THE SL₂(ℂ) CASSON INVARIANT FOR SEIFERT FIBERED HOMOLOGY SPHERES AND SURGERIES ON TWIST KNOTS

Abstract

We derive a simple closed formula for the SL 2 (C) Casson invariant for Seifert fibered homology 3-spheres using the correspondence between SL 2 (C) character varieties and moduli spaces of parabolic Higgs bundles of rank two. These results are then used to deduce the invariant for Dehn surgeries on twist knots by combining computations of the Culler-Shalen norms with the surgery formula for the SL 2 (C) Casson invariant.

Gauge theory and knot homologies

“…In particular, here we will be interested in the most basic type of surface operators, which correspond to the singular behavior of the gauge field A and the Higgs field φ of the form [23]: For example, in complex structure I it corresponds to counting parabolic Higgs bundles, a fact that has already been used e.g. in [62] for studying the SL(2,C) Casson invariant for Seifert fibered homology spheres.…”

The SL(2, C) Casson Invariant for Knots and the Â-polynomial

“…On closed 3-manifolds Σ, the invariant λ SL(2,C) (Σ) ≥ 0 is nonnegative, satisfies λ SL(2,C) (−Σ) = λ SL(2,C) (Σ) under orientation reversal, and is additive under connected sum of Z/2-homology 3-spheres (cf. Theorem 3.1, [2]). If Σ is hyperbolic, then λ(Σ) > 0 by Proposition 3.2 of [12].…”

“…We briefly recall some useful properties of the SL(2, C) Casson invariant and we refer to [12] and [2] for further details.…”

State invariants of 2-bridge knots

“…Figure 1. The surfaces S and S corresponding to [2,3] We remark that any Seifert matrix V for a nontrivial knot is not symmetric. The asymmetry in V arises precisely in the off-diagonal entries v ij = v ji whenever the oriented intersection number of x i and x j is ±1.…”