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

Abstract

We reformulate the twisted 1-loop invariant in terms of Ptolemy coordinates. In addition, we prove that the twisted 1-loop invariant is equal to the adjoint twisted Alexander polynomial for all hyperbolic once-punctured torus bundles. This shows that the 1-loop conjecture proposed by Dimofte and Garoufalidis holds for all hyperbolic once-punctured torus bundles.