On symmetric matrices associated with oriented link diagrams

TL;DR

This paper introduces a symmetric matrix invariant derived from oriented link diagrams, which potentially relates to the Tristram-Levine signature function and offers a new algebraic perspective in knot theory.

Contribution

It defines a novel symmetric matrix invariant for oriented links based on a modified S-equivalence, connecting algebraic properties to classical link invariants.

Findings

The matrix invariant is well-defined under certain equivalences.

Conjecturally, the negative signature relates to the Tristram-Levine signature.

Provides a new algebraic tool for studying link signatures.

Abstract

Let $D$ be an oriented link diagram with the set of regions $\operatorname{r}_{D}$. We define a symmetric map (or matrix) $\operatorname{\tau}_{D}\colon\operatorname{r}_{D}\times \operatorname{r}_{D} \to \mathbb{Z}[x]$ that gives rise to an invariant of oriented links, based on a slightly modified $S$-equivalence of Trotter and Murasugi in the space of symmetric matrices. In particular, for real $x$, the negative signature of $\operatorname{\tau}_{D}$ corrected by the writhe is conjecturally twice the Tristram--Levine signature function, where $2x=\sqrt{t}+\frac1{\sqrt{t}}$ with $t$ being the indeterminate of the Alexander polynomial.