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

TL;DR

This paper extends the $SL_2(\mathbb{C})$ Casson invariant to all knots in homology 3-spheres, relates it to the $\widehat{A}$-polynomial, and explores their properties under connected sum, revealing limitations in detecting the unknot.

Contribution

It introduces a generalized $SL_2(\mathbb{C})$ Casson invariant for knots, establishes a product formula for the $\widehat{A}$-polynomial, and demonstrates cases where these invariants fail to detect the unknot.

Findings

Proved a product formula for the $\widehat{A}$-polynomial of connected sums.

Showed the $SL_2(\mathbb{C})$ Casson knot invariant is additive under connected sum for many knots.

Provided an example of a nontrivial knot with trivial $\widehat{A}$-polynomial and Casson invariant.

Abstract

In this paper, we extend the definition of the $SL_2(\Bbb C)$ Casson invariant to arbitrary knots $K$ in integral homology 3-spheres and relate it to the $m$-degree of the $\widehat{A}$-polynomial of $K$. We prove a product formula for the $\widehat{A}$-polynomial of the connected sum $K_1 \# K_2$ of two knots in $S^3$ and deduce additivity of $SL_2(\Bbb C)$ 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 $\widehat{A}$-polynomial and trivial $SL_2(\Bbb C)$ Casson knot invariant, showing that neither of these invariants detect the unknot.