Knot invariants from the combinatorics of triangulations.
by
Rinat Kashaev
University of Geneva

The usual approach to construction of (quantum) knot invariants uses the combinatorics of knot diagrams combined with the theorem of Reidemeister. Algebraically the problem is reduced to the theory of the (quantum) Yang-Baxter equation which realizes the most important of the Reidemeister moves. I will discuss a similar approach to construction of knot invariants based on the combinatorics of triangulations: either ideal triangulations of knot complements or triangulations of the three-sphere where a knot is realized as a one-dimensional subcomplex. The counterpart of the theorem of Reidemeister in this case is the theorem of Pachner and its relative versions. Algebraically one has to deal with the theory of the Pentagon equation which realizes the most important of the Pachner moves.

In relation to Pentagon equation I will also touch upon the following subjects: Hopf algebras and their representation theory, algebras with families of comultiplications, operator generalizations of the dilogarithm. I should remark that the form of the Pentagon equation to be discussed is seemingly very different from the Pentagon equation within the theory of quasi-Hopf algebras.

Date received: May 25, 2009