On symmetric matrices associated with oriented link diagrams

Rinat Kashaev

Published 2018-01-15Version 1

Let $D$ be an oriented link diagram with $\operatorname{c}_{+}$ (respectively $\operatorname{c}_{-}$) positive (respectively negative) crossings and 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. Namely, denoting by $\operatorname{SM}(R)$ the set of all symmetric square matrices of all sizes (including the empty matrix $[]$) over a commutative ring $R$, let $SR$ be the equivalence relation in the set $\mathbb{Z}^2\times\operatorname{SM}(R)$ generated by the congruences $(m,A)\sim (m, M^{\mathrm{T}}AM)$ for all invertible matrices $M$ of the same size as that of $A$ and the modified $S$-equivalences $\left(m, \begin{bmatrix} 0&1 1&r \end{bmatrix}\boxplus A \right)\sim\left(m+(1,1),A\right)$ for all $r\in R$, and $\left(m, [\epsilon] \boxplus A\right)\sim\left(m+\frac{(1+\epsilon,1-\epsilon)}{1+\epsilon^2},A\right)$ for all $\epsilon \in \{0,\pm1\}$. Then, the $SR$-equivalence class of $(-\operatorname{c}_{+},-\operatorname{c}_{-},\operatorname{\tau}_{D})$, where $R= \mathbb{Z}\!\left[x,\frac1{2x+1}\right]$ or any quotient ring thereof, is an invariant of oriented links. In the particular case $R=\mathbb{R}$, $(-\operatorname{c}_{+},-\operatorname{c}_{-},\operatorname{\tau}_{D})$ is $SR$-equivalent to a unique $(\kappa_0(x),\kappa_1(x),[])$ for all $x\ne-1/2$. Identifying $2x= \sqrt{t}+\frac1{\sqrt{t}}$, where $t$ is the indeterminate of the Alexander polynomial, $(\kappa_1(x)-\kappa_0(x))/2$ is conjecturally the Tristram--Levine signature function.