The SL(2,C) Casson invariant for knots and the $\hat{A}$-polynomial

Abstract

In this paper, we extend the definition of the $S{L}_2(\mathbb{C})$ Casson invariant to arbitrary knots $K$ in integral homology 3-spheres and relate it to the $m$-degree of the $\hat{A}$-polynomial of $K$. We prove a product formula for the $\hat{A}$-polynomial of the connected sum $K_1\#K_2$ of two knots in $S^3$ and deduce additivity of $S{L}_2(\mathbb{C})$ Casson knot invariant under connected sum for a large class of knots in $S^3$. We also present an example of a nontrivial knot $K$ in $S^3$ with trivial $\hat{A}$-polynomial and trivial $S{L}_2(\mathbb{C})$ Casson knot invariant, showing that neither of these invariants detect the unknot.

Final version; new material includes a discussion of the character variety of Whitehead link

References in corpus (5)

• On representation varieties of 3-manifold groups
• On the AJ conjecture for knots
• The SL(2,C) Casson invariant for Dehn surgeries on two-bridge knots
• Closed surfaces and character varieties
• Non-triviality of the $M$-degree of the $A$-polynomial

Cited by in corpus (1)

• A sheaf-theoretic SL(2,C) Floer homology for knots