On the potential functions for a link diagram

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.

22 pages

References in corpus (3)

• Octahedral developing of knot complement I: pseudo-hyperbolic structure
• On the Hikami-Inoue conjecture
• The volume and Chern-Simons invariant of a Dehn-filled manifold

Cited by in corpus (1)

• On the Potential Function of the Colored Jones Polynomial with Arbitrary Colors