Deforming abelian elliptic SL(2,ℝ)--representations of knot groups

2025/10/22 by Yi Liu, Liu, Yi
Mathematics · #57K14 #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Primary 57K31 #Secondary 12D10

paper · pdf · doi:10.48550/arxiv.2510.19748

openalex publication_date 2025/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

The following criterion is proved in this paper. If the Alexander polynomial of a knot K⊂ S³ has a zero of odd order on the complex unit circle, then there exists a continuous family of irreducible representations π₁(S³∖ K)→ SL(2,ℝ) converging to an abelian representation of noncentral elliptic type. As an application, the author shows that the Alexander polynomial of any nontrivial L-space knot satisfies the condition of the criterion. In particular, it follows that the fundamental group of any nontrivial L-space knot complement admits an irreducible SL(2,ℝ)--representation.

Citations

• Fibered 3-manifolds with unique incompressible surfaces
• Knot Colorings: Coloring and Goeritz matrices
• The Levine-Tristram signature: a survey
• A note on orderability and Dehn filling
• On L-spaces and left-orderable fundamental groups
• On L-spaces and left-orderable fundamental groups
• A survey of classical knot concordance
• On knot Floer homology and lens space surgeries
• Orderable 3-manifold groups

Related

• On the twisted Alexander polynomial for representations into SL₂(C)
• Deformations of reducible representations of knot groups into\n \SL(n,\C)
• Deformations of metabelian representations of knot groups into SL(3,ℂ)
• Twisted Alexander polynomials and character varieties of 2-bridge knot groups
• Twisted Alexander polynomials of 2-bridge knots associated to metabelian representations