The volume and Chern-Simons invariant of a representation

TL;DR

This paper presents an efficient simplicial formula to compute the volume and Chern-Simons invariant of boundary-parabolic PSL(2,C)-representations of tame 3-manifolds, enabling direct calculations from ideal triangulations.

Contribution

It introduces a novel simplicial formula that allows direct computation of volume and Chern-Simons invariants from ideal triangulations without extra combinatorial steps.

Findings

Formula computes volume and Chern-Simons invariant efficiently

Applicable to boundary-parabolic PSL(2,C)-representations

Direct calculation from ideal triangulations

Abstract

We give an efficient simplicial formula for the volume and Chern-Simons invariant of a boundary-parabolic PSL(2,C)-representation of a tame 3-manifold. If the representation is the geometric representation of a hyperbolic 3-manifold, our formula computes the volume and Chern-Simons invariant directly from an ideal triangulation with no use of additional combinatorial topology. In particular, the Chern-Simons invariant is computed just as easily as the volume.