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The SL(2, C) Casson Invariant for Knots and the Â-polynomial
Published online by Cambridge University Press: 20 November 2018
Abstract
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.
- Type
- Research Article
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- Copyright
- Copyright © Canadian Mathematical Society 2016
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