On symmetric matrices associated with oriented link diagrams

Abstract

Let $D$ be an oriented link diagram with the set of regions $r_D$. We define a symmetric map (or matrix) ${\tau }_D$ : \( \operatorname{r}_{D}$$\times \) $r_D$ $\to$ $\mathbb{Z}[x]$ 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 $2x=\sqrt{t}+\frac{1}{\sqrt{t}}$ with $t$ being the indeterminate of the Alexander polynomial.