Contact cosmetic surgery conjecture

Prove that any Legendrian knot in the standard tight contact 3-sphere (S^3, ξ_std) that is not smoothly isotopic to the unknot admits no cosmetic contact surgeries; specifically, establish that for such a Legendrian knot L, there do not exist two distinct contact surgeries on L that yield contactomorphic contact 3-manifolds.

Background

The classical smooth cosmetic surgery conjecture posits that non-trivial knots in S^3 admit no truly cosmetic surgeries (orientation-preserving diffeomorphic results). In the contact category, contact (r)-surgery on a Legendrian knot L in (S^3, ξ_std) replaces a standard neighborhood of L and extends the contact structure over the glued-in solid torus; for most r, there are multiple tight contact structures to choose from, creating the possibility of distinct contact surgeries yielding the same contact manifold.

This conjecture is the natural contact-geometric analogue of the smooth cosmetic surgery conjecture. The paper proves the contact cosmetic surgery conjecture for all non-trivial Legendrian knots except possibly for a small exceptional family (±2 surgery on certain Lagrangian slice knots in knot types with τ=0, tb̄=-1, genus 2, and quasi-positive). The authors also classify cosmetic contact surgeries on Legendrian unknots, showing a mix of unique and infinitely-many cosmetic cases. The conjecture remains as the overarching open problem outside the resolved cases.