### 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 language | English |
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Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Experimental Mathematics |

DOIs | |

Publication status | Accepted/In press - 2016 Oct 31 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Combinatorial Decompositions, Kirillov–Reshetikhin Invariants, and the Volume Conjecture for Hyperbolic Polyhedra.** / Kolpakov, Alexander; Murakami, Jun.

Research output: Contribution to journal › Article

}

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

UR - http://www.scopus.com/inward/record.url?scp=84994158300&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84994158300&partnerID=8YFLogxK

U2 - 10.1080/10586458.2016.1242441

DO - 10.1080/10586458.2016.1242441

M3 - Article

SP - 1

EP - 15

JO - Experimental Mathematics

JF - Experimental Mathematics

SN - 1058-6458

ER -