The holonomy braiding for U_ξ(\mathfraksl₂) in terms of geometric quantum dilogarithms

2025/09/02 by Calvin McPhail-Snyder, Nicolai Reshetikhin, McPhail-Snyder, Calvin +1
Mathematics · #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2509.02354

openalex publication_date 2025/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive an explicit formula for the holonomy R-matrix of quantum \mathfraksl₂ at a root of unity. We show it factorizes into a product of four quantum dilogarithms and satisfies a holonomy Yang-Baxter equation. This factorization extends previously known results and we collect many existing results needed for our computation.

Citations

• Octahedral coordinates from the Wirtinger presentation
• The descendant colored Jones polynomials
• Evaluation of state integrals at rational points
• Optimistic limits of Kashaev invariants and complex volumes of hyperbolic links
• Braiding for the quantum gl₂ at roots of unity
• Invariants of tangles with flat connections in their complements.I. Invariants and holonomy R-matrices
• The colored Jones polynomials and the simplicial volume of a knot
• Heisenberg Double and Pentagon Relation

Related

• Holonomy braidings, biquandles and quantum invariants of links with SL₂(\mathbb C) flat connections
• Braiding for the quantum gl₂ at roots of unity
• SL₂(ℂ)-holonomy invariants of links
• Invariants of tangles with flat connections in their complements.I. Invariants and holonomy R-matrices
• Unified invariant of knots from homological braid action on Verma modules