On symmetric matrices associated with oriented link diagrams

Monday, 15 January, 2018

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Let D be an oriented link diagram with the set of regions rD. 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 τD 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.


Rinat Kashaev