The SL(2,C) Casson invariant for knots and the Aˆ-polynomial

Abstract

In this paper, we extend the definition of the SL(2, C) Casson invariant to arbitrary knots K in integral homology 3-spheres and relate it to the m-degree of the (A) over cap -polynomial of K. We prove a product formula for the (A) over cap -polynomial of the connected sum K-1#K-2 of two knots in S-3 and deduce additivity of the SL(2, C) Casson knot invariant under connected sums for a large class of knots in S-3. We also present an example of a nontrivial knot K in S-3 with trivial (A) over cap -polynomial and trivial SL(2, C) Casson knot invariant, showing that neither of these invariants detect the unknot.