The twisted 1-loop invariant and the Jacobian of Ptolemy coordinates

TL;DR

This paper introduces a new way to define the twisted 1-loop invariant using Ptolemy coordinates and proves its equivalence to the adjoint twisted Alexander polynomial for hyperbolic once-punctured torus bundles, confirming a conjecture.

Contribution

It provides an alternative definition of the twisted 1-loop invariant and establishes its equality with the adjoint twisted Alexander polynomial for a class of 3-manifolds.

Findings

Twisted 1-loop invariant equals the adjoint twisted Alexander polynomial for hyperbolic once-punctured torus bundles.

The 1-loop conjecture by Dimofte and Garoufalidis is verified for these bundles.

The paper links Ptolemy coordinates with quantum invariants in 3-manifold topology.

Abstract

We present an alternative definition of the twisted 1-loop invariant in terms of the Jacobian of Ptolemy coordinates. As an application, we prove that the twisted 1-loop invariant is equal to the adjoint twisted Alexander polynomial for all hyperbolic once-punctured torus bundles. This implies that the 1-loop conjecture proposed by Dimofte and Garoufalidis holds for all hyperbolic once-punctured torus bundles.