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Torus knots, the A-polynomial, and SL(2,C)

John A. Baldwin, Steven Sivek

TL;DR

The paper shows that the enhanced A-polynomial $\tilde{A}_K(M,L)$ detects torus knots and, together with $\deg\Delta_K(t)$, distinguishes all torus knots. It proves that knots with infinitely many $SL(2,\mathbb{C})$-abelian surgeries are exactly torus knots, using instanton Floer techniques, L-space knot structure, and satellite analysis. By connecting representation varieties, boundary slopes via Newton polygons, and Dehn-surgery phenomena, the work characterizes torus knots within the broader knot landscape. It also establishes that thin knots are precisely torus knots and advances understanding of when A-polynomials reflect knot type.

Abstract

The A-polynomial of a knot is defined in terms of SL(2,C) representations of the knot group, and encodes information about essential surfaces in the knot complement. In 2005, Dunfield-Garoufalidis and Boyer-Zhang proved that it detects the unknot using Kronheimer-Mrowka's work on the Property P conjecture. Here we use more recent results from instanton Floer homology to prove that a version of the A-polynomial detects whether a knot is a torus knot. We moreover completely determine which individual torus knots are detected by this A-polynomial. These results enable progress towards a folklore conjecture about boundary slopes of non-torus knots. Finally, we use similar ideas to prove that a knot in the 3-sphere admits infinitely many SL(2,C)-abelian Dehn surgeries if and only if it is a torus knot, affirming a variant of a conjecture due to Sivek-Zentner.

Torus knots, the A-polynomial, and SL(2,C)

TL;DR

The paper shows that the enhanced A-polynomial detects torus knots and, together with , distinguishes all torus knots. It proves that knots with infinitely many -abelian surgeries are exactly torus knots, using instanton Floer techniques, L-space knot structure, and satellite analysis. By connecting representation varieties, boundary slopes via Newton polygons, and Dehn-surgery phenomena, the work characterizes torus knots within the broader knot landscape. It also establishes that thin knots are precisely torus knots and advances understanding of when A-polynomials reflect knot type.

Abstract

The A-polynomial of a knot is defined in terms of SL(2,C) representations of the knot group, and encodes information about essential surfaces in the knot complement. In 2005, Dunfield-Garoufalidis and Boyer-Zhang proved that it detects the unknot using Kronheimer-Mrowka's work on the Property P conjecture. Here we use more recent results from instanton Floer homology to prove that a version of the A-polynomial detects whether a knot is a torus knot. We moreover completely determine which individual torus knots are detected by this A-polynomial. These results enable progress towards a folklore conjecture about boundary slopes of non-torus knots. Finally, we use similar ideas to prove that a knot in the 3-sphere admits infinitely many SL(2,C)-abelian Dehn surgeries if and only if it is a torus knot, affirming a variant of a conjecture due to Sivek-Zentner.
Paper Structure (13 sections, 32 theorems, 91 equations)

This paper contains 13 sections, 32 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Proof sketch
  3. Organization
  4. Acknowledgments
  5. Preliminaries
  6. Instanton L-space knots
  7. $\mathrm{SU}(2)$-averse knots
  8. The A-polynomial
  9. L-space satellite knots with torus knot pattern
  10. Thin knots: the proof of Theorem \ref{['thm:thin-torus']}
  11. Boundary slopes
  12. $\mathrm{PSL}(2,\mathbb{C})$-averse knots
  13. $\mathrm{SL}(2,\mathbb{C})$-averse knots: the proof of Theorem \ref{['thm:SLabeliantorus']}

Key Result

Theorem 1.1

Let $K\subset S^3$ be a knot. Then $S^3_r(K)$ is $\mathrm{SL}(2,\mathbb{C})$-abelian for infinitely many $r\in \mathbb{Q}$ if and only if $K$ is a torus knot.

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.7
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • ...and 49 more