Jacob Rasmussen, Cambridge University

Title: SL_2(R) representations of knot groups and an extended Lin invariant

Abstract: I'll discuss some work in progress with Nathan Dunfield. In the 90's, X.S. Lin defined a Casson-style invariant of knots by counting SU(2) representations of the knot group with fixed holonomy along the meridian. This invariant was subsequently shown to be equivalent to the Levine-Tristam signature. I'll describe a variant of Lin's construction which counts both SU(2) and SL_2(R) representations of the knot group. Lifting to the universal cover widetilde{SL}_2(R) allows us to define an enhanced Lin invariant, which is a Laurent polynomial rather than just an integer. I'll give some applications (including a new proof of the Riley conjecture) and discuss a conjecture about the enhanced invariant of L-space knots.