$C^0$-limits of Legendrian knots and contact non-squeezing

TL;DR

This paper investigates the behavior of Legendrian knots under $C^0$-limits of contactomorphisms, showing that Legendrian knots remain Legendrian in the limit, and introduces a contact non-squeezing property distinguishing Legendrian from non-Legendrian knots.

Contribution

It establishes that Legendrian knots are preserved under $C^0$-limits of contactomorphisms and introduces a contact non-squeezing theorem based on the Thurston--Bennequin inequality.

Findings

Legendrian knots remain Legendrian under $C^0$-limits of contactomorphisms.

Non-Legendrian knots can be contact-squeezed onto transverse knots.

Legendrian knots cannot be contact-squeezed onto transverse knots.

Abstract

Take a sequence of contactomorphisms of a contact three-manifold that $C^0$-converges to a homeomorphism. If the images of a Legendrian knot limit to a smooth knot under this sequence, we show that it is Legendrian. We prove this by establishing that, on one hand, non-Legendrian knots admit a type of contact-squeezing onto transverse knots while, on the other, Legendrian knots do not admit such a squeezing. The non-trivial input from contact topology that is needed is (a local version of) the Thurston--Bennequin inequality.