On contact cosmetic surgery

TL;DR

This paper proves the contact cosmetic surgery conjecture for most Legendrian knots, explores surgeries on Legendrian unknots, and reveals surprising phenomena regarding contact surgeries and their cosmetic pairs.

Contribution

It establishes the conjecture for all non-trivial Legendrian knots except possibly Lagrangian slice knots and uncovers unique behaviors of Legendrian unknots in contact surgeries.

Findings

The contact cosmetic surgery conjecture holds for all non-trivial Legendrian knots.

Some Legendrian unknots have contact surgeries with no cosmetic pair.

Other contact surgeries on Legendrian unknots are contactomorphic to infinitely many others.

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.