Reidemeister torsion of a 3-manifold obtained by an integral Dehn-surgery along the figure-eight knot

Abstract

Let M be a 3-manifold obtained by a Dehn-surgery along the figure-eight knot. We give a formula of the Reisdemeisiter torsion of M for any SL(2; C)-irreducible representation. It has a rational expression of the trace of the image of the meridian.2010 Mathematics Subject Classification. 57M27.

Reidemeister Torsion and Dehn Surgery on Twist Knots

“…(2) Theorem 2 generalizes the formula for the Reidemeister torsion of the 3-manifold obtained by a p q -surgery on the figure eight knot by Kitano [Ki1]. Example 1.2.…”

“…In a recent paper Kitano [Ki1] gives a formula for the Reidemeister torsion of the 3manifold obtained by a Dehn surgery on the figure eight knot. In this paper we generalize his result to all twist knots.…”

“…Recall that λ is the canonical longitude corresponding to the meridian µ = a. If tr ρ(λ) = 2, then by [Ki1] (see also [Ki2,Ki3]) the Reidemeister torsion of M is given by (5.1) τ ρ (M) = τ ρ (E K ) 2 − tr ρ(λ) .…”

Twisted Alexander polynomials of genus one two-bridge knots

“…Recall that λ is the canonical longitude corresponding to the meridian µ = a. If tr ρ(λ) = 2, then by [Ki1] (see also [Ki2,Ki3]) the Reidemeister torsion of M is given by (4.3) τ ρ (M) = τ ρ (K) 2 − tr ρ(λ) .…”

Reidemeister torsion, hyperbolic three-manifolds, and character varieties

“…This proposition can be proved using Mayer-Vietoris (Proposition 2.10) to the pair (M 3 , D 2 × S 1 ). The proof can be found for instance in [49] for the figure eight knot, and in [94] for twist knots, but it applies to every cusped manifold, and it is essentially done too in [82,Proposition 4.10]. The only computation required is the torsion of the solid torus added by Dehn filling, or its core geodesic, which is (by Example 2.16):…”