On the potential functions for a link diagram

TL;DR

This paper extends the potential function for link diagrams to a broader class of $ ext{PSL}(2, ext{C})$-representations, enabling a combinatorial approach to compute volume and Chern-Simons invariants of closed 3-manifolds.

Contribution

It generalizes the potential function concept beyond boundary parabolic representations, providing a new combinatorial formula for invariants of closed 3-manifolds.

Findings

Extended potential function to non-boundary parabolic representations.

Derived a combinatorial formula for volume and Chern-Simons invariants.

Applicable to a wider class of 3-manifold representations.

Abstract

For an oriented diagram of a link $L$ in the 3-sphere, Cho and Murakami defined the potential function whose critical point, slightly different from the usual sense, corresponds to a boundary parabolic $\mathrm{PSL}(2,\mathbb{C})$-representation of $\pi_1(S^3 \setminus L)$. They also showed that the volume and Chern-Simons invariant of such a representation can be computed from the potential function with its partial derivatives. In this paper, we extend the potential function to a $\mathrm{PSL}(2,\mathbb{C})$-representation that is not necessarily boundary parabolic. Under a mild assumption, it leads us to a combinatorial formula for computing the volume and Chern-Simons invariant of a $\mathrm{PSL}(2,\mathbb{C})$-representation of a closed 3-manifold.