Title: Twisted Alexander polynomials of hyperbolic knots
Speaker: Nathan Dunfield
Speaker Info: UIUC
Special Note: There will not be a preseminar talk this week.
I will discuss a twisted Alexander polynomial naturally associated to a hyperbolic knot in the 3-sphere via a lift of its holonomy representation to SL(2, C). It is an unambiguous symmetric Laurent polynomial whose coefficients lie in a number field coming from the hyperbolic geometry. The polynomial can be defined as the Reidmeister torsion of a certain acyclic chain complex, namely the first homology of the knot exterior with coefficients twisted by the holonomy representation tensored with the abelianization map. This polynomial contains much topological information, for instance about the simplest surface bounded by the knot. I will present computations showing that for all 313,209 hyperbolic knots in S^3 with at most 15 crossings it in fact gives perfect such information, in contrast with a related polynomial coming from the adjoint representation of SL(2, C) on it's Lie algebra. No familiarity with low-dimensional topology will be assumed. This is joint work with Stefan Friedl and Nicholas Jackson.Date: Monday, October 10, 2011