On symmetric matrices associated with oriented link diagrams

Let $D$ be an oriented link diagram with the set of regions ${\mathrm{r}}_D$. 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 ${\mathrm{\tau }}_D$ 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.