The SL\((2,C)\) Casson invariant for knots and the \(\widehat{A}\)-polynomial (Q2786449)

The authors avoid the limitation to small knots and, using limits, present a version of the \(SL(2,\mathbb{C})\) Casson Invariant [\textit{C. L. Curtis}, Topology 40, No. 4, 773--787 (2001; Zbl 0979.57006); erratum ibid. 42, 929 (2003)] for knots in integral homology spheres. They state their results are applicable to those knots in integral homology spheres where the character scheme of the knot's complement is reduced (a condition they note is not satisfied for all 3-manifolds).NEWLINENEWLINEThey relate their results to the m-degree of the \( \widehat{A}\)-polynomial [\textit{S. Boyer} and \textit{X. Zhang}, J. Differ. Geom. 59, No. 1, 87--176 (2001; Zbl 1030.57024)].NEWLINENEWLINEThey exhibit a non-trivial knot in \(S^3\) with trivial \(SL(2,\mathbb{C})\) Casson invariant, and trivial \(\widehat{A}\)-polynomial. Formulae for the invariants of certain knot sums are also presented.