Mahler's measure and the volume of hyperbolic manifolds

The logarithmic Mahler measure m(P) of a polynomial P(x,y) is the average of \log|P(x,y)| over the unit 2-torus. Given a knot K, Cooper, Culler, Gilet, Long and Shalen have defined a certain polynomial invariant A_K(x,y) of K which is defined in terms of representations of the fundamental group of K. For hyperbolic knots, the complement of K can be given the structure of a 3-manifold which has a well-defined volume V(K), and it appears that \pi m(A_K) has an interesting relationship with V(K). For certain simple knots, \pi m(A_K) = V(K), but the general picture is considerably more interesting. We will describe how m(A_K) seems to be related to another invariant of the knot, the Borel regulator, an arithmetic invariant introduced by Neumann and Yang. We will present a precisely formulated conjecture supported by a number of highly accurate computations.

Refreshments will be served in Math Annex Room 1115, 3:15 p.m.