Octahedral coordinates from the Wirtinger presentation

2024/04/29 by Calvin McPhail-Snyder, McPhail-Snyder, Calvin · 1 citation
Engineering · Physics and Astronomy · #57K32 (Primary) #57M05 (Secondary) #FOS: Mathematics #Geometric Topology (math.GT) #Geophysics and Sensor Technology #Scientific Research and Discoveries

paper · pdf · doi:10.48550/arxiv.2404.19155

openalex publication_date 2024/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ρ be a representation of a knot group (or more generally, the fundamental group of a tangle complement) into SL₂(ℂ) expressed in terms of the Wirtinger generators of a diagram D. This diagram also determines an ideal triangulation of the complement called the octahedral decomposition. ρ induces a hyperbolic structure on the complement of D, and in this note we give a direct algebraic formula for the geometric parameters of the octahedral decomposition induced by this structure. Our formula gives a new, explicit criterion for whether ρ occurs as a critical point of the diagram's Neumann-Zagier--Yokota potential function.

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

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