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2012 The $\mathit{SL}(2,{\mathbb C})$ Casson invariant for Dehn surgeries on two-bridge knots
Hans Boden, Cynthia Curtis
Algebr. Geom. Topol. 12(4): 2095-2126 (2012). DOI: 10.2140/agt.2012.12.2095


We investigate the behavior of the SL(2,) Casson invariant for 3–manifolds obtained by Dehn surgery along two-bridge knots. Using the results of Hatcher and Thurston, and also results of Ohtsuki, we outline how to compute the Culler–Shalen seminorms, and we illustrate this approach by providing explicit computations for double twist knots. We then apply the surgery formula of Curtis [Topology 40 (2001), 773–787] to deduce the SL(2,) Casson invariant for the 3–manifolds obtained by (pq)–Dehn surgery on such knots. These results are applied to prove nontriviality of the SL(2,) Casson invariant for nearly all 3–manifolds obtained by nontrivial Dehn surgery on a hyperbolic two-bridge knot. We relate the formulas derived to degrees of A–polynomials and use this information to identify factors of higher multiplicity in the –polynomial, which is the A–polynomial with multiplicities as defined by Boyer–Zhang.


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Hans Boden. Cynthia Curtis. "The $\mathit{SL}(2,{\mathbb C})$ Casson invariant for Dehn surgeries on two-bridge knots." Algebr. Geom. Topol. 12 (4) 2095 - 2126, 2012.


Received: 21 May 2012; Revised: 5 July 2012; Accepted: 28 July 2012; Published: 2012
First available in Project Euclid: 19 December 2017

zbMATH: 1267.57015
MathSciNet: MR3020202
Digital Object Identifier: 10.2140/agt.2012.12.2095

Primary: 57M27
Secondary: 57M05 , 57M25

Keywords: Casson invariant , character variety , two-bridge knot

Rights: Copyright © 2012 Mathematical Sciences Publishers


Vol.12 • No. 4 • 2012
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