The SL(2,C) Casson invariant for Dehn surgeries on two-bridge knots

Boden, Hans; Curtis, Cynthia

doi:10.2140/agt.2012.12.2095

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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

DOI: 10.2140/agt.2012.12.2095

Abstract

We investigate the behavior of the $SL (2, ℂ)$ 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 $SL (2, ℂ)$ Casson invariant for the $3$ –manifolds obtained by $(p ∕ q)$ –Dehn surgery on such knots. These results are applied to prove nontriviality of the $SL (2, ℂ)$ 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 $Â$ –polynomial, which is the $A$ –polynomial with multiplicities as defined by Boyer–Zhang.

Keywords

Casson invariant, character variety, two-bridge knot

References

Publication

Received: 21 May 2012

Revised: 5 July 2012

Accepted: 28 July 2012

Published: 3 December 2012

Authors

Hans U Boden
Mathematics & Statistics McMaster University 1280 Main St. W. Hamilton, Ontario L8S 4K1 Canada
http://www.math.mcmaster.ca/~boden/index.html
Cynthia L Curtis
Mathematics & Statistics The College of New Jersey PO Box 7718 Ewing, NJ 08628 USA

Volume 12, issue 4 (2012)