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Heegaard Floer homology and chirally cosmetic surgeries

Konstantinos Varvarezos

TL;DR

This work develops immersed-curve techniques in Heegaard Floer theory to obstruct chirally cosmetic surgeries on knots. By relating Dehn surgeries to intersections of knot-curve invariants with slope lines, it derives explicit rank and grading constraints that must hold for opposite-sign cosmetic pairs. The authors apply these obstructions to families such as odd alternating pretzel knots and Whitehead doubles, obtaining classification and exclusion results, and prove a strong complementary statement: nontrivial L-space knots do not admit cosmetic surgeries with slopes of opposite signs. The results significantly constrain possible chirally cosmetic phenomena and shed light on when symmetry in surgery cannot occur. The methods combine Floer theory, bordered/immersed-curve formalisms, and finite-type invariants to produce concrete, testable obstructions.

Abstract

A pair of surgeries on a knot is chirally cosmetic if they result in homeomorphic manifolds with opposite orientations. We find new obstructions to the existence of such surgeries coming from Heegaard Floer homology; in particular, we make use of immersed curve formulations of knot Floer homology and the corresponding surgery formula. As an application, we completely classify chirallly cosmetic surgeries on odd alternating pretzel knots, and we rule out such surgeries for a large class of Whitehead doubles. Furthermore, we rule out cosmetic surgeries for L-space knots along slopes with opposite signs.

Heegaard Floer homology and chirally cosmetic surgeries

TL;DR

This work develops immersed-curve techniques in Heegaard Floer theory to obstruct chirally cosmetic surgeries on knots. By relating Dehn surgeries to intersections of knot-curve invariants with slope lines, it derives explicit rank and grading constraints that must hold for opposite-sign cosmetic pairs. The authors apply these obstructions to families such as odd alternating pretzel knots and Whitehead doubles, obtaining classification and exclusion results, and prove a strong complementary statement: nontrivial L-space knots do not admit cosmetic surgeries with slopes of opposite signs. The results significantly constrain possible chirally cosmetic phenomena and shed light on when symmetry in surgery cannot occur. The methods combine Floer theory, bordered/immersed-curve formalisms, and finite-type invariants to produce concrete, testable obstructions.

Abstract

A pair of surgeries on a knot is chirally cosmetic if they result in homeomorphic manifolds with opposite orientations. We find new obstructions to the existence of such surgeries coming from Heegaard Floer homology; in particular, we make use of immersed curve formulations of knot Floer homology and the corresponding surgery formula. As an application, we completely classify chirallly cosmetic surgeries on odd alternating pretzel knots, and we rule out such surgeries for a large class of Whitehead doubles. Furthermore, we rule out cosmetic surgeries for L-space knots along slopes with opposite signs.
Paper Structure (11 sections, 32 theorems, 59 equations, 15 figures)

This paper contains 11 sections, 32 theorems, 59 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Obstructions from Heegaard Floer Homology
  3. Heegaard/Knot Floer Floer homology and immersed curves
  4. The main obstruction
  5. Combining other obstructions
  6. Applications
  7. Pretzel knots
  8. Whitehead doubles
  9. L-space knots
  10. Gradings
  11. Proof of Theorem \ref{['thm:lspace']}

Key Result

Theorem 1.2

Let $K$ be a knot and suppose $S^3_r(K)\cong \pm S^3_{r'}(K)$ for two distinct slopes $r,r'.$ Then either $r$ and $r'$ have opposite signs or $S^3_r(K)$ is an L-space (hence $K$ is an L-space knot).

Figures (15)

  • Figure 1: The pretzel knot $P(-2k_1-1,-2k_2-1,-2k_3-1,\dots,-2k_{2g+1}-1).$ Here the boxes represent $k_i$ full left-handed twists of the strands passing through.
  • Figure 2: The positive Whitehead pattern inside a solid torus.
  • Figure 3: An example of immersed curves in $\overline{T}_{\bullet}$ corresponding to some knot-like complex. The left and right edges are identified. We can see that this knot would have genus 3, tau 2 and epsilon 1.
  • Figure 4: Schematics of the kinds of immersed curve pairings occurring in the proof of Lemma \ref{['lem:pos']}.
  • Figure 5: Schematics of the kinds of immersed curve pairings occurring in the proof of Lemma \ref{['lem:neg']}.
  • ...and 10 more figures

Theorems & Definitions (61)

  • Theorem 1.2: Theorem 9.8 of OSzRat
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 2.1: Theorem 2 of HRW
  • Remark 2.2
  • Theorem 2.3
  • proof : Note:
  • ...and 51 more