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

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 ${\pi }_1(N)\to \mathrm{PSL}(2,\mathbb{C})$. 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.

References in corpus (2)

• The Ptolemy field of $3$-manifold-representations
• Octahedral developing of knot complement I: pseudo-hyperbolic structure

Cited by in corpus (2)

• On the potential functions for a link diagram
• The twisted 1-loop invariant and the Jacobian of Ptolemy coordinates