Hyperbolic structures on link complements, octahedral decompositions, and quantum \mathfraksl₂

2022/03/11 by Calvin McPhail-Snyder, McPhail-Snyder, Calvin
Mathematics · #20G42 #57K10 #57K32 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2203.06042

openalex publication_date 2022/03/11 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28

Abstract

Hyperbolic structures on link complements (equivalently, representations of the fundamental group into SL₂(ℂ)) can be described algebraically by using the octahedral decomposition determined by a link diagram. The decomposition (like any ideal triangulation) gives a set of gluing equations in shape parameters whose solutions are hyperbolic structures. We show that these equations can be obtained from Kashaev-Reshetikhin's braiding on the Kac-de Concini quantum group U_ξ(\mathfraksl₂) at a root of unity ξ. This braiding gives coordinates on the SL₂(ℂ) representation variety of a link and our work shows how to interpret these geometrically.

Related

• Holonomy braidings, biquandles and quantum invariants of links with SL₂(\mathbb C) flat connections
• The Kashaev and quantum hyperbolic link invariants
• A quantization of the SL₂(ℂ) Chern-Simons invariant of tangle exteriors
• Kashaev--Reshetikhin Invariants of Links
• Octahedral coordinates from the Wirtinger presentation