Combinatorial Decompositions, Kirillov–Reshetikhin Invariants, and the Volume Conjecture for Hyperbolic Polyhedra. Issue 2 (3rd April 2018)

• Record Type: Journal Article
• Authors: Kolpakov, Alexander; Murakami, Jun
• Is Part Of: Experimental mathematics. Volume 27:Issue 2(2018)
• Page Start: 193
• Page End: 207
• Publication Date: 2018-04-03
• Subjects: hyperbolic polyhedron -- quantum 6-j symbol -- Volume Conjecture; 52B10 -- 52A38 -- 57M27; Mathematics -- Periodicals; Mathematics -- Research -- Periodicals; 510.724
• DOI: 10.1080/10586458.2016.1242441
• Languages: English
• ISSNs: 1058-6458

Abstract

ABSTRACT: We suggest a method of computing volume for a simple polytope P in three-dimensional hyperbolic space . 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.