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

ABSTRACT

We suggest a method of computing volume for a simple polytope P in three-dimensional hyperbolic space . 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 generalized tetrahedra. 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 associate the above-mentioned sequence of combinatorial moves with the sequence of moves required for computing the Kirillov–Reshetikhin invariants of the trivalent 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 behavior of the Kirillov–Reshetikhin invariants of Γ, which is colloquially known as the Volume Conjecture.

Keywords:

• hyperbolic polyhedron
• quantum 6-j symbol
• Volume Conjecture

2010 AMS Subject Classification:

• 52B10
• 52A38
• 57M27

Notes

1 We suppose that the sequences a^(r), …, f^(r) are such that the corresponding 6j-symbol is well-defined.

2 However, only two truncating polar planes determined by its ultra-ideal vertices intersect.

3 We suppose that the sequences a^(r), …, f^(r) are such that the corresponding 6j-symbol is well-defined.

Additional information

Funding

A.K. was supported by the Swiss National Science Foundation (SNSF project no. P300P2-151316) and the Japan Society for the Promotion of Science (Invitation Programs for Research project no. S-14021). A.K. is thankful to Waseda University for hospitality during his visit. J.M. was supported by Waseda University (Grant for Special Research Projects no. 2014A-345) and the Japan Society for the Promotion of Science (Grant-in-Aid projects no. 25287014, no. 25610022).