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Heegaard Floer Surgery Formula and Cosmetic Surgeries

Alan Du

TL;DR

This work investigates cosmetic Dehn surgeries on a class of null-homologous knots in arbitrary closed orientable 3-manifolds using the mapping cone formula that relates Dehn surgery to knot Floer data. It provides a new, conceptually simpler proof of the integer-surgery formula for Spin$^c$ structures with non-torsion first Chern class and extends the surgery framework to rational coefficients, enabling precise rank-based obstructions for cosmetic surgeries. The main results show that for this knot family, any pair of purely cosmetic surgeries with positive slopes must have both slopes exceeding 1, and that $Y_{1/q}(K)$ is not homeomorphic to the original manifold for all $q\ge 2$ (with analogous statements for blow-up unknotted/bounding knots). These findings offer uniform obstructions across arbitrary 3-manifolds and illustrate the power of Heegaard Floer theory, via the mapping cone construction, in controlling when Dehn surgeries can yield homeomorphic manifolds.

Abstract

Two Dehn surgeries on a knot are called cosmetic if they yield homeomorphic three-manifolds. We show for a certain family of null-homologous knots in any closed orientable three-manifold, if the knot admits cosmetic surgeries with a pair of positive surgery coefficients, then the coefficients are both greater than $1$. In addition, for this family of knots, we show that $1/q$ Dehn surgery for $q$ at least $2$ is not homeomorphic to the original three-manifold. The proofs of these results use the mapping cone formula for the Heegaard Floer homology of Dehn surgery in terms of the knot Floer homology of the knot; we provide a new proof of this formula for integer surgeries in $\text{Spin}^c$ structures with nontorsion first Chern class.

Heegaard Floer Surgery Formula and Cosmetic Surgeries

TL;DR

This work investigates cosmetic Dehn surgeries on a class of null-homologous knots in arbitrary closed orientable 3-manifolds using the mapping cone formula that relates Dehn surgery to knot Floer data. It provides a new, conceptually simpler proof of the integer-surgery formula for Spin structures with non-torsion first Chern class and extends the surgery framework to rational coefficients, enabling precise rank-based obstructions for cosmetic surgeries. The main results show that for this knot family, any pair of purely cosmetic surgeries with positive slopes must have both slopes exceeding 1, and that is not homeomorphic to the original manifold for all (with analogous statements for blow-up unknotted/bounding knots). These findings offer uniform obstructions across arbitrary 3-manifolds and illustrate the power of Heegaard Floer theory, via the mapping cone construction, in controlling when Dehn surgeries can yield homeomorphic manifolds.

Abstract

Two Dehn surgeries on a knot are called cosmetic if they yield homeomorphic three-manifolds. We show for a certain family of null-homologous knots in any closed orientable three-manifold, if the knot admits cosmetic surgeries with a pair of positive surgery coefficients, then the coefficients are both greater than . In addition, for this family of knots, we show that Dehn surgery for at least is not homeomorphic to the original three-manifold. The proofs of these results use the mapping cone formula for the Heegaard Floer homology of Dehn surgery in terms of the knot Floer homology of the knot; we provide a new proof of this formula for integer surgeries in structures with nontorsion first Chern class.
Paper Structure (5 sections, 26 theorems, 90 equations, 5 figures)

This paper contains 5 sections, 26 theorems, 90 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on Heegaard Floer Homology
  3. Integer Surgeries
  4. Rational Surgeries
  5. Cosmetic Surgeries

Key Result

Theorem 1.4

Let $K\subset Y$ be a blow-up unknotted knot, and suppose $0<r<s$ are distinct positive rational numbers with the property that $Y_r(K)\cong\pm Y_s(K)$. Then $r,s>1$ or $K$ is the unknot.

Figures (5)

  • Figure 1: Commutative diagrams of the maps $v_k^+$ and $h_k^+$
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:

Theorems & Definitions (41)

  • Definition 1.1
  • Conjecture 1.2: kirby-problem-list-1995 Problem 1.81
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: oz-sz-4-mflds Theorem 3.6
  • Theorem 2.2: oz-sz-integer Theorem 3.1
  • Lemma 2.3
  • Theorem 2.4
  • Lemma 2.5
  • ...and 31 more