The SL$(2, C)$ Casson Invariant for Knots and the $\hat{A}$-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 Â-polynomial of K . We prove a product formula for the Â-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 Â-polynomial and trivial SL (2,ℂ) Casson knot invariant, showing that neither of these invariants detect the unknot.