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Deforming abelian elliptic $\mathrm{SL}(2,\mathbb{R})$--representations of knot groups

Yi Liu

TL;DR

This work establishes a criterion linking zeros of the knot's Alexander polynomial on the unit circle to deformations of abelian elliptic representations of knot groups into $SL(2,\mathbb{R})$, producing continuous families of irreducible representations that approach the abelian limit and lift to the universal cover $\widetilde{SL}(2,\mathbb{R})$. The authors develop a unified quaternionic framework and a detailed trace-slice/intersection approach, leveraging equivariant transversality to obtain local modulo-$2$ intersection numbers that guarantee deformation when the zero is of odd order. As a key application, every nontrivial $L$-space knot has an Alexander polynomial with an odd-order unit-circle zero, yielding irreducible $SL(2,\mathbb{R})$ representations for its knot group and, via small-slope surgery, potential implications for left-orderability of surgery groups. The paper also provides a coefficient-based criterion (via Konvalina–Matache) that ensures the existence of such odd-order unit-circle zeros, broadening accessible checks beyond simple zeros. Overall, the results connect knot invariants, representation varieties, and 3-manifold topology, offering a general framework to detect nonabelian real representations from Alexander-polynomial data and to relate these to orderability phenomena after Dehn surgery.

Abstract

The following criterion is proved in this paper. If the Alexander polynomial of a knot $K\subset S^3$ has a zero of odd order on the complex unit circle, then there exists a continuous family of irreducible representations $π_1(S^3\setminus K)\to \mathrm{SL}(2,\mathbb{R})$ 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 $\mathrm{SL}(2,\mathbb{R})$--representation.

Deforming abelian elliptic $\mathrm{SL}(2,\mathbb{R})$--representations of knot groups

TL;DR

This work establishes a criterion linking zeros of the knot's Alexander polynomial on the unit circle to deformations of abelian elliptic representations of knot groups into , producing continuous families of irreducible representations that approach the abelian limit and lift to the universal cover . The authors develop a unified quaternionic framework and a detailed trace-slice/intersection approach, leveraging equivariant transversality to obtain local modulo- intersection numbers that guarantee deformation when the zero is of odd order. As a key application, every nontrivial -space knot has an Alexander polynomial with an odd-order unit-circle zero, yielding irreducible representations for its knot group and, via small-slope surgery, potential implications for left-orderability of surgery groups. The paper also provides a coefficient-based criterion (via Konvalina–Matache) that ensures the existence of such odd-order unit-circle zeros, broadening accessible checks beyond simple zeros. Overall, the results connect knot invariants, representation varieties, and 3-manifold topology, offering a general framework to detect nonabelian real representations from Alexander-polynomial data and to relate these to orderability phenomena after Dehn surgery.

Abstract

The following criterion is proved in this paper. If the Alexander polynomial of a knot has a zero of odd order on the complex unit circle, then there exists a continuous family of irreducible representations 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 --representation.
Paper Structure (13 sections, 25 theorems, 71 equations, 1 table)

This paper contains 13 sections, 25 theorems, 71 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Braid presentations
  4. Quaternionic representations
  5. Quaternion algebras
  6. Spaces of representations
  7. Spaces of characters
  8. The analytic topology
  9. Trace slices
  10. Equivariant transversality
  11. Deforming an abelian elliptic representation
  12. Inferring an odd-order unitary zero
  13. Further discussion

Key Result

Theorem 1.1

For any knot $K\subset S^3$, if the Alexander polynomial $\Delta_K$ of $K$ has a zero of odd order on the complex unit circle, then $\pi_1(S^3\setminus K)$ admits a continuous family of irreducible $\mathrm{SL}(2,{\mathbb R})$--representations which converges to an abelian $\mathrm{SL}(2,{\mathbb R}

Theorems & Definitions (51)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Conjecture 1.5
  • Theorem 2.1
  • Corollary 2.2
  • Example 2.3
  • Theorem 2.4
  • Remark 2.5
  • ...and 41 more