The \(SL(2, \mathbb C)\) Casson invariant for Dehn surgeries on two-bridge knots. (English) Zbl 1267.57015
In the paper under review the goal is to provide computations of the \(SL(2,\mathbb C)\) Casson invariant for 3-manifolds obtained by Dehn surgery on a two-bridge knot. To do it, the authors consider a formula given by C. L. Curtis [Topology 40, No.4, 773–787 (2001); erratum ibid. 42, 929 (2003; Zbl 0979.57006)] which involves computing the Culler-Shalen seminorms. As an application, using the classification of exceptional Dehn surgeries on two-bridge knots given by M. Brittenham and Y.-Q. Wu [Commun. Anal. Geom. 9, No. 1, 97–113 (2001; Zbl 0964.57013)], it is proved that nearly all 3-manifolds given by a nontrivial \(p/q\)-Dehn surgery on a hyperbolic two-bridge knot \(K\) have nontrivial \(SL(2,\mathbb C)\) Casson invariant. A second application is to \(A\)-polynomials. For a given knot \(K\), let \(A_K(M,L)\) be the polynomial originally defined by D. Cooper et al., [Invent. Math. 118, No.1, 47-84 (1994; Zbl 0842.57013)], and let \(\widehat A_K(M,L)\) be the polynomial defined by S. Boyer and X. Zhang [J. Differ. Geom. 59, No. 1, 87–176 (2001; Zbl 1030.57024)]. By using the relationships between the \(SL(2,\mathbb C)\) Casson invariant, Culler-Shalen seminorms and the \(\widehat A\) polynomial, closed formulas are given for the \(L\)-degree of \(\widehat A_K(M,L)\) for all two bridge knots and the \(M\)-degree of \(\widehat A_K(M,L)\) for double twist knots. Some knots are identified for which of \(A_K(M,L) \not= \widehat A_K(M,L)\).
Reviewer: Lorena Armas-Sanabria (México D. F.)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 57M05 | Fundamental group, presentations, free differential calculus |
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