Abstract view
The SL$(2, C)$ Casson Invariant for Knots and the $\hat{A}$polynomial


Published:20151029
Printed: Feb 2016
Hans Ulysses Boden,
Mathematics & Statistics, McMaster University, Hamilton, Ontario
Cynthia L Curtis,
Mathematics & Statistics, The College of New Jersey, Ewing, NJ
Abstract
In this paper, we extend the definition of the ${SL(2, {\mathbb C})}$ Casson
invariant
to arbitrary knots $K$ in integral homology 3spheres 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 ${SL(2, {\mathbb C})}$ 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 $\widehat{A}$polynomial and trivial
${SL(2, {\mathbb C})}$ Casson knot invariant, showing that neither of these invariants
detect the unknot.