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A diagrammatic computation of abelian link invariants

David Cimasoni, Livio Ferretti, Jessica Liu

TL;DR

The paper provides a diagrammatic method to compute abelian link invariants for μ-colored links from a single symmetric matrix $\tau_D(x)$ derived from a colored diagram. By studying the evaluated matrix $\widetilde{\tau}_D(\omega)$ and $\widetilde{\tau}_D(t^2)$, it derives explicit formulas for the multivariable signature $\sigma_L(\omega)$, nullity $\eta_L(\omega)$, and the Conway function $\nabla_L$, unifying and extending Kashaev's single-variable approach to the multivariable case. It also establishes a multivariable Kauffman determinantal model linking $\tau_D(t^2)$ to a product of region- and crossing-labeled matrices, and discusses connections to Zibrowius's state-sum models. Finally, the work analyzes the Alexander module, showing a square presentation in the μ=1 case and explaining obstructions in higher colors, thereby clarifying the diagrammatic computability of these abelian invariants from C-complex data.

Abstract

We show how the multivariable signature and Alexander polynomial of a colored link can be computed from a single symmetric matrix naturally defined from a colored link diagram. In the case of a single variable, it coincides with the matrix introduced by Kashaev in [arXiv:1801.04632], which was recently proven to compute the Levine-Tristram signature and the Alexander polynomial of oriented links [arXiv:2311.01923, arXiv:2310.16729]. As a corollary, we obtain a multivariable extension of Kauffman's determinantal model of the Alexander polynomial, recovering a result of Zibrowius [arXiv:1601.04915v1].

A diagrammatic computation of abelian link invariants

TL;DR

The paper provides a diagrammatic method to compute abelian link invariants for μ-colored links from a single symmetric matrix derived from a colored diagram. By studying the evaluated matrix and , it derives explicit formulas for the multivariable signature , nullity , and the Conway function , unifying and extending Kashaev's single-variable approach to the multivariable case. It also establishes a multivariable Kauffman determinantal model linking to a product of region- and crossing-labeled matrices, and discusses connections to Zibrowius's state-sum models. Finally, the work analyzes the Alexander module, showing a square presentation in the μ=1 case and explaining obstructions in higher colors, thereby clarifying the diagrammatic computability of these abelian invariants from C-complex data.

Abstract

We show how the multivariable signature and Alexander polynomial of a colored link can be computed from a single symmetric matrix naturally defined from a colored link diagram. In the case of a single variable, it coincides with the matrix introduced by Kashaev in [arXiv:1801.04632], which was recently proven to compute the Levine-Tristram signature and the Alexander polynomial of oriented links [arXiv:2311.01923, arXiv:2310.16729]. As a corollary, we obtain a multivariable extension of Kauffman's determinantal model of the Alexander polynomial, recovering a result of Zibrowius [arXiv:1601.04915v1].
Paper Structure (10 sections, 7 theorems, 65 equations, 10 figures)

This paper contains 10 sections, 7 theorems, 65 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Generalized Seifert surfaces and matrices
  4. Signatures and nullities of colored links
  5. The Conway function of a colored link
  6. Proofs of the main results
  7. Signatures and nullities
  8. The Conway function
  9. The multivariate Kauffman model
  10. The Alexander module

Key Result

Theorem 1

Let $D$ be an arbitrary $\mu$-colored diagram for a $\mu$-colored link $L$.

Figures (10)

  • Figure 1: A $2$-colored diagram $D$ for a $2$-colored link $L=L_1\cup L_2$. The crossings are labelled 1 through 5 and the regions are labelled $a$ through $g$. Crossings $1$ to $4$ are positive and bichromatic, while crossing $5$ is negative and monochromatic. The marked point on $L_1$ will serve a further purpose.
  • Figure 2: A crossing $v$ together with the corresponding $4\times 4$ minor of $\tau_v(x)$. The incoming left strand is of color $j$, the incoming right strand of color $k$, and the four adjacent regions are $a, b, c,$ and $d$.
  • Figure 3: The labels in the definition of $K_D$ around a vertex $v$ with $s=\operatorname{sgn}(v)$.
  • Figure 4: A positive clasp intersection (left), and a negative one (right).
  • Figure 7: The five cycles of $S$ near a monochromatic crossing (left, one image) and the four cycles of $S$ near a bichromatic crossing (right, four images). The labels $a, b, c, d, v$ for the regions are used for the local linking matrices in Figure \ref{['fig:locallinking']}.
  • ...and 5 more figures

Theorems & Definitions (19)

  • Definition 1
  • Example 1
  • Theorem 1
  • Example 2
  • Corollary 1
  • Example 3
  • Definition 2
  • Definition 3: CimFlo08
  • Proposition 1
  • proof
  • ...and 9 more