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

Boden, Hans U. ; Curtis, Cynthia L.

Canadian journal of mathematics, Tome 68 (2016) no. 1, pp. 3-23

Résumé

In this paper, we extend the definition of the $SL\left( 2,\,\mathbb{C} \right)$ 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 the $SL\left( 2,\,\mathbb{C} \right)$ 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 $\widehat{A}$ -polynomial and trivial $SL\left( 2,\,\mathbb{C} \right)$ Casson knot invariant, showing that neither of these invariants detect the unknot.

Détail

DOI : 10.4153/CJM-2015-025-5

Mots-clés : 57M27, 57M25, 57M05, knots, 3-manifolds, character variety, Casson invariant, A-polynomial