A volume formula for hyperbolic tetrahedra in terms of edge lengths
Jun Murakami, Akira Ushijima
TL;DR
This paper presents a closed-form formula for calculating the volume of hyperbolic tetrahedra using edge lengths, based on the volume conjecture and involving dilogarithm functions.
Contribution
It introduces a novel explicit volume formula for hyperbolic tetrahedra derived from quantum invariants and the volume conjecture.
Findings
Provides a closed-form volume formula in terms of edge lengths
Involves dilogarithm functions with specified branches for accuracy
Connects hyperbolic geometry with quantum invariants
Abstract
We give a closed formula for volumes of generic hyperbolic tetrahedra in terms of edge lengths. The cue of our formula is by the volume conjecture for the Turaev-Viro invariant of closed 3-manifolds, which is defined from the quantum 6j-symbols. This formula contains the dilogarithm functions, and we specify the adequate branch to get the actual value of the volumes.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Numerical Analysis Techniques · Geometric and Algebraic Topology