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

Alexander Kolpakov, Jun Murakami

Research output: Contribution to journalArticle

Abstract

We sugges. method of computing volume fo. simple polytop. in three-dimensional hyperbolic space H3. This method combines the combinatorial reduction o. 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 associat. potential function Φ such that the volume o. can be expressed throug. 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 o. might be used in order to establis. link between the volume o. and the asymptotic behavior of the Kirillov–Reshetikhin invariants of Γ, which is colloquially known as the Volume Conjecture.

Original languageEnglish
Pages (from-to)193-207
Number of pages15
JournalExperimental Mathematics
Volume27
Issue number2
DOIs
Publication statusPublished - 2018 Apr 3

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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