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

Alexander Kolpakov, Jun Murakami

    Research output: Contribution to journalArticle

    Abstract

    We suggest a method of computing volume for a simple polytope P in three-dimensional hyperbolic space (Formula presented.). 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.

    Original languageEnglish
    Pages (from-to)1-15
    Number of pages15
    JournalExperimental Mathematics
    DOIs
    Publication statusAccepted/In press - 2016 Oct 31

    Fingerprint

    Polyhedron
    Decompose
    Invariant
    Computing
    Hyperbolic Space
    Triangular pyramid
    Shrinking
    Graph in graph theory
    Potential Function
    Numerics
    Skeleton
    Polytope
    Critical value
    Triangular
    Asymptotic Behavior
    Face
    Three-dimensional
    Experiment

    Keywords

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

    ASJC Scopus subject areas

    • Mathematics(all)

    Cite this

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    title = "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 (Formula presented.). 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.",
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