On contact cosmetic surgery

TL;DR

This work establishes a contact-geometry analogue of the cosmetic surgery conjecture for Legendrian knots: for all nontrivial Legendrian knots in the standard tight $S^3$ manifold, there are no cosmetic contact surgeries except for a narrowly described exceptional family involving Lagrangian-slice knots. The authors develop and combine a toolbox of methods—classifications of tight contact structures via Farey-graph paths, a reduction of general contact surgery to sequences of $(\pm 1)$-surgeries, and $d_3$-invariant calculations on the resulting 4-manifolds—to rule out contact cosmetic pairs. They provide a complete treatment for Legendrian unknots and perform an extensive obstruction analysis across tb-values $-1$, $-2$, and $<-2$ (showing no cosmetic pairs occur in these ranges except the stated potential exception), thereby confirming the conjecture in broad generality and identifying prime candidates for any remaining counterexample. The results connect contact surgery representations with classical smooth surgery theory and leverage $d_3$-invariants to distinguish contactomorphic possibilities, highlighting a deep interplay between topology, contact geometry, and 4-manifold invariants.

Abstract

We demonstrate that the contact cosmetic surgery conjecture holds true for all non-trivial Legendrian knots, with the possible exception of Lagrangian slice knots. We also discuss the contact cosmetic surgeries on Legendrian unknots and make the surprising observation that there are some Legendrian unknots that have a contact surgery with no cosmetic pair, while all other contact surgeries are contactomorphic to infinitely many other contact surgeries on the knot.

Figures (7)

• Figure 1: On the left we see the result of contact $(-3/2)$-surgery on $L$. The blue path describes the contact structure on $S^3_L$ and the red path describes the contact structure on $S_r$. In the middle figure, we see the same manifold after applying the coordinate change $\phi_1$. On the right, we see that the image of the $S^3_L$ can be split into a solid torus (with upper meridian $0$ and dividing slope $-1)$ and a thickened torus (with dividing slopes $-1$ and $\infty$). Attaching the thickened torus to the image of $S_r$ under $\phi_1$ shows that this manifold is the result of contact $(8/5)$-surgery on $L$ and different such surgeries are giving this manifold as there are two choices for the sign describing the contact structure on the thickened torus.
• Figure 2: For a Legendrian knot $K$ with $\mathop{\mathrm{tb}}\nolimits=-1$ we see a smooth $-1/2$ surgery (that is contact $(1/2)$ surgery) and $-1/n$ surgery for $n>2$ (that is contact $((n-1)/n)$ surgery) on the upper and lower left, respectively, and a smooth $1/n$ surgery (that is a contact $((n+1)/n)$ surgery) on the right. On the right $a$ and $b$ are non-negative integers so that $a+b=n$ (that is the second knot is the Legendrian push-off of the first with $n$ stabilizations of one sign or the other).
• Figure 3: Computing the signature for $X_n$.
• Figure 4: For a Legendrian knot $K$ with $\mathop{\mathrm{tb}}\nolimits=-1$ we see a smooth $-2$ surgery (that is contact $(-1)$ surgery) on the left and a smooth $2$ surgery (that is a contact $(+3)$ surgery) on the right (the stabilization can be either positive or negative.
• Figure 5: For a Legendrian knot $K$ with $\mathop{\mathrm{tb}}\nolimits=-2$ we see a smooth $-1$ surgery (that is contact $(1)$ surgery) and $-1/n$ surgery for $n>1$ (that is contact $((2n-1)/n)$ surgery) on the upper and lower left, respectively (the stabilizations in the lower diagram can be of any sign), and a smooth $1/n$ surgery (that is a contact $((2n+1)/n)$ surgery) on the right. On the right $a$ and $b$ are non-negative integers so that $a+b=n-1$ (that is the second knot is the Legendrian push-off of the first with $n-1$ stabilizations of one sign or the other).

Theorems & Definitions (32)

• Conjecture 1.1: Contact cosmetic surgery conjecture
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 2.1: Hanselman, 2023 Hanselman2023
• Theorem 2.2: Plamenevskaya 2004, Plamenevskaya2004
• Theorem 2.3: Ni and Wu, 2013NiWu2015
• proof : Proof of Theorem \ref{['unknotsurg']}
• Proposition 4.1
• proof