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Filtered instanton homology and cosmetic surgery

Aliakbar Daemi, Mike Miller Eismeier, Tye Lidman

TL;DR

The paper develops a filtration-based approach to the cosmetic surgery problem using filtered instanton Floer homology. By introducing IP-modules and the Chern–Simons filtration, it constructs a distance-two surgery triangle and a monotone invariant $\ell$, enabling sharp obstructions to cosmetic surgery, including the elimination of $\pm 1/n$-surgeries and constraints for $\pm 2$-surgeries on genus $2$ knots with $\Delta_K=1$. It establishes non-diffeomorphism results for surgeries on $S^2\times S^1$ and proves a cascade of exact triangles to compare invariants across surgeries, ultimately connecting three-manifold invariants to classical knot invariants. The work provides a robust framework to bound cosmetic surgeries and yields new structural insights into how instanton gauge theory interacts with Dehn surgery.

Abstract

The cosmetic surgery conjecture predicts that for a non-trivial knot in the three-sphere, performing two different Dehn surgeries results in distinct oriented three-manifolds. Hanselman reduced the problem to $\pm 2$ or $\pm 1/n$ surgeries being the only possible cosmetic surgeries. We remove the case of $\pm 1/n$-surgeries using the Chern-Simons filtration on Floer's original irreducible-only instanton homology, reducing the conjecture to the case of $\pm 2$ surgery on genus $2$ knots with trivial Alexander polynomial. We also prove some similar results for surgeries on knots in $S^2 \times S^1$. As key steps in establishing these results, we define invariants of the oriented homeomorphism type of three-manifolds derived from filtered instanton Floer homology and introduce a new surgery relationship for Floer's instanton homology.

Filtered instanton homology and cosmetic surgery

TL;DR

The paper develops a filtration-based approach to the cosmetic surgery problem using filtered instanton Floer homology. By introducing IP-modules and the Chern–Simons filtration, it constructs a distance-two surgery triangle and a monotone invariant , enabling sharp obstructions to cosmetic surgery, including the elimination of -surgeries and constraints for -surgeries on genus knots with . It establishes non-diffeomorphism results for surgeries on and proves a cascade of exact triangles to compare invariants across surgeries, ultimately connecting three-manifold invariants to classical knot invariants. The work provides a robust framework to bound cosmetic surgeries and yields new structural insights into how instanton gauge theory interacts with Dehn surgery.

Abstract

The cosmetic surgery conjecture predicts that for a non-trivial knot in the three-sphere, performing two different Dehn surgeries results in distinct oriented three-manifolds. Hanselman reduced the problem to or surgeries being the only possible cosmetic surgeries. We remove the case of -surgeries using the Chern-Simons filtration on Floer's original irreducible-only instanton homology, reducing the conjecture to the case of surgery on genus knots with trivial Alexander polynomial. We also prove some similar results for surgeries on knots in . As key steps in establishing these results, we define invariants of the oriented homeomorphism type of three-manifolds derived from filtered instanton Floer homology and introduce a new surgery relationship for Floer's instanton homology.

Paper Structure

This paper contains 22 sections, 39 theorems, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Cosmetic surgeries
  3. The strategy and surgery exact triangles in instanton Floer homology
  4. The strategy to prove Theorem \ref{['thm:S2timesS1']}
  5. The strategy to prove Theorem \ref{['thm:1/n']} for $n=\pm 1$
  6. Filtered groups and numerical invariants
  7. Instanton Floer homology
  8. IP-modules from instantons
  9. Exact triangles
  10. Proof of the main theorems
  11. Proof of Theorem \ref{['thm:S2timesS1']}
  12. Proof of Theorem \ref{['thm:1/n']}
  13. The distance-two surgery triangle
  14. Homological algebra of surgery exact triangles
  15. Cobordisms and families of metrics
  16. ...and 7 more sections

Key Result

Theorem 1.2

If $p/q$, $p'/q'$ are a cosmetic pair for a non-trivial knot $K \subset S^3$, then $p/q = -p'/q'$. Furthermore, $p/q = \pm 2$ or $\pm 1/n$ for some non-zero integer $n$.

Figures (3)

  • Figure 1: A representation of the cobordism $X = W^{-1}_{-2}$. The three boundary components are labeled by their oriented diffeomorphism types. The embedded surface $\mathfrak c_- = [-3,-1] \times S^1 \times \{0\}$ is represented as the darker red curve, while $\mathfrak c_+$ is represented as the lighter blue curve. We visualize $c_+$ and $c_-$ by the same picture, with the understanding that the intersections with integer homology spheres are capped off by Seifert surfaces.
  • Figure 2: A representation of the cobordism $W^2_{-1}$, with its five boundary components labeled. The more thin arcs represent the five submanifolds used in the construction of the family of metrics $G^2_{-1}$, and each arc is labeled by the diffeomorphism type of the corrresponding submanifold. The surface $\hat{c}^2_{-1}$ is represented as the lighter blue curve; the surface $\check c^2_{-1}$ is the union of the darker red and lighter blue curves. The middle end $\mathbb{RP}^3$ is drawn differently than the $S^3$ middle ends to make the asymmetry more apparent. The pictures for other $W^i_{i-3}$ are similar; for instance, the bundle data can on $W^i_{i-3}$ obtained by cyclically permuting the three depicted pieces.
  • Figure 3: The three pentagons of metrics. Each boundary edge corresponds to an interval of metrics broken along a given submanifold, and each edge in the diagram is labeled by the diffeomorphism type of the breaking submanifold. The vertex $v_{0,1}$ is the topmost vertex, and the edges $e_j$ proceed counter-clockwise.

Theorems & Definitions (84)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • proof
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Corollary 1.9
  • ...and 74 more