Combinatorial decompositions, Kirillov-Reshetikhin invariants and the Volume Conjecture for hyperbolic polyhedra

TL;DR

This paper introduces a combinatorial method to compute hyperbolic polytope volumes using graph decompositions and potential functions, linking geometric volume to Kirillov-Reshetikhin invariants and the Volume Conjecture.

Contribution

It proposes a novel approach connecting polytope volume computation with graph invariants and the Volume Conjecture through combinatorial decompositions and potential functions.

Findings

Numeric experiments support the link between combinatorial moves and invariants.

Potential functions can express polytope volume at critical points.

The method suggests a connection between volume and asymptotics of invariants.

Abstract

We suggest a method of computing volume for a simple polytope $P$ in three-dimensional hyperbolic space $\mathbb{H}^3$. This method combines the combinatorial reduction of $P$ as a trivalent graph $\Gamma$ (the $1$-skeleton of $P$) by $I-H$, or Whitehead, moves (together with shrinking of triangular faces) aligned with its geometric splitting into generalised tetrahedra. With each decomposition (under some conditions) we associate a potential function $\Phi$ such that the volume of $P$ can be expressed through a critical values of $\Phi$. The results of our numeric experiments with this method suggest that one may associated the above mentioned sequence of combinatorial moves with the sequence of moves required for computing the Kirillov-Reshetikhin invariants of the trivalent graph $\Gamma$. Then the corresponding geometric decomposition of $P$ might be used in order to establish a link between the volume of $P$ and the asymptotic behaviour of the Kirillov-Reshetikhin invariants of $\Gamma$, which is colloquially know as the Volume Conjecture.