1-loop equals torsion for two-bridge knots

TL;DR

This paper proves that for hyperbolic 2-bridge knots, the 1-loop term of a certain power series associated with ideal triangulations equals the adjoint Reidemeister torsion, confirming a conjecture linking quantum invariants and topology.

Contribution

It establishes the conjecture for hyperbolic 2-bridge knots by combining existing work with explicit computations, advancing understanding of quantum invariants in knot theory.

Findings

Confirmed the conjecture for hyperbolic 2-bridge knots

Connected the 1-loop term to the adjoint Reidemeister torsion

Used explicit computations and prior theoretical results

Abstract

Motivated by the conjectured asymptotics of the Kashaev invariant, Dimofte and the first author introduced a power series associated to a suitable ideal triangulation of a cusped hyperbolic 3-manifold, proved that its constant (1-loop) term is a topological invariant and conjectured that it equals to the adjoint Reidemeister torsion. We prove this conjecture for hyperbolic 2-bridge knots by combining the work of Ohtsuki--Takata with an explicit computation.