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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

Hans U. Boden
Mathematics & Statistics, McMaster University, Hamilton, ON e-mail:
Cynthia L. Curtis
Mathematics & Statistics, The College of New Jersey, Ewing, NJ, USA e-mail:
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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.

Research Article
Copyright © Canadian Mathematical Society 2016


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