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Combinatorial decompositions, Kirillov-Reshetikhin invariants, and the volume conjecture for hyperbolic polyhedra. (English) Zbl 1391.52015
Summary: 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 generalized 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 associate 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 behavior of the Kirillov-Reshetikhin invariants of $$\Gamma$$, which is colloquially known as the Volume Conjecture.

##### MSC:
 52B10 Three-dimensional polytopes 52A38 Length, area, volume and convex sets (aspects of convex geometry) 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)
Mathematica; Orb
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