Don Zagier, Max Planck Insitute Bonn and ICTP
Title: Knot invariants, Nahm sums, and units coming from algebraic $K$-theory
Abstract: Some years ago I formulated a conjecture--based at that time purely on numerical evidence--giving a surprising modularity property of the Kashaev invariant of a hyperbolic knot. This conjecture has now been proved for the original case (figure 8 knot) and verified numerically for several other knots in joint work with Stavros Garoufalidis. We also found an unexpected connection between the asymptotic series occurring here and the radial asymptotics of certain Nahm sums near roots of unity. In both cases one sees mysterious algebraic numbers occurring as coefficients in the expansions, and in particular a pre-factor to the entire expansion that is a root of a unit in a cyclotomic extension of the ground field. In joint work with Frank Calegari, we have developed an
independent way to construct such units from an arbitrary element of the Bloch group of the ground field that agree (provably in the Nahm sum case and conjecturally in the Kashaev invariant case) with the units that we had found. I will describe some of these results and, if time permits, also generalizations to higher $K$-groups and a possible application to Nahm's conjecture relating the modularity of his sums to algebraic $K$ theory.