On symmetric matrices associated with oriented link diagrams

Abstract

Let $D$ be an oriented link diagram with the set of regions $\operatorname{r}_{D}$. We define a symmetric map (or matrix) $\tau_D$ : \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 $\tau_D$ 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.