Research ArticleNo Access

On the potential functions for a link diagram

Departament de Matemátiques, Universitat Autónoma de Barcelona, Spain

https://doi.org/10.1142/S0218216521500565Cited by:1 (Source: Crossref)

Abstract

Cho and Murakami defined the potential function for a link $L$ in $S^{3}$ whose critical point, slightly different from the usual sense, corresponds to a boundary-parabolic representation $ρ : π_{1} (S^{3} ∖ L) \to {PSL}_{2} (ℂ)$ . They also showed that the volume and Chern–Simons invariant of $ρ$ can be computed from the potential function with its partial derivatives. In this paper, we extend the potential function to a representation that is not necessarily boundary-parabolic. We show that under a mild assumption it leads us to a combinatorial formula for computing the volume and Chern–Simons invariant of a ${PSL}_{2} (ℂ)$ -representation of a closed 3-manifold.

Keywords:

AMSC: 57K10, 57K31, 57K32

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