Alexander Stoimenov
On some relations between hyperbolic volume and combinatorial knot invariants
Abstract: There have been so far at least two situations, in which the hyperbolic volume exhibits a relation to a combinatorily defined quantum algebra structure. The most important one is Kashaev's conjecture on values of colored Jones polynomials. Another correspondence was observed by Dunfield, namely that the volume linearly approximates a logarithm of the determinant of alternating knots. (Khovanov suggested an extension to non-alternating knots in terms of his generalization of the Jones polynomial.) I will start by briefly explaining a proof of Dunfield's conjecture (in the alternating case) using inequalities of Lackenby-Agol-Thurston. In weaker form these inequalities were preceeded by Brittenham, who showed that the canonical genus bounds the hyperbolic volume of the knot. The maximal volume can now be numerically calculated up to canonical genus 4, using a characterization in terms of volumes of links assigned to trivalent graphs, similar to Habiro's claspers. A new weight system-volume-conjecture claims a direct relation between these volumes and the $sl_N$ weight system polynomials of Vassiliev invariants. Latter have in turn some relevance in the enumeration of alternating knots by genus.