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A volumish theorem for alternating virtual links

Abhijit Champanerkar, Ilya Kofman

TL;DR

This work generalizes the Dasbach–Lin volumish bounds from classical alternating knots to alternating links on surfaces and virtual links by introducing the homological twist number $\tau_F(K)$, which is captured by specific coefficients of the reduced Jones–Krushkal polynomial $J_K(t,z)$. Central to the approach are the Krushkal polynomial $p_G(x,y,u,v)$ and its surface-embedded graph interpretations, which yield invariant coefficients $\mu$, $\lambda$, and, when present, $\gamma$ that encode the cycle ranks of reduced Tait graphs. The authors prove that $\tau_F(K)$ is a link invariant under suitable conditions and derive linear volume bounds for hyperbolic volume in terms of $\tau_F(K)$, with genus-dependent adjustments, thereby extending volumish results to alternating virtual links. The results leverage the interplay between graph polynomials on surfaces and 3-manifold hyperbolicity, providing a computable bridge from diagrammatic invariants to geometric volume estimates in thickened surfaces. Examples on torus diagrams illustrate the precise extraction of invariants from $J_K(t,z)$ and confirm the theoretical bounds.

Abstract

Dasbach and Lin proved a "volumish theorem" for alternating links. We prove the analogue for alternating link diagrams on surfaces, which provides bounds on the hyperbolic volume of a link in a thickened surface in terms of coefficients of its reduced Jones-Krushkal polynomial. Along the way, we show that certain coefficients of the 4-variable Krushkal polynomial express the cycle rank of the reduced Tait graph on the surface.

A volumish theorem for alternating virtual links

TL;DR

This work generalizes the Dasbach–Lin volumish bounds from classical alternating knots to alternating links on surfaces and virtual links by introducing the homological twist number , which is captured by specific coefficients of the reduced Jones–Krushkal polynomial . Central to the approach are the Krushkal polynomial and its surface-embedded graph interpretations, which yield invariant coefficients , , and, when present, that encode the cycle ranks of reduced Tait graphs. The authors prove that is a link invariant under suitable conditions and derive linear volume bounds for hyperbolic volume in terms of , with genus-dependent adjustments, thereby extending volumish results to alternating virtual links. The results leverage the interplay between graph polynomials on surfaces and 3-manifold hyperbolicity, providing a computable bridge from diagrammatic invariants to geometric volume estimates in thickened surfaces. Examples on torus diagrams illustrate the precise extraction of invariants from and confirm the theoretical bounds.

Abstract

Dasbach and Lin proved a "volumish theorem" for alternating links. We prove the analogue for alternating link diagrams on surfaces, which provides bounds on the hyperbolic volume of a link in a thickened surface in terms of coefficients of its reduced Jones-Krushkal polynomial. Along the way, we show that certain coefficients of the 4-variable Krushkal polynomial express the cycle rank of the reduced Tait graph on the surface.

Paper Structure

This paper contains 5 sections, 7 theorems, 47 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The Krushkal polynomial
  3. The homological twist number
  4. The Jones-Krushkal polynomial
  5. Examples

Key Result

Theorem 1.1

For a closed orientable surface $F$ of genus $g\geq 1$, let $K$ be a non-split oriented link in $F\times I$ that admits a cellularly embedded, strongly reduced WGA diagram $D$ on $F\times\{0\}$. Let $\tau_F(K)$ be the homological twist number of $D$. Let $J_K(t,0)=a_nt^n+\cdots +a_mt^m$, with sub-ex $\tau_F(K)$ is an invariant of $K$ in $F\times I$, and $F\times I-K$ is hyperbolic with

Figures (7)

  • Figure 1: An alternating link diagram (left) and its Tait graph $G_A$ (right) on the torus Krushkal.
  • Figure 2: Different kinds of cycles in the Tait graph are shown in different colors. From left to right: loop giving the diagram representativity 2 (cyan), nugatory crossing (green), null-homologous 2-cycle (pink), genus-generating loops with representativity 4 (blue and red).
  • Figure 3: Two alternating link diagrams are projected on $F$, partly shown. In both cases, the red crossing and the blue crossing are homologically twist-equivalent. One Tait graph has homologous loops (left) or a null-homologous $2$--cycle (right). Neither pair of crossings forms a twist region on $F$.
  • Figure 4: For $D$ on the torus (left), states $s_A$ (middle) and $s_B$ (right) are shown. Here, $|s_A|=2,\, |s_B|=1,\, r(s_A)=r(s_B)=1,\, k(s_A)=1,\, k(s_B)=0$.
  • Figure 5: The $2\times 2$ square weave on the torus as a virtual link, its diagram $D$ with $\tau_F(D)=4$, and self-dual Tait graphs $G_A=G_B$ shown as a ribbon graph.
  • ...and 2 more figures

Theorems & Definitions (21)

  • Theorem 1.1
  • Definition 2.1: Krushkal
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 3.1
  • Definition 3.2
  • ...and 11 more