)$ from [13]. Equation 5 allows for $\Xi_{p}(K,\rho)$ to be computed in terms of $K$ and its coloring using [4] and [2]. Equation 6 expresses $\Xi_{p}(K,\rho)$ in terms of a singular branched cover of $S^{4}$ in the more general case where the branching set is a possibly non-orientable surface.

Refocusing for a moment on the case where the dihedral branched cover $X$ of $S^{4}$ is a manifold, we note that there exist many infinite families of knots $K\subset S^{3}$ whose irregular dihedral covers are homeomorphic to $S^{3}$. For example, this is a property shared by all $p$-colorable two-bridge knots (a well-known fact recalled in the proof of [13, Lemma 3.3]). By definition, if a $p$-admissible knot $K$ has $S^{3}$ as its dihedral cover, then it is in fact strongly $p$-admissible. We are then able study invariants of $K$ using four-dimensional techniques such as trisections. Criteria for admissibility of singularities are discussed in more detail in [3] where we also use the invariant $\Xi_{p}(K)$ to give a homotopy ribbon obstruction for strongly $p$-admissible knots $K$. A generalization of this ribbon obstruction to all $p$-admissible knots appears in [9].

We conclude this section by reviewing the formula for computing the invariant $\Xi_{p}$ given in [13]. Let $p$ be an odd integer and $K$ a $p$-admissible knot. Let $V$ be a Seifert surface for $K$ and $V^{\circ}$ the interior of $V$. Denote by $\beta\subset V^{\circ}$ a mod $p$ characteristic knot222Precisely, if $K$ admits multiple $p$-colorings, one must work with a characteristic knot corresponding to the coloring in question. The sense in which a characteristic knot determines a coloring is laid out in [5, Proposition 1.1]. The examples we construct always admit a unique $p$-coloring, up to permuting the colors, and therefore a unique equivalence class of mod $p$ characteristic knots. for $K$, as defined in [5]. Also denote by $L_{V}$ the symmetrized linking form for $V$ and by $\sigma_{\zeta^{i}}$ the Tristram-Levine $\zeta^{i}$-signature, where $\zeta$ is a primitive $p^{\text{th}}$ root of unity. Finally, let $W(K,\beta)$ be the cobordism constructed in [5] between the $p$-fold cyclic cover of $S^{3}$ branched along $\beta$ and the $p$-fold dihedral cover of $S^{3}$ branched along $K$ and determined by $\rho$. We briefly describe the manifold $W(K,\beta)$. Let $\Sigma$ be the $p$-fold cyclic branched cover of $\beta$ and let $\Sigma_{p}(\beta)\times[0,1]\to S^{3}\times[0,1]$ be the induced cyclic cover branched along $\beta\times[0,1]$. Letting $\mathbb{Z}/2\mathbb{Z}$ act on an appropriate subset of $\Sigma_{p}(\beta)\times\{0\}$, one obtains $W(K,\beta)$ as a quotient of $\Sigma_{p}(\beta)\times[0,1]$ by this action. One boundary of this quotient, namely $\Sigma_{p}(\beta)\times\{1\}$, is clearly the $p$-fold cyclic cover of $\beta$. The other boundary component, that is, the image of $\Sigma_{p}(\beta)\times\{0\}$ under the $\mathbb{Z}/2\mathbb{Z}$ action, is the dihedral cover of $\alpha$ as shown in [5, Proposition 1.1]. By [13, Theorem 1.4],

[TABLE]

The Novikov signature $\sigma(W(K,\beta))$ can be computed in terms of linking numbers in the dihedral cover of $K$ [13, Proposition 2.5]. Thus, the above formula allows $\Xi_{p}(K)$ to be evaluated directly from a $p$-colored diagram of $K$, without direct reference to a four-dimensional construction. An explicit algorithm for performing this computation is outlined in [4]. Note also that when a knot $K$ is realized as the only singularity on an embedded surface $F\subset S^{4}$ and moreover this surface is presented by a Fox $p$-colored singular triplane diagram, [3] gives a method for computing $\Xi_{p}(K)$ from this data, via the signature of the associated cover of $S^{4}$. This technique is reviewed and applied in Section 3 below.

We also review the context in which Equations 1 and 2 arise, allowing us to relate $\Xi_{p}(K,\rho)$ to the signature of a singular branched cover $X$ of $S^{4}$. Consider an irregular dihedral cover $f:X\to S^{4}$ whose branching set $F$ is an embedded surface, not necessarily orientable, locally flat away from one singularity $z\in F$ of type $K$. The induced coloring of $F$ is, as always, an extension of $\rho$. Once again we denote by $X^{\prime}$ the dihedral cover of $B^{4}$ branched along the complement in $F$ of a neighborhood of the singular point $z$. Note also that $X^{\prime}$ is obtained by deleting from $X$ a small open neighborhood of $f^{-1}(z)$. We have

[TABLE]

where $e(F)$ denotes the self-intersection number of $F$. This is a special case of the signature formula for dihedral branched covers over an arbitrary base333The reference [9] is written in the language of intersection homology. In the case of a singular branched cover $f:X\to S^{4}$, this is equivalent to the Novikov signature $\sigma(X^{\prime},\partial X^{\prime})$ since $X$ has only an isolated singularity. given in [9, Theorem 7]. Note that, when $F$ is orientable and $X$ a manifold, Equation 6 reduces to Equation 1, that is, $\Xi_{p}(K)=-\sigma(X)$. In this case, the $\Xi_{p}$ invariant of a singularity can be understood entirely in terms of the signature of the branched cover and, in particular, can be computed using four-manifold techniques. We further note that it is possible to realize all connected sums $\#n\mathbb{C}P^{2}$ as 3-fold dihedral covers of $S^{4}$ with one knot singularity on a connected, embedded branching set, if one allows the branching set to be non-orientable [1]. By contrast, we see in Theorem 5 that orientability of the branching set, together with a single singular point, imply that the signature of such a cover is odd.

3. Knots with equal topological and dihedral genera

In this section we construct families of knots for which the topological, ribbon and 3-dihedral genus are equal. We use trisections of four-manifolds [8], tri-plane diagrams [14], and singular tri-plane diagrams [3], all of which we review informally for the reader’s convenience.

Given a smooth, oriented, 4-manifold $X$, a $(g,k_{1},k_{2},k_{3})$-trisection of $X$ is a decomposition of $X=X_{1}\cup X_{2}\cup X_{3}$ into three 4-handlebodies with boundary, such that

• •

$X_{i}\cong\natural k_{i}(B^{3}\times S^{1})$

• •

$X_{1}\cap X_{2}\cap X_{3}\cong\Sigma_{g}$ is a closed, oriented surface of genus $g$

• •

$Y_{ij}=\partial(X_{i}\cup X_{j})\cong\#k_{l}(S^{2}\times S^{1})$ where $i$, $j$ and $l$ $\in\{1,2,3\}$ are distinct

• •

$\Sigma_{g}\subset Y_{ij}$ is a Heegaard surface for $Y_{ij}$.

Every embedded surface $F\subset S^{4}$ can be described combinatorially by a $(b;c_{1},c_{2},c_{3})$-tri-plane diagram [14]. This is a set of three $b$-strand trivial tangles $(A,B,C)$, such that each boundary union of tangles $A\cup\overline{B}$, $B\cup\overline{C}$, and $C\cup\overline{A}$ is a $c_{i}$-component unlink, for $i=1,2,3$ respectively. Here $\overline{T}$ denotes the mirror image of $T$. To obtain $F$ from $(A,B,C)$, one views each of $A\cup\overline{B}$, $B\cup\overline{C}$, and $C\cup\overline{A}$ as unlinks in bridge position in the spokes $Y_{12}$, $Y_{23}$, and $Y_{31}$ of the standard genus-0 trisection of $S^{4}$, glues $c_{i}$ disks to the components of each of these unlinks, and pushes these disks into the $X_{i}$ to obtain an embedded surface.

The authors introduce singular tri-plane diagrams and their colorings in [3]. A $(b;1,c_{2},c_{3})$ singular tri-plane diagram is a triple of $b$-strand trivial tangles $(A,B,C)$. As above $B\cup\overline{C}$ and $C\cup\overline{A}$ are $c_{2}$- and $c_{3}$-component unlinks. $A\cup\overline{B}$ is a knot $K$. To build a surface with one singularity of type $K$, one again views each of $A\cup\overline{B}$, $B\cup\overline{C}$, and $C\cup\overline{A}$ in bridge position in the three spokes $Y_{12}$, $Y_{23}$, and $Y_{31}$ of the standard genus-0 trisection of $S^{4}$ and glues $c_{2}$ and $c_{3}$ disks to the components of each of the two unlinks. Rather than glue disks to $A\cup\overline{B}$, one attaches the cone on $K$. Note that by interchanging the order of the tangles $A$ and $B$, one obtains a surface with singularity $\overline{K}$, the mirror of $K$.

A $p$-colored singular tri-plane diagram is a singular tri-plane diagram together with an assignment of values in $\{1,2,\dots,p\}$ to the arcs of the diagram, such that on each tangle, the assignment is a Fox $p$-coloring, and such that the colors along the endpoints of each tangle agree. Such a coloring induces a coloring on the corresponding singular surface.

We use 3-colored singular triplane diagrams to construct a family of 3-fold dihedral covers of $S^{4}$ which realize the knots $K_{m}$ given in Figure 1 as singularities on the branching sets. This construction allows us to compute the values of $\Xi_{3}(K_{m})$ using the induced trisections of the corresponding branched cover. As a corollary, we obtain Theorem 5, which establishes the range of the invariant $\Xi_{3}$.

Proposition 4**.**

Each knot $K_{m}$ in Figure 1 arises as the only singularity on a 3-fold dihedral branched cover $f_{m}:\#(2m+1)\overline{\mathbb{CP}}^{2}\rightarrow S^{4}$ whose branching set $F_{m}$ is an oriented surface of genus $m$, embedded smoothly in $S^{4}$ away from the one singular point. Equivalently, each knot $\overline{K}_{m}$ arises as the only singularity on a 3-fold dihedral branched cover $\bar{f}_{m}:\#(2m+1)\mathbb{CP}^{2}\rightarrow S^{4}$, also with an embedded oriented branching set of genus $m$.

Remark. By deleting a small neighborhood of the singularity on the branching set in $S^{4}$, one obtains an oriented, 3-colored surface in $F_{m}^{\prime}\subset B^{4}$ with $\partial F_{m}^{\prime}=K_{m}$. In Section 4, we prove that the genus of $F_{m}^{\prime}$ is minimal, that is, equal to $g_{4}(K_{m})$. Moreover, by construction, each surface $F^{\prime}_{m}$ is ribbon.

Proof of Proposition 4.

We will construct the surface $F_{m}$ and will give its Fox coloring using a colored (singular) tri-plane diagram. From this information, we will produce a trisection of the dihedral cover of $S^{4}$ determined by this coloring. We will identify this cover as $\#n\overline{\mathbb{CP}}^{2}$, where $n=2m+1$.

The colored tri-plane diagram $(A_{n},B_{n},C_{n})$ for $F_{m}$, where $m=(n-1)/2$, is shown in Figure  3. We write the value $i\in\{1,2,3\}$ next to an arc of a tangle or knot if the homotopy class of the meridian of that arc is mapped to the reflection in $D_{3}$ fixing $i$.

The union $A_{n}\cup\overline{B}_{n}$ is the knot $\overline{K}_{m}$, while $B_{n}\cup\overline{C}_{n}$ and $C_{n}\cup\overline{A}_{n}$ are each 2-component unlinks; see Figure  4 for a verification of this fact when $n=3$. A tri-plane diagram with $b$ bridges and $c_{i}$ components in each link diagram has Euler characteristic $c_{1}+c_{2}+c_{3}-b$; hence, the surface $F_{m}$ with singularity $\overline{K}_{m}$ has Euler characteristic $3-n$ and genus $m=(n-1)/2$ since $F_{m}$ is connected and orientable, and since the tangles $A_{n}$, $B_{n}$ and $C_{n}$ have $b=n+2$ bridges.

The fact that $F_{m}$ is orientable requires a careful check. Consider the cell structure on $F_{m}$ corresponding to its tri-plane structure. To show that $F_{m}$ is orientable, we show that it is possible to coherently orient the faces of this cell structure so that each edge (a bridge in one of the three tangles $A_{n}$, $B_{n}$, or $C_{n}$) inherits two different orientations from the two faces adjacent to it. This is shown in Figure  4 in the case $m=1$ (or $n=3$).

An Euler characteristic computation shows that the 3-fold dihedral branched cover of the bridge sphere $S^{2}$, branched along the $2(n+2)$ endpoints of the bridges, is a surface $\Sigma_{n}$ of genus $n$. We now show this 3-colored tri-plane diagram $(A_{n},B_{n},C_{n})$ gives rise to a genus $n$ trisection of $\#n\overline{\mathbb{C}P}^{2}$ with central surface $\Sigma_{n}$ following a method explained in [3]. The branching set $F_{m}$ is orientable and has one singularity of type $K_{m}$, so it will follow from Equation 6 that $\Xi_{3}(K_{m})=-\sigma(\#_{n}\overline{\mathbb{C}P}^{2})=n$.

If a properly embedded $b$-strand tangle $(T,\partial T)\subset(B^{3},S^{2})$ with arcs $t_{1},t_{2},\dots t_{b}$ is trivial, then by definition there exists a collection of disjoint arcs $d_{1},d_{2},\dots,d_{b}$ in $S^{2}$ such that the boundary unions $t_{i}\cup d_{i}$ bound a collection of disjoint disks in $B_{3}$. We refer to the $d_{i}$ as disk bottoms. The existence of such a collection of disks is equivalent to the arcs of $T$ being simultaneously isotopic to a collection of disjoint arcs (the $d_{i}$) in $S^{2}$.

To determine the trisection diagram, we must first find the disk bottoms for the three tangles $A_{n}$, $B_{n}$ and $C_{n}$, then lift them from the bridge sphere $S^{2}$ to its irregular dihedral cover $\Sigma_{n}$. The curves in the trisection diagram are formed by certain lifts of these disk bottoms; we identify these lifts later.

The disk bottoms for each tangle $A_{n}$, $B_{n}$, and $C_{n}$ are depicted in Figure 5, in the case $n=3$. In Figure 7, we draw just three of the disk bottoms for each of $A_{n}$ (blue), $B_{n}$ (red), and $C_{n}$ (green) on the same copy of $S^{2}$.

In the next step of the proof, we use a construction of the irregular 3-fold dihedral cover $\Sigma_{n}\rightarrow S^{2}$, branched along $2(n+2)$ points in $S^{2}$, due to Hilden [11]. We review this construction now; the reader should refer to Figure  6 for an example in the case $n=3$. In this construction, the meridians of two branch points map to the transposition $(23)$ (equivalently, are colored ‘1’), and the meridians of the remaining $2n+2$ branch points map to the transposition $(13)$ (equivalently, are colored ‘2’). One first constructs the 6-fold regular dihedral cover $R_{n}\rightarrow S^{2}$ branched along $2(n+2)$ points determined by this coloring. The resulting surface has genus $3n+1$. The 3-fold irregular dihedral cover $\Sigma_{n}$ is obtained from this regular one by an involution, namely $180^{\circ}$ rotation about the vertical axis.

Next, we lift the disk bottoms from the bridge sphere to $\Sigma_{n}$, where $\Sigma_{n}$ is constructed as above. Each disk bottom has three lifts to $\Sigma_{n}$, two of which fit together to form a closed curve. Not all of these closed curves are necessarily essential curves on $\Sigma_{n}$; see [3] for further examples. However, we may choose $n-2$ disk bottoms for each tangle $(A_{n},B_{n},C_{n})$ whose lifts are essential. These lifts are shown in Figure  8, again in the case $n=3$.

The resulting curves form a trisection diagram for $\#n\overline{\mathbb{C}P}^{2}$. Moreover, the standard trisection of $S^{4}$, branched along $F_{m}$, lifts to a $(n;0,0,0)$-trisection of $\#n\overline{\mathbb{C}P}^{2}$. This can be found by analyzing the lifts of the three pieces of the trisection of $(S^{4},F_{m})$; for details see Theorem 8 of [3].∎

We use the above construction to establish the range of the invariant $\Xi_{3}$.

Theorem 5**.**

Let $n$ denote an integer. There exists a strongly 3-admissible singularity $K_{n}$ and a 3-coloring $\rho_{n}$ of $K_{n}$ such that $\Xi_{3}(K_{n},\rho_{n})=n$ if and only if $n\in 2\mathbb{Z}+1$.

Remark. The proof of Theorem 5 is slightly more general than what the theorem statement requires. That is, we establish that $\Xi_{p}(K,\rho)$ is odd whenever $p\equiv 3\mod 4$. Realizability of all odd integers by $\Xi_{p}$ is open for $p\neq 3$.

Proof of Theorem 5.

We have given a construction realizing each of the knots $K_{m}$ as the only singularity on a branched cover $\#({2m+1})\overline{\mathbb{CP}}^{2}\rightarrow S^{4}$ whose branching set is oriented. By Equation 6, it follows that $\Xi_{3}(K_{m})=-\sigma(\#(2m+1)\overline{\mathbb{CP}}^{2})=2m+1$, where $m\geq 0$. Note also that $\Xi_{p}(\overline{K}_{m})=-\Xi_{p}(K_{m})$ as proved in [3], where $\overline{K}$ denotes the mirror image of $K$. Of course, $K$ is (strongly) $p$-admissible if and only if $\overline{K}$ is. This proves that all odd integers are contained in the range of the invariant $\Xi_{3}$ on strongly 3-admissible knots.

Conversely, we will verify that for any $p$-coloring $\rho$ of any strongly $p$-admissible singularity $K$, the integer $\Xi_{p}(K,\rho)$ is odd. It suffices to assume that $p\equiv 3\mod 4$. We use Equation 5. Since $p$ is odd, $p^{2}\equiv 1\mod 4$, so $\dfrac{p^{2}-1}{6p}L_{V}(\beta,\beta)$ is even. It follows from [13, Equation 2.20] that, if $p\equiv 3\mod 4$, the rank of $H_{2}(W(K,\beta);\mathbb{Z})$ is odd, hence so is the signature. Lastly, each $\sigma_{\zeta^{i}}$ is an even integer. It follows that $\Xi_{p}(K)$ is odd. ∎

Remark. The knot $K_{m}$ has bridge number 2, showing that two-bridge knots realize the full range of $\Xi_{p}$ when $p=3$. This answers a question posed in [12]. It is not known whether the full range of $\Xi_{p}$ is realized by two-bridge knots when $p\neq 3$. It would be of interest to establish that it is “sufficient” to consider two-bridge knots when constructing singular dihedral covers of four-manifolds since $p$-admissibility is particularly easy to detect for two-bridge singularities [kjorr2017admissible].

4. Proofs of Theorems 1,  2 and  3

Proof of Theorem 1.

(A) Given a $p$-admissible knot $K$ with $p$-coloring $\rho$, we wish to prove the inequality:

[TABLE]

where $M$ denotes the dihedral cover of $S^{3}$ branched along $K$ corresponding to $\rho$. Throughout, all homology groups are with $\mathbb{Z}$ coefficients.

For $K$ a $p$-admissible knot, by the definition of homotopy ribbon dihedral genus, there exists a topologically locally flat orientable homotopy ribbon surface $F^{\prime}$ for $K$ such that the genus of $F^{\prime}$ equals $\mathfrak{g}_{p}(K,\rho)$. (If $\rho$ does not extend over any locally flat, orientable, homotopy ribbon surface, $\mathfrak{g}_{p}(K,\rho)=\infty$ and the inequality is trivial.) Recall that, since $F^{\prime}$ is orientable, $|\Xi_{p}(K,\rho)|=\sigma(X^{\prime},\partial X^{\prime})$; the righthand side denotes the Novikov signature of $X^{\prime}$ as a manifold with boundary. We will find an upper bound for $|\sigma(X^{\prime},\partial X^{\prime})|$ in terms of the Euler characteristic of $X=X^{\prime}\cup c(\partial X^{\prime})$.

Let $\bar{\rho}:\pi_{1}(B^{4}-F^{\prime})\rightarrow D_{p}$ be the homomorphism which extends the coloring $\rho:\pi_{1}(S^{3}-K)\rightarrow D_{p}$ and induces the cover $X^{\prime}\to B^{4}$ branched over $F^{\prime}$. Let $\hat{M}$ be the unbranched irregular dihedral cover of $S^{3}-K$ corresponding to $\rho$, and $M$ the induced branched cover. Denote by $F\subset S^{4}$ the singular surface which is the boundary union of $F^{\prime}$ and the cone on $K$, so that $X$ is the dihedral cover of $S^{4}$ with branching set $F$.

We will show that $X$ is simply-connected. Consider the commutative diagram below. All maps in the diagram are either induced by inclusions or by covering maps. Clearly $p_{*}$ and $q_{*}$ are injective, as they are induced by covering maps, and $\mathsf{i}_{M*}$ and $\mathsf{i}_{X*}$ are surjective, as they are induced by inclusions of unbranched covering spaces into their branched counterparts. The homomorphisms $\rho$ and $\bar{\rho}$ are surjective by definition. Finally, since $F^{\prime}$ is a homotopy-ribbon surface for $K$, $i_{*}$ is surjective.

{\pi_{1}(M)}$${\pi_{1}(X^{\prime})}$${\pi_{1}(\hat{M})}$${\pi_{1}(\hat{X}^{\prime})}$${\pi_{1}(S^{3}-K)}$${\pi_{1}(B^{4}-F^{\prime})}$${D_{p}}$$\scriptstyle{\mathsf{i}_{*}}$$\scriptstyle{j_{*}}$$\scriptstyle{p_{*}}$$\scriptstyle{\mathsf{i}_{M*}}$$\scriptstyle{q_{*}}$$\scriptstyle{\mathsf{i}_{X*}}$$\scriptstyle{i_{*}}$$\scriptstyle{\rho}$$\scriptstyle{\bar{\rho}}

We now show that $j_{*}$ is surjective as well. Consider an element $\gamma\in\pi_{1}(\hat{X}^{\prime})$. Since $i_{*}$ is surjective, there exists an element $\delta\in\pi_{1}(S^{3}-K)$ such that $i_{*}(\delta)=q_{*}(\gamma)$. We have that $\bar{\rho}\circ q_{*}(\gamma)\in\mathbb{Z}/2\mathbb{Z}\subset D_{p}$, the reflection subgroup which determines the cover $\hat{X}^{\prime}$ of $B^{4}-F^{\prime}$. By commutativity of the lower triangle, $\rho(\delta)=\bar{\rho}\circ q_{*}(\gamma)\in\mathbb{Z}/2$, so $\delta\in\text{Im }p_{*}$. Take $\tilde{\delta}\in\pi_{1}(\hat{M})$ such that $p_{*}(\tilde{\delta})=\delta.$ Consider $q_{*}\circ j_{*}(\tilde{\delta})$, which by commutativity is equal to $i_{*}\circ p_{*}(\tilde{\delta})$. Now $q_{*}\circ j_{*}(\tilde{\delta})=i_{*}(\delta)=q_{*}(\gamma)$. By injectivity of $q_{*}$, we have $j_{*}(\tilde{\delta})=\gamma$, so $j_{*}$ is indeed surjective.

Observe that, since $j_{*}$ and $\mathsf{i}_{X*}$ are both surjective, $\mathsf{i}_{*}:\pi_{1}(M)\rightarrow\pi_{1}(X^{\prime})$ is surjective as well .

By Seifert-van Kampen’s theorem, we have $\pi_{1}(X)\simeq\pi_{1}(c(M))*_{\pi_{1}(M)}\pi_{1}(X^{\prime})$. The cone $c(M)$ is contractible so $\pi_{1}(c(M))$ is trivial. Hence

[TABLE]

The quotient $\pi_{1}(X^{\prime})/\text{im }\mathsf{i}_{*}$ is trivial since $\mathsf{i}_{*}$ is surjective. Hence $X$ is simply connected and, in particular, $H_{1}(X)=0$. We also know that $H_{0}(X)=1$ since $X$ is path-connected.

Next, consider

[TABLE]

Since $X=X^{\prime}\cup c(\partial X^{\prime})$,

[TABLE]

By Lefschetz Duality and Universal Coefficients, we see that

[TABLE]

and

[TABLE]

hence

[TABLE]

By [13, Equation 1.1] we have

[TABLE]

Recall that the Novikov signature $\sigma(X,\partial X^{\prime})$ is the signature of the intersection form defined on the image of the map $\mathfrak{i}_{\ast}:H_{2}(X^{\prime})\rightarrow H_{2}(X^{\prime},\partial X^{\prime})$ induced by inclusion. Thus,

[TABLE]

The result follows by combining this inequality with the two formulas for $\chi(X)$ above. We substitute

[TABLE]

into Equation 7, and

[TABLE]

into 8, which gives

[TABLE]

Simplifying, we obtain

[TABLE]

Finally, since the inclusion $\mathsf{i}:M\rightarrow X^{\prime}$ induces a surjection on fundamental groups, we also know that $\text{rk }H_{1}(M)\geq\text{rk }H_{1}(X^{\prime})$. Hence

[TABLE]

(B) Let $K$ be a $p$-admissible knot with respect to a coloring $\rho$ and let $F^{\prime}\subset B^{4}$ be a homotopy ribbon, locally flat oriented surface with boundary $K$ such that $\rho$ extends over $F^{\prime}$. Denote by $F\subset S^{4}$ the surface with singularity $K$ obtained as a boundary union of $F^{\prime}$ and the cone on $K$, and denote by $X$ the dihedral cover of $S^{4}$ determined by the induced coloring of $F$. We assume that $X$ is a definite manifold. In particular, $K$ is in fact strongly admissible with respect to the coloring $\rho$, and the corresponding branched cover $M$ of $S^{3}$ along $K$ is again $S^{3}$. We then wish to show that the inequality (3) is sharp:

[TABLE]

Precisely, we will show that the right hand-side of this equation equals the genus of $F^{\prime}$. That is, $F^{\prime}$ will be seen to realize the lower bound from (A) on the dihedral genus $\mathfrak{g}_{{p}}(K,\rho)$ of $K$.

Since $X$ is a definite manifold, $\text{rk }H_{2}(X)=|\sigma(X)|$. By the proof of (A), $X$ is simply connected; by Poincaré duality we have $\chi(X)=2+\text{rk }H_{2}(X)$ and hence

[TABLE]

On the other hand, denoting by $g(F)$ the genus of $F$, by Equation 7 we have

[TABLE]

Putting these two equations together, we conclude that

[TABLE]

By assumption, the coloring $\rho$ extends over $F^{\prime}$, so

[TABLE]

Thus, $F^{\prime}$ realizes the $p$-dihedral genus of $K$.

In the second part of the theorem, we assume in addition that $|\sigma(K)|=\frac{2|\Xi_{p}(K,\rho)|}{p-1}-1$, where $\sigma(K)$ is the signature of the knot $K$. We wish to show that the topological four-genus and the topological homotopy ribbon $p$-dihedral genus of $K$ both equal $\frac{|\sigma(K)|}{2}$.

The additional assumption here can be rewritten as $|\sigma(K)|=2\mathfrak{g}_{p}(K,{\rho})$. Murasugi’s signature bound [15, Theorem 9.1] states that $g_{4}(K)\geq|\sigma(K)|/2$. Thus, we have $g_{4}(K)\geq\mathfrak{g}_{p}(K,{\rho})$. But $g_{4}(K)\leq\mathfrak{g}_{p}(K)\leq{\mathfrak{g}_{p}(K,\rho)}$ in general, so $g_{4}(K)=\mathfrak{g}_{p}(K)$. ∎

Proof of Theorem 2.

Our aim is to show that the equalities

[TABLE]

hold for the 3-colored knots $K_{m}$ introduced in the previous section. In particular, it will follow that the generalized topological Slice-Ribbon Conjecture holds for these knots.

By (B) of Theorem 1, it suffices to show that

1. (1)

Each $K_{m}$ is the boundary of a homotopy-ribbon surface $F_{m}^{\prime}$ such that $\mathfrak{g}_{3}(K)=g(F_{m}^{\prime})$, and 2. (2)

The signature $\sigma(K_{m})$ satisfies the equality

[TABLE]

for $p=3$.

We first address (1). Surfaces $F_{m}^{\prime}$ realizing the lower bound on dihedral homotopy-ribbon genus for the knots $K_{m}$ are constructed in the proof of Proposition 4: we have shown $g(F_{m}^{\prime})=m$ and $|\Xi_{3}(K_{m})|=2m+1$, so

[TABLE]

We note that, since the knots $K_{m}$ are two-bridge, each of them has a unique $3$-coloring (up to permuting the colors), so there is no distinction between $\mathfrak{g}_{p}(K_{m},\rho_{m})$ and $\mathfrak{g}_{p}(K_{m})$. By construction, the surface $F_{m}^{\prime}\subset B^{4}$ obtained by deleting a small neighborhood of the singularity $K_{m}$ is ribbon since $A_{m}\cup\overline{B}_{m}$ only bounds the cone on $K_{m}$, while the unlinks $B_{m}\cup\overline{C}_{m}$ and $C_{m}\cup\overline{A}_{m}$ bound standard unknotted disks in $B^{4}$.

We now address (2). We will compute the signature $\sigma(K_{m})$, and show it is equal to $2m=\frac{2|\Xi_{p}(K)|}{p-1}-1$.

The signature of $K$ can be computed using the Goeritz matrix $G(K)$, the matrix of a quadratic form associated to a knot diagram via a checkerboard coloring, and hence a (not necessarily orientable) spanning surface; this technique was introduced by Gordon and Litherland [10]. The advantage of this technique is that the dimension of the Goeritz matrix associated to a projection of a knot may be much smaller than the dimension of the corresponding Seifert matrix; indeed, the dimension of $G(K_{m})$ is 4 for all $m$.

Gordon and Litherland proved that the signature of a knot is equal the signature of the Goeritz matrix of a diagram of the knot plus a certain correction term: $\sigma(K)=\sigma(G(K))-\mu$. We start by computing the Goeritz matrix $G(K_{m})$ and its signature.

One first computes the unreduced Goeritz matrix. To do this, one chooses a checkerboard coloring of the knot diagram, and labels the “white” regions $X_{1},\dots,X_{k}$. Such a labelling for the $K_{m}$ is shown in Figure  9. The entries $g_{ij}$ of the unreduced Goeritz matrix are computed as follows:

[TABLE]

The signs $\eta(c)$ are computed as in Figure  10; shaded area corresponds to “black” regions of the checkerboard coloring.

The unreduced Goeritz matrix of $K_{m}$ is

[TABLE]

The Goeritz matrix $G(K_{m})$ is obtained by deleting the first row and column of $G^{\prime}(K_{m})$. The characteristic polynomial of this matrix is

[TABLE]

Hence $\lambda=-3$ is an eigenvalue. In addition, since $m\geq 0$, it is straightforward to verify that any root of the cubic factor must be negative (if $\lambda$ is nonnegative, the cubic, as written above, is a sum of three nonnegative terms). Hence, $\sigma(G(K_{m}))=-4$.

The correction term $\mu(K)$ in Gordon and Litherland’s formula for $\sigma(K)$ is computed as follows. Each crossing $c$ of $K$ can be classified as type I or type II, as shown in Figure  10. Let $\mu(K)=\sum_{c}\eta(c)$ where the sum is taken over all type II crossings.

The knot $K_{m}$ has $4+2m$ type II crossings, each of negative sign; see Figure  9. Hence $\sigma(K_{m})=-4+(4+2m)=2m$. ∎

Proof of Theorem 3.

Given any non-negative integer $m$, our goal is to construct an infinite family of knots whose 3-dihedral and topological 4-genus are both equal to $m$. Let $K_{m}$ denote the knot given in Theorem 2 whose 3-dihedral and topological 4-genus equal $m$. We will prove that, given a nontrivial ribbon knot $\gamma$, the knot $K_{m}\#\gamma$ has the desired property. The theorem follows by taking repeated connect sums of $K_{m}$ with $\gamma$.

Let $\gamma$ denote any ribbon knot and let $D\subset B^{4}$ be a ribbon disk with $\partial D=\gamma$. The knot $K_{m}\#\gamma$ has 3-dihedral genus and topological four-genus equal to $m$, as we now show. It is clear that the smooth and topological four-genera of $K_{m}\#\gamma$ are both equal to $m$ since the knot is smoothly concordant to $K_{m}$. Next, remark that the given 3-coloring $\rho_{m}$ of $K_{m}$ induces a 3-coloring $\rho_{\gamma}$ of $K_{m}\#\gamma$ which restricts trivially to $\gamma$. Moreover, since $\rho_{m}$ extends over $F^{\prime}_{m}$, $\rho_{\gamma}$ extends over the ribbon surface $F_{m}^{\prime}\natural D$, where $\natural$ denotes boundary connected sum. Therefore, the ribbon 3-dihedral genus of $K_{m}\#\gamma$ is at most $m$. Since $g_{4}$ is a lower bound for the topological 3-dihedral genus, which in turn is a lower bound for the ribbon 3-dihedral genus, it follows that these genera are equal, as claimed. ∎

Patricia Cahn

Smith College

pcahn@smith.edu

Alexandra Kjuchukova

Max Planck Institute for Mathematics–Bonn

sashka@mpim-bonn.mpg.de