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

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 [18] 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 Apolynomials and use this information to identify factors of higher multiplicity in the y A-polynomial, which is the A-polynomial with multiplicities as defined by Boyer-Zhang. 57M27; 57M25, 57M05

Chirally cosmetic surgeries and Casson invariants

“…In this section, we recall a surgery formula of the SL(2, C) Casson invariant, denoted by λ SL(2,C) , based on [3].…”

“…Since no root of unity is a root of the Alexander polynomial ∆ K (t) = 2t 2 − 3t + 2, all slopes except the boundary slopes 0, −4, −10, are admissible. By [3] the total Culler-Shalen seminorm ||p/q|| of K is given by the formula…”

“…Let v 3 (K) be the primitive finite type invariant of degree three of a knot K in S 3 , normalized so that it takes the value 1 4 on the right-handed trefoil. Using the derivatives of the Jones polynomial V K (t) of K, we see that v 3 (K) is written by…”

Some numerical computations on Reidemeister torsion for homology 3-spheres obtained by Dehn surgeries along the figure-eight knot

“…Here we mention the Casson invariant and the SL(; C)-Casson invariant. Please see [1] and [4,2,3] for precise definitions and properties.…”

State invariants of 2-bridge knots

“…The boundary slopes of these surfaces have become increasingly important computationally, for example in the computation of Culler-Gordon-Luecke-Shalen semi-norms (see [8], [5], [9] and [13]), SL(2, C)-Casson invariants (see [9], [2], [3], and [4]), and A-polynomials (see [7], [5] and [4]). A motivating observation is that the Hatcher-Thurston formula for computing the boundary slope of an essential surface S with boundary a 2-bridge knot is 2(M −M 0 ), where M is a quantity computed from S and M 0 is an analogous quantity computed from a Seifert surface for the knot.…”