TY - JOUR

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

AU - Kolpakov, Alexander

AU - Murakami, Jun

N1 - Funding Information:
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).
Publisher Copyright:
© 2018 Taylor & Francis.

PY - 2018/4/3

Y1 - 2018/4/3

N2 - 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.

AB - 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.

KW - Volume Conjecture

KW - hyperbolic polyhedron

KW - quantum 6-j symbol

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U2 - 10.1080/10586458.2016.1242441

DO - 10.1080/10586458.2016.1242441

M3 - Article

AN - SCOPUS:84994158300

VL - 27

SP - 193

EP - 207

JO - Experimental Mathematics

JF - Experimental Mathematics

SN - 1058-6458

IS - 2

ER -