The SL(2, C) Casson Invariant for Knots and the Â-polynomial

Abstract

AbstractIn 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 Â-polynomial of K. We prove a product formula for the Â-polynomial of the connected sum K1#K2 of two knots in S3 and deduce additivity of the SL(2,ℂ) Casson knot invariant under connected sums for a large class of knots in S3. We also present an example of a nontrivial knot K in S3 with trivial Â-polynomial and trivial SL(2,ℂ) Casson knot invariant, showing that neither of these invariants detect the unknot.