The Dihedral Genus of a Knot
Patricia Cahn Alexandra Kjuchukova


    Benno Stein
    December 30, 2023
=====================




 
 Let K⊂ S^3 be a Fox p-colored knot and assume K bounds a locally flat surface S⊂ B^4 over which the given p-coloring extends. This coloring of S induces a dihedral branched cover X→ 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.











§ INTRODUCTION


The Slice-Ribbon Conjecture of Fox <cit.>  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'⊂ B^4 is homotopy ribbon if the fundamental group of its complement is generated by meridians of ∂ F' 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'⊂ B^4 with ∂ F'=K over which some p-coloring of K extends, in the sense defined in Section <ref>. 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' for K can always be chosen to be a disk. This is essentially a consequence of <cit.>; a detailed explanation can be found in <cit.>. 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': X'→ B^4 branched along a locally flat properly embedded surface F' with ∂ F'=K; that is, F' is a properly embedded topological submanifold of B^4. We now construct a new cover f: X→ S^4 by taking the cones of ∂ X', 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 𝔷, the pre-image of the singularity on F. The link of 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 𝔷 in X.


When  f: X→ S^4 in the above construction is a p-fold irregular dihedral cover (Definition <ref>), an invariant of (p-colored) knots, Ξ_p, is extracted from this construction. This invariant is our main tool. In a general setting, Ξ_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 Ξ_p(K,ρ). This term depends only on the isotopy class of the knot K and its Fox p-coloring ρ, but not on the locally flat part of the branching set. 



Given a dihedral cover  f: X→ S^4 whose branching set is orientable with one singularity, we in fact have

    Ξ_p(K,ρ)=-σ(X)

by <cit.>,  when X is a manifold, and 

    Ξ_p(K,ρ)=-σ(X', ∂ X')

when X has a singularity <cit.>. 
In the latter formula, 
∂ X' is the dihedral cover of K induced by f_|, and σ(·, ·) denotes the Novikov signature of a manifold with boundary. Of course, the first formula for Ξ_p(K,ρ) 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 Ξ_p(K,ρ).  In Equation <ref> we recall an explicit formula <cit.>  for Ξ_p which does not rely on constructing the cover X. We also note that  Ξ_p(K,ρ) can be computed algorithmically from a colored diagram of K <cit.>. We often suppress notation and write Ξ_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 Ξ_p(K).  
The main result of this paper, Theorem <ref>, obtains a certain genus bound for K from Ξ_p(K).



As implied by the above, the signature defect Ξ_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 <cit.>. A knot K is called p-admissible over S^4, or simply p-admissible, if there exists a p-fold dihedral cover  f: X→ 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 knots[Like the invariant Ξ_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 <cit.>. 



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

In Theorem <ref> and Theorem <ref>, we construct, for any integer m≥ 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 <ref>.  The various four-genera of these knots are computed with the help of Theorem <ref>.  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  <ref>. 
 
The technique we apply is the following.  Given a strongly p-admissible knot K, one can evaluate Ξ_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 
    f_m: #(2m+1)ℂP^2→ S^4.
 The branching set of f_m is the boundary union of the cone on K_m with the surface F_m' realizing the four-genus of K_m. We construct these covering maps explicitly using singular triplane diagrams, a technique introduced in <cit.>.  Equivalently, we construct a family of covers #(2m+1)ℂP^2→ S^4, again with oriented, connected branching sets, with the mirror images of the knots K_m as singularities. This construction appears in Section  <ref>.  As a corollary of this construction, we realize all odd integers as values of Ξ_3. In Theorem  <ref>, we prove that the range of values of Ξ_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' 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 <cit.>. The examples used in Theorem <ref> were inspired by joint work with Ryan Blair <cit.>.



§ DIHEDRAL FOUR-GENUS AND THE MAIN THEOREMS





 §.§ 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⊂ 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' in B^4 over which the p-coloring of K extends. Precisely, a given p-coloring ρ of K extends over F' if there exists a homomorphism ρ̅ which makes the following diagram commute (where i_∗ the map induced by inclusion):




	
		π_1(S^3-K)[r,"i_*"][d,"ρ"]   π_1(B^4-F')[ld, dashed, "ρ̅"]

		D_p
		
		


The p-dihedral genus above is defined for a knot K with a fixed coloring ρ, hence we denote it g_p(K,ρ) 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 ρ 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' as above. In <cit.>, 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 ρ of K extends, we define g_p(K,ρ) 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' in B^4 with boundary K, such that F' 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' in B^4 with boundary K such that i_∗: π_1(S^3-K)↠π_1(B^4-F'), 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-genus[d,"≤"][r,"≤"]   hom. ribbon genus[d,"≤"][r,"≤"]   ribbon genus[d,"≤"]

		p-dihedral genus[r,"≤"]   p-dihedral hom. ribbon genus[r,"≤"]   p-dihedral ribbon genus

		
		


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




 §.§ The Main Theorems


Denote by g_4(K) the topological 4-genus of a knot K, and by 𝔤_p(K,ρ) the topological homotopy-ribbon p-dihedral genus of a knot K with coloring ρ. Again, the minimum such genus over all colorings ρ of K is 𝔤_p(K). Let σ(K) be the (Murasugi) signature of the knot K. We relate 𝔤_p(K, ρ), Ξ_p(K, ρ) and σ(K). Here, Ξ_p(K, ρ) 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 <ref>. 

  

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


	
	
    𝔤_p(K, ρ)≥|Ξ_p(K,ρ)| -  H_1(M; ℤ)p-1-12

	
		 
(B)	Let K be a p-admissible knot and F'⊂ B^4 a locally flat homotopy ribbon oriented surface for K over which a given p-coloring ρ 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'∪_K c(K) is a definite manifold, then the inequality (<ref>) is sharp. In particular, F' realizes the dihedral genus 𝔤_p(K, ρ) of K. If, in addition, the equality 
    |σ(K)| = 2|Ξ_p(K, ρ)|/p-1 -1
 holds, then the topological four-genus and the topological homotopy ribbon p-dihedral genus of K coincide and equal |σ(K)|/2, so the generalized topological Slice-Ribbon Conjecture holds for K.


 
  Remark. If K has multiple p-colorings, denote by min_p(K) the minimum value of 
    |Ξ_p(K,ρ)|- H_1(M;ℤ)
 over all such colorings of K. Theorem <ref> implies:
 		 
    𝔤_p(K )≥min_p(K)p-1-12.





 For every integer m≥ 0, there exists a knot K_m and corresponding 3-coloring ρ_m, such that: 
	
    g_4(K_m)=𝔤_3(K_m)=|Ξ_3(K_m, ρ_m)|2 -1/2=m.
 
	That is, the inequality (<ref>) 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.





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


 §.§ Singular dihedral covers of S^4 and the invariant Ξ_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 Ξ_p arises, as well as a couple of techniques for its calculation.



Let Y be a manifold and B⊂ Y a codimension-two submanifold with the property that there exists a surjection φ: π_1(Y-B) ↠ D_p. Denote by X the covering space of Y-B corresponding to the conjugacy class of subgroups φ^-1(ℤ/2ℤ) in π_1(Y-B), where ℤ/2ℤ⊂ D_p is any reflection subgroup. The completion of X  to a branched cover f: X→ 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 Ξ_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 <cit.>.

Recall the following construction from the introduction. Let F' be a surface with connected boundary K, properly embedded in B^4 and locally flat. Given a branched cover of manifolds with boundary f': X'→ B^4, one constructs a singular branched cover of S^4 by coning off ∂ X', ∂ B^4 and the map f'. The resulting covering map, f: X→ S^4, has total space X:=X'∪_∂ X' c(∂ X'), where c(∂ X') denotes the cone on ∂ X'. The branching set is a closed surface F:=F'∪_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 ∂ X'≅ S^3. 

Denote by σ(X',∂ X') the Novikov signature of the manifold with boundary given as a cover of B^4 branched along F'. When ∂ X' = S^3, denote by σ(X) the signature of the manifold X. In this case, we have σ(X) = σ(X', ∂ X'). 



Given f': X'→ B^4 as before with f' an (irregular) dihedral covering map, we always assume that the associated homomorphism ρ: π_1(S^3-K)→ D_p is surjective or, equivalently, that ∂ X' is connected. In this case, assuming F' is orientable, Ξ_p(K, ρ)=-σ(X', ∂ X') by <cit.>. In particular, when X is a manifold, this equation reduces to the earlier result Ξ_p(K, ρ)=-σ(X) <cit.>.
 
  Below, we recall two formulas for Ξ_p(K, ρ) from <cit.>. Equation <ref> allows for Ξ_p(K, ρ) to be computed in terms of K and its coloring using <cit.>  and <cit.>. Equation <ref> expresses Ξ_p(K, ρ) 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⊂ 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 <cit.>). 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 <cit.> where we also use the invariant Ξ_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 <cit.>.









We conclude this section by reviewing the formula for computing the invariant Ξ_p given in <cit.>. Let p be an odd integer and K a p-admissible knot. Let V be a Seifert surface for K and V^∘ the interior of V. Denote by β⊂ V^∘ a mod p characteristic knot[Precisely, 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 <cit.>.  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 <cit.>. 
Also denote by L_V the symmetrized linking form for V and by σ_ζ^i the Tristram-Levine ζ^i-signature, where ζ is a primitive p^th root of unity.  Finally, let W(K,β) be the cobordism constructed in <cit.> between the p-fold cyclic cover of S^3 branched along β and the p-fold dihedral cover of S^3 branched along K and determined by ρ. We briefly describe the manifold W(K,β).  
Let Σ be the p-fold cyclic branched cover of β and let Σ_p(β)×[0, 1]→ S^3×[0, 1] be the induced cyclic cover branched along β×[0, 1]. Letting ℤ/2ℤ act on an appropriate subset of Σ_p(β)×{0}, one obtains W(K,β) as a quotient of Σ_p(β)×[0, 1] by this action. One boundary of this quotient, namely Σ_p(β)×{1}, is clearly the p-fold cyclic cover of β. The other boundary component, that is, the image of Σ_p(β)×{0} under the ℤ/2ℤ action, is the dihedral cover of α as shown in <cit.>.   By <cit.>, 


    Ξ_p(K, ρ)=p^2-16pL_V(β, β)+σ(W(K, β))+∑_i=1^p-1σ_ζ^i(β).


The Novikov signature σ(W(K,β)) can be computed in terms of linking numbers in the dihedral cover of K <cit.>. Thus, the above formula allows Ξ_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 <cit.>.  Note also that when a knot K is realized as the only singularity on an embedded surface F⊂ S^4 and moreover this surface is presented by a Fox p-colored singular triplane diagram, <cit.> gives a method for computing Ξ_p(K) from this data, via the signature of the associated cover of S^4. This technique is reviewed and applied in Section <ref> below.




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

    Ξ_p(K)=-p-14e(F)-σ(X', ∂ X'),

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 base[The reference <cit.> is written in the language of intersection homology. In the case of a singular branched cover f: X→ S^4, this is equivalent to the Novikov signature σ(X', ∂ X')  since X has only an isolated singularity.] given in <cit.>. Note that, when F is orientable and X a manifold, Equation <ref> reduces to Equation <ref>, that is, Ξ_p(K)=-σ(X).  
In this case,  the Ξ_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 ℂP^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 <cit.>. By contrast, we see in Theorem <ref> that orientability of the branching set, together with a single singular point, imply that the signature of such a cover is odd. 





§ 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 <cit.>, tri-plane diagrams <cit.>, and singular tri-plane diagrams <cit.>, 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∪ X_2∪ X_3 into three 4-handlebodies with boundary, such that 

	
  * X_i≅♮ k_i (B^3× S^1)
	
  * X_1∩ X_2∩ X_3 ≅Σ_g is a closed, oriented surface of genus g
	
  * Y_ij=∂ (X_i∪ X_j)≅# k_l (S^2× S^1) where i, j and l ∈{1,2,3} are distinct
	
  * Σ_g⊂ Y_ij is a Heegaard surface for Y_ij.


Every embedded surface F⊂ S^4 can be described combinatorially by a (b;c_1,c_2,c_3)-tri-plane diagram <cit.>.  This is a set of three b-strand trivial tangles (A,B, C), such that each boundary union of tangles  A∪B,  B∪C, and C∪A is a c_i-component unlink, for i=1,2,3 respectively.  Here T denotes the mirror image of T.  To obtain F from (A,B,C), one views each of A∪B,  B∪C, and C∪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 <cit.>.  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∪C  and C∪A are c_2- and c_3-component unlinks.  A∪B is a knot K.  To build a surface with one singularity of type K, one again views each of A∪B,  B∪C, and C∪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∪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 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, …,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 <ref> as singularities on the branching sets.  This construction allows us to compute the values of Ξ_3(K_m)  using the induced trisections of the corresponding branched cover. As a corollary, we obtain Theorem <ref>, which establishes the range of the invariant Ξ_3. 



	
	Each knot K_m in Figure <ref> arises as the only singularity on  a 3-fold dihedral branched cover f_m: #(2m+1) ℂℙ^2→ 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 K_m arises as the only singularity on  a 3-fold dihedral branched cover f̅_m: #(2m+1)ℂℙ^2→ 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'⊂ B^4 with  ∂ F_m' = K_m. In Section <ref>, we prove that the genus of F_m' is minimal, that is, equal to g_4(K_m).  Moreover, by construction, each surface F'_m is ribbon. 




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 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  <ref>.  We write the value i∈{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∪B_n is the knot K_m, while B_n∪C_n and C_n∪A_n are each 2-component unlinks; see Figure  <ref> 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 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  <ref> 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 Σ_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 #nℂP^2 with central surface Σ_n following a method explained in <cit.>.  The branching set F_m is orientable and has one singularity of type K_m, so it will follow from Equation <ref> that Ξ_3(K_m)=-σ(#_nℂP^2)=n.
	
	
If a properly embedded b-strand tangle (T,∂ T)⊂ (B^3,S^2) with arcs t_1,t_2,… t_b is trivial, then by definition there exists a collection of disjoint arcs d_1,d_2,…, d_b in S^2 such that the boundary unions t_i∪ 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 Σ_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 <ref>, in the case n=3.  In Figure <ref>, 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 Σ_n→ S^2, branched along 2(n+2) points in S^2, due to Hilden <cit.>.  We review this construction now; the reader should refer to Figure  <ref> 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→ 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 Σ_n is obtained from this regular one by an involution, namely 180^∘ rotation about the vertical axis. 




 Next, we lift the disk bottoms from the bridge sphere to Σ_n, where Σ_n is constructed as above. Each disk bottom has three lifts to Σ_n, two of which fit together to form a closed curve.  Not all of these closed curves are necessarily essential curves on Σ_n; see <cit.> 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  <ref>, again in the case n=3.

	

The resulting curves form a trisection diagram for #nℂP^2.  Moreover, the standard trisection of S^4, branched along F_m, lifts to a (n;0,0,0)-trisection of #nℂP^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 <cit.>.

We use the above construction to establish the range of the invariant Ξ_3.

 

Let n denote an integer. There exists a strongly 3-admissible singularity K_n and a 3-coloring ρ_n of K_n such that  Ξ_3(K_n, ρ_n)=n if and only if n∈ 2ℤ+1.


Remark. The proof of Theorem <ref> is slightly more general than what the theorem statement requires. That is, we establish that Ξ_p(K, ρ) is odd whenever p≡ 3 4. Realizability of all odd integers by Ξ_p is open for p≠ 3. 



	We have given a construction realizing each of the knots K_m as the only singularity on a branched cover #(2m+1) ℂℙ^2→ S^4 whose branching set is oriented. By Equation <ref>, it follows that Ξ_3(K_m)=-σ(#(2m+1)ℂℙ^2)=2m+1, where m≥ 0. Note also that Ξ_p(K_m)=-Ξ_p(K_m) as proved in <cit.>, where K denotes the mirror image of K. Of course, K is (strongly) p-admissible if and only if K is. This proves that all odd integers are contained in the range of the invariant Ξ_3 on strongly  3-admissible knots. 
	
Conversely, we will verify that for any p-coloring ρ of any strongly p-admissible singularity K, the integer  Ξ_p(K, ρ) is odd. It suffices to assume that p≡ 3 4. We use Equation <ref>. Since p is odd, p^2≡ 14, so p^2-16pL_V(β, β) is even.  It follows from <cit.> that, if p≡ 3 4, the rank of H_2(W(K,β); ℤ) is odd, hence so is the signature. Lastly, each σ_ζ^i is an even integer.  It follows that Ξ_p(K) is odd. 

Remark. The knot K_m has bridge number 2, showing that two-bridge knots realize the full range of Ξ_p when p=3. This answers a question posed in <cit.>. It is not known whether the full range of Ξ_p is realized by two-bridge knots when p≠ 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 <cit.>. 








§ PROOFS OF THEOREMS <REF>,  <REF> AND  <REF>






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



	
	
    𝔤_p(K, ρ)≥|Ξ_p(K,ρ)|- H_1(M)p-1-12,

	where M denotes the dihedral cover of S^3 branched along K corresponding to ρ.	Throughout, all homology groups are with 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' for K such that the genus of F' equals 𝔤_p(K, ρ). (If ρ does not extend over any locally flat, orientable, homotopy ribbon surface, 𝔤_p(K, ρ)=∞ and the inequality is trivial.) Recall that, since F' is orientable, |Ξ_p(K,ρ)|=σ(X',∂ X'); the righthand side denotes the Novikov signature of X' as a manifold with boundary.  We will find an upper bound for |σ(X',∂ X')| in terms of the Euler characteristic of X=X'∪ c(∂ X').
	
	
	Let ρ̅ :π_1(B^4-F')→ D_p be the homomorphism which extends the coloring ρ:π_1(S^3-K)→ D_p and induces the cover X'→ B^4 branched over F'.  Let M̂ be the unbranched irregular dihedral cover of S^3-K corresponding to ρ, and M the induced branched cover.  Denote by F⊂ S^4 the singular surface which is the boundary union of F' 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 𝗂_M* and 𝗂_X* are surjective, as they are induced by inclusions of unbranched covering spaces into their branched counterparts.  The homomorphisms ρ and ρ̅ are surjective by definition.  Finally, since F' is a homotopy-ribbon surface for K, i_* is surjective. 
 	
	
	

	
			π_1(M)[r,"𝗂_*"]   π_1(X')

			π_1(M̂)[r,"j_*"][d,"p_*"][u,"𝗂_M*"]   π_1(X̂')[d,"q_*"][u,"𝗂_X*"]

			π_1(S^3-K)[r,"i_*"][d,"ρ"]   π_1(B^4-F')[ld,"ρ̅"]

			D_p
			
			

	 We now show that j_* is surjective as well.  Consider an element γ∈π_1(X̂'). Since i_* is surjective, there exists an element δ∈π_1(S^3-K) such that i_*(δ)=q_*(γ).  We have that ρ̅∘ q_*(γ)∈ℤ/2ℤ⊂ D_p, the reflection subgroup which determines the cover X̂' of B^4-F'. By commutativity of the lower triangle, ρ(δ)=ρ̅∘ q_*(γ)∈ℤ/2, so  δ∈Im p_*. Take δ̃∈π_1(M̂) such that p_*(δ̃)=δ.  Consider q_*∘ j_*(δ̃), which by commutativity is equal to i_*∘ p_*(δ̃).  Now q_*∘ j_*(δ̃)=i_*(δ)=q_*(γ). By injectivity of q_*, we have j_*(δ̃)=γ, so j_* is indeed surjective.
	 
	Observe that, since j_* and 𝗂_X* are both surjective, 𝗂_*:π_1(M)→π_1(X') is surjective as well . 
	 			
	
			
	By Seifert-van Kampen's theorem, we have π_1(X)≃π_1(c(M))*_π_1(M)π_1(X'). The cone c(M) is contractible so π_1(c(M)) is trivial.  Hence
	
    π_1(c(M))*_π_1(M)π_1(X')≃π_1(X')/im 𝗂_*(π_1(M)).

The quotient π_1(X')/im 𝗂_* is trivial since 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 
    χ(X)= H_4(X)- H_3(X)+  H_2(X)+1.
  
Since X=X'∪ c(∂ X'), 
    H_n(X)=H_n(X'/∂X')=H̃_n(X',∂ X').
   By Lefschetz Duality and Universal Coefficients, we see that

    H_3(X)= H_3(X',∂ X')= H^1( X')= H_1(X'),

and

    H_4(X)= H_4(X',∂ X')= H^0(X')= H_0(X')=1,

hence 
    χ(X)=2+ H_2(X)- H_1(X').


  By <cit.> we have 
  	
    χ(X)=2p-p-12χ(F)-p-12.

	
	
Recall that the Novikov signature σ(X,∂ X') is the signature of the intersection form defined on the image of the map 𝔦_∗: H_2(X')→ H_2(X',∂ X') induced by inclusion. Thus,

    |Ξ_p(K,ρ)|=σ(X,∂ X')≤im 𝔦_∗≤ H_2(X',∂ X')= H_2(X).
 

	
The result follows by combining this inequality with the two formulas for χ(X) above. We substitute 
    χ(F)=2-2g(F)
 into Equation <ref>, and 
    H_2(X)= χ(X)-2+ H_1(X')
 into <ref>, which gives

    |Ξ_p(K,ρ)|≤ 2p-p-12(2-2g(F))-p-12-2+ H_1(X').


Simplifying, we obtain 


    g(F)≥|Ξ_p(K,ρ)|- H_1(X')p-1-12.

Finally, since the inclusion 𝗂:M→ X' induces a surjection on fundamental groups, we also know that H_1(M)≥ H_1(X').  Hence 

    g(F)≥|Ξ_p(K,ρ)|- H_1(M)p-1-12.


 
	
	
	
	(B) 		Let K be a p-admissible knot with respect to a coloring ρ and let F'⊂ B^4 be a homotopy ribbon, locally flat oriented surface with boundary K such that ρ extends over F'. Denote by F⊂ S^4 the surface with singularity K obtained as a boundary union of F' 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 ρ, and the corresponding branched cover M of S^3 along K is again S^3. We then wish to show that the inequality (<ref>) is sharp:
		
    𝔤_p(K, ρ) = |Ξ_p(K,ρ)|p-1-12.

		
		Precisely, we will show that the right hand-side of this equation equals the genus of F'. That is, F' will be seen to realize the lower bound from (A) on the dihedral genus 𝔤_p(K, ρ) of K. 	
		
	Since X is a definite manifold,  rk  H_2(X)=|σ(X)|. By the proof of (A), X is simply connected; by Poincaré duality we have χ(X)=2+ H_2(X) and hence 
    |Ξ_p(K, ρ)|=|σ(X)|= χ(X)-2.
  On the other hand, denoting by g(F) the genus of F, by  Equation <ref> we have 
    χ(X)=2p-p-12(2-2g(F))-p-12.
 Putting these two equations together, we conclude that 
    g(F)= |Ξ_p(K,ρ)|p-1-12.
 
	By assumption, the coloring ρ extends over F', so  
    g(F')≥𝔤_p(K, ρ)≥|Ξ_p(K,ρ)|p-1-12.
 Thus, F' realizes the p-dihedral genus of K.
	
	
		In the second part of the theorem, we assume in addition that |σ(K)| = 2|Ξ_p(K, ρ)|/p-1 -1, where σ(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 |σ(K)|/2.

	
	The additional assumption here can be rewritten as |σ(K)|=2g_p(K,ρ).  Murasugi's signature bound <cit.> states that g_4(K)≥ |σ(K)|/2.
	Thus, we have g_4(K)≥g_p(K,ρ).  But g_4(K)≤g_p(K)≤g_p(K,ρ) in general, so g_4(K)= g_p(K).
	

	
Our aim is to show that the equalities		
    g_4(K_m)=𝔤_3(K_m)=|Ξ_3(K_m, ρ_m)|2 -1/2=m
 	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 <ref>, it suffices to show that 
	 
		
  * Each K_m is the boundary of a homotopy-ribbon surface F_m' such that g_3(K)=g(F_m'), and
		 
  * The signature σ(K_m) satisfies the equality 
    |σ(K)|=2|Ξ_p(K, ρ)|/p-1 -1
 for p=3.
		 
		
	We first address (1).	Surfaces F_m' realizing the lower bound on dihedral homotopy-ribbon genus for the knots K_m are constructed in the proof of Proposition <ref>: we have shown g(F_m')=m and |Ξ_3(K_m)|=2m+1, so
		
		
    |Ξ_p(K_m)|p-1-1/2=m.

		 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 g_p(K_m,ρ_m) and g_p(K_m). By construction, the surface F_m'⊂ B^4 obtained by deleting a small neighborhood of the singularity K_m is ribbon since A_m∪B_m only bounds the cone on K_m, while the unlinks B_m∪C_m and C_m∪A_m bound standard unknotted disks in B^4.
		
We now address (2).	We will compute the signature σ(K_m), and show it is equal to 2m=2|Ξ_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 <cit.>. 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: σ(K)=σ(G(K))-μ. 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,…,X_k.  Such a labelling for the K_m is shown in Figure  <ref>. The entries g_ij of the unreduced Goeritz matrix are computed as follows:
	
	
    g_ij= {[                                                        -∑η(c) i≠ j  and c  a double point incident to regions  X_i  and X_j;                                         -∑_s∈{1,…,k}∖{i} g_is                                                           i=j ]..

	
	The signs η(c) are computed as in Figure  <ref>; shaded area corresponds to “black” regions of the checkerboard coloring.
 	
	
	
	The unreduced Goeritz matrix of K_m is

    G'(K_m)=[ -2m-3    -2   -2m     0    -1;    -2    -3    -1     0     0;   -2m    -1 -2m-2    -1     0;     0     0    -1    -2    -1;    -1     0     0    -1    -2 ].



The Goeritz matrix G(K_m) is obtained by deleting the first row and column of G'(K_m).  The characteristic polynomial of this matrix is

    p_G(K_m)(λ)=(λ+3)(λ(λ+3)^2+2(λ+1)(λ+3)m+3).

Hence λ=-3 is an eigenvalue.  In addition, since m≥ 0, it is straightforward to verify that any root of the cubic factor must be negative (if λ is nonnegative, the cubic, as written above, is a sum of three nonnegative terms).  Hence, σ(G(K_m))=-4. 

 The correction term μ(K) in Gordon and Litherland's formula for σ(K) is computed as follows.  Each crossing c of K can be classified as type I or type II, as shown in Figure  <ref>.  Let μ(K)=∑_c η(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  <ref>. Hence σ(K_m)=-4+(4+2m)=2m. 

 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 <ref> whose 3-dihedral and topological 4-genus equal m. We will prove that, given a nontrivial ribbon knot γ, the knot K_m#γ has the desired property. The theorem follows by taking repeated connect sums of K_m with γ. 
	
Let γ denote any ribbon knot and let D⊂ B^4 be a ribbon disk with ∂ D=γ. The knot K_m#γ 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#γ are both equal to m since the knot is smoothly concordant to K_m. Next, remark that the given 3-coloring ρ_m of K_m  induces a 3-coloring ρ_γ of K_m#γ which restricts trivially to γ. Moreover, since ρ_m extends over F'_m, ρ_γ extends over the ribbon surface F_m'♮ D, where ♮ denotes boundary connected sum. 	Therefore, the ribbon 3-dihedral genus of K_m#γ 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




amsplain


