TY - JOUR

T1 - Combinatorial Decompositions, Kirillov–Reshetikhin Invariants, and the Volume Conjecture for Hyperbolic Polyhedra

AU - Kolpakov, Alexander

AU - Murakami, Jun

PY - 2016/10/31

Y1 - 2016/10/31

N2 - 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.

AB - 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.

KW - hyperbolic polyhedron

KW - quantum 6-j symbol

KW - Volume Conjecture

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U2 - 10.1080/10586458.2016.1242441

DO - 10.1080/10586458.2016.1242441

M3 - Article

AN - SCOPUS:84994158300

SP - 1

EP - 15

JO - Experimental Mathematics

JF - Experimental Mathematics

SN - 1058-6458

ER -