Combinatorial Decompositions, Kirillov-Reshetikhin Invariants, and the Volume Conjecture for Hyperbolic Polyhedra

Alexander Kolpakov (1) , Jun Murakami (2)

1 Department of Mathematics [University of Toronto]
2 Departement of Mathematics [Waseda]

We suggest a method of computing volume for a simple polytope P in three-dimensional hyperbolic space H 3. This method combines the combinatorial reduction of P as a trivalent graph Γ (the 1-skeleton of P) by I − H, or Whitehead, moves (together with shrinking of triangular faces) aligned with its geometric splitting into generalised tetra-hedra. With each decomposition (under some conditions) we associate a potential function Φ such that the volume of P can be expressed through a critical values of Φ. 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 triva-lent graph Γ. 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 Γ, which is colloquially know as the Volume Conjecture.

Mots clés: hyperbolic polyhedron; quantum invariant; I-H move; Whitehead move; volume

Domaines: Topologie algébrique [math.AT]; Topologie géométrique [math.GT]; Combinatoire [math.CO]

Alexander Kolpakov, Jun Murakami. Combinatorial Decompositions, Kirillov-Reshetikhin Invariants, and the Volume Conjecture for Hyperbolic Polyhedra. Experimental Mathematics, 2016, pp.1 - 15. ⟨10.1080/10586458.2016.1242441⟩. ⟨hal-01396113⟩