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Cosmetic operations and Khovanov multicurves

Artem Kotelskiy, Tye Lidman, Allison H. Moore, Liam Watson, Claudius Zibrowius

Abstract

We prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots. Our proof combines a recent result of Hanselman with the Khovanov multicurve invariants $\widetilde{\operatorname{Kh}}$ and $\widetilde{\operatorname{BN}}$. We apply the same techniques to reprove a result of Wang about the Cosmetic Crossing Conjecture and split links. Along the way, we show that $\widetilde{\operatorname{Kh}}$ and $\widetilde{\operatorname{BN}}$ detect if a Conway tangle is split.

Cosmetic operations and Khovanov multicurves

Abstract

We prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots. Our proof combines a recent result of Hanselman with the Khovanov multicurve invariants and . We apply the same techniques to reprove a result of Wang about the Cosmetic Crossing Conjecture and split links. Along the way, we show that and detect if a Conway tangle is split.

Paper Structure

This paper contains 14 sections, 18 theorems, 36 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Cosmetic surgeries
  3. Cosmetic Crossings
  4. Review of the Khovanov multicurve invariants
  5. The construction of the multicurves
  6. A gluing theorem
  7. Gradings
  8. Geography of components of Khr
  9. A dimension formula
  10. Split tangle detection for BNr and Khr
  11. Proof of Theorem \ref{['thm:intro:cosmeticrational']}
  12. The Generalized Cosmetic Crossing Conjecture holds asymptotically
  13. Non-trivial band detection
  14. Towards split closure detection

Key Result

Theorem 1.1

Given a non-trivial strongly invertible knot $(K,h)$ and $r,r'\in\operatorname{\mathbb{Q}P}^1$, suppose that there exists an orientation-preserving diffeomorphism $f\colon S^3_r(K)\rightarrow S^3_{r'}(K)$ such that $h_{r'}\circ f= f \circ h_r$. Then $r=r'$.

Figures (11)

  • Figure 1: A crossing circle $c$ (a), some examples of rational tangles (b--d), and the $p/q$-rational filling $T(p/q)$ of a Conway tangle $T$ (e)
  • Figure 2: The multicurve invariants for the pretzel tangle $P_{2,-3}$. Under the covering $\mathbb{R}^2\smallsetminus \mathbb{Z}^2\rightarrow S^2_{4,\ast}$, the shaded regions in (b+c) correspond to the shaded regions in (d+e).
  • Figure 3: Two immersed curves and their corresponding chain complexes (a+b) and their Lagrangian Floer homology (c); cf KWZ
  • Figure 4: Two tangle decompositions defining the link $T_1\cup T_2$. The tangle $T_2$ is the result of rotating $T_2$ around the vertical axis. By rotating the entire link on the right-hand side around the vertical axis, we can see that $T_1\cup T_2=T_2\cup T_1$.
  • Figure 5: The curves $\mathbf{r}_n(0)$ and $\mathbf{s}_{2n}(0)$ (a--c) and their lifts to $\mathbb{R}^2\smallsetminus \mathbb{Z}^2$ (d). While not visually apparent, the curves $\mathbf{r}_n(0)$ are invariant under the Dehn twist interchanging the lower two punctures.
  • ...and 6 more figures

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Wang Wang
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 2.1
  • Example 2.2
  • Theorem 2.3
  • Definition 2.4
  • ...and 30 more