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.