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

  Published:2015-10-29
 Printed: Feb 2016
  • Hans Ulysses Boden,
    Mathematics & Statistics, McMaster University, Hamilton, Ontario
  • Cynthia L Curtis,
    Mathematics & Statistics, The College of New Jersey, Ewing, NJ
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Abstract

In this paper, we extend the definition of the ${SL(2, {\mathbb C})}$ 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 ${SL(2, {\mathbb C})}$ 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.
Keywords: Knots, 3-manifolds, character variety, Casson invariant, $A$-polynomial Knots, 3-manifolds, character variety, Casson invariant, $A$-polynomial
MSC Classifications: 57M27, 57M25, 57M05 show english descriptions Invariants of knots and 3-manifolds
Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Fundamental group, presentations, free differential calculus
57M27 - Invariants of knots and 3-manifolds
57M25 - Knots and links in $S^3$ {For higher dimensions, see 57Q45}
57M05 - Fundamental group, presentations, free differential calculus
 

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