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Zero-surgery characterizes infinitely many knots

John A. Baldwin, Steven Sivek

TL;DR

The paper proves that $0$ is a characterizing slope for infinitely many genus-1 knots by combining a recent classification of genus-1 nearly fibered knots with both Heegaard-Floer and perturbative 3-manifold invariants. The authors show that for genus-1 knots with $\\dim \\widehat{HFK}(K,1)=2$, $0$-surgery determines the knot among a finite candidate set, and they treat the determinant-$7$ and determinant-$9$ cases separately, using JSJ decompositions, Casson–Gordon invariants, and Ohtsuki perturbative invariants to distinguish $0$-surgeries. In particular, they establish $0$ as a characterizing slope for $15n_{43522}$, $\\mathrm{Wh}^-(T_{2,3},2)$, $\\mathrm{Wh}^+(T_{2,3},2)$ and the pretzel family $P(-3,3,2n+1)$ (and their mirrors), extending previously known examples beyond $5_2$ and a few small knots. The work advances understanding of how Dehn surgery, knot Floer data, and perturbative 3-manifold invariants interact to characterize knots, with implications for the study of $0$-traces and potential exotic $4$-spheres.

Abstract

We prove that 0 is a characterizing slope for infinitely many knots, namely the genus-1 knots whose knot Floer homology is 2-dimensional in the top Alexander grading, which we classified in recent work and which include all $(-3,3,2n+1)$ pretzel knots. This was previously only known for $5_2$ and its mirror, as a corollary of that classification, and for the unknot, trefoils, and the figure eight by work of Gabai from 1987.

Zero-surgery characterizes infinitely many knots

TL;DR

The paper proves that is a characterizing slope for infinitely many genus-1 knots by combining a recent classification of genus-1 nearly fibered knots with both Heegaard-Floer and perturbative 3-manifold invariants. The authors show that for genus-1 knots with , -surgery determines the knot among a finite candidate set, and they treat the determinant- and determinant- cases separately, using JSJ decompositions, Casson–Gordon invariants, and Ohtsuki perturbative invariants to distinguish -surgeries. In particular, they establish as a characterizing slope for , , and the pretzel family (and their mirrors), extending previously known examples beyond and a few small knots. The work advances understanding of how Dehn surgery, knot Floer data, and perturbative 3-manifold invariants interact to characterize knots, with implications for the study of -traces and potential exotic -spheres.

Abstract

We prove that 0 is a characterizing slope for infinitely many knots, namely the genus-1 knots whose knot Floer homology is 2-dimensional in the top Alexander grading, which we classified in recent work and which include all pretzel knots. This was previously only known for and its mirror, as a corollary of that classification, and for the unknot, trefoils, and the figure eight by work of Gabai from 1987.
Paper Structure (5 sections, 13 theorems, 45 equations, 1 figure)

This paper contains 5 sections, 13 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Zero-surgery on genus-one knots
  3. The determinant-7 case
  4. The determinant-9 case, part 1
  5. The determinant-9 case, part 2

Key Result

Theorem 1.1

Let $K$ be any of the knots or their mirrors. Then $0$ is a characterizing slope for $K$.

Figures (1)

  • Figure 1: The knots that Theorem \ref{['thm:main']} says are characterized by their $0$-surgeries.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2: bs-nonfibered
  • Theorem 1.3: bs-characterizing
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • ...and 13 more