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The $\mathit{SL}(2,{\mathbb C})$ Casson invariant for Dehn surgeries on two-bridge knots

Hans U Boden and Cynthia L Curtis

Algebraic & Geometric Topology 12 (2012) 2095–2126

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.

Casson invariant, character variety, two-bridge knot
Received: 21 May 2012
Revised: 5 July 2012
Accepted: 28 July 2012
Published: 3 December 2012
Hans U Boden
Mathematics & Statistics
McMaster University
1280 Main St. W.
Hamilton, Ontario L8S 4K1 Canada
Cynthia L Curtis
Mathematics & Statistics
The College of New Jersey
PO Box 7718
Ewing, NJ 08628 USA