The volume and Chern-Simons invariant of a Dehn-filled manifold

TL;DR

This paper extends Zickert's combinatorial formula to compute the volume and Chern-Simons invariant for a broader class of representations of closed 3-manifolds, using deformed Ptolemy varieties.

Contribution

It introduces deformed Ptolemy varieties and generalizes Zickert's formula to non-boundary parabolic representations of closed 3-manifolds.

Findings

Extended formula for volume and Chern-Simons invariant to non-boundary parabolic representations

Defined deformed Ptolemy varieties for closed 3-manifolds

Enabled computation of invariants for a wider class of 3-manifold representations

Abstract

For a compact 3-manifold $N$ with non-empty boundary, Zickert gave a combinatorial formula for computing the volume and Chern-Simons invariant of a boundary parabolic representation $\pi_1(N)\rightarrow \mathrm{PSL}(2,\mathbb{C})$. In this paper, we introduce a notion of deformed Ptolemy varieties and extend the formula of Zickert to a representation that is not necessarily boundary parabolic. This allows us to compute the volume and Chern-Simons invariant of a $\mathrm{PSL}(2,\mathbb{C})$-representation of a closed 3-manifold.