Geometric structures and PSL₂(ℂ) representations of knot groups from knot diagrams

2025/05/17 by Kathleen L. Petersen, Petersen, Kathleen L., Anastasiia Tsvietkova +1
Mathematics · #57K10 #57K31 #57K32 #57K35 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2505.11779

openalex publication_date 2025/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe a new method of producing equations for the canonical component of representation variety of a knot group into PSL₂(ℂ). Unlike known methods, this one does not involve any polyhedral decomposition or triangulation of the knot complement, and uses only a knot diagram satisfying a few mild restrictions. This gives a simple algorithm that can often be performed by hand, and in many cases, for an infinite family of knots at once. The algorithm yields an explicit description for the hyperbolic structures (complete or incomplete) that correspond to geometric representations of a hyperbolic knot. As an illustration, we give the formulas for the equations for the variety of closed alternating braids (σ₁(σ₂)^-1)ⁿ that depend only on n.

Citations

• Generalization of the Thistlethwaite--Tsvietkova Method
• Intercusp Geodesics and Cusp Shapes of Fully Augmented Links
• Waist size for cusps in hyperbolic 3-manifolds II
• On the geometry of the slice of trace--free SL(2,C)-characters of a knot group
• Strong rigidity ofQ-rank 1 lattices

Related

• Hyperbolic structures on link complements, octahedral decompositions, and quantum \mathfraksl₂
• On the fundamental 3-classes of knot group representations
• Algorithm for computing Representations of the Braid Group and Temperley-Lieb algebra
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• Symplectic quandles and parabolic representations of 2-bridge Knots and Links