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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)
Software:
Mathematica; Orb
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References:
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