Legendrian Lavrentiev links

TL;DR

This paper introduces Legendrian Lavrentiev links, a special class of curves in contact geometry, and proves their isotopy classes align with smooth classes, bridging geometric and topological perspectives.

Contribution

It defines Legendrian Lavrentiev curves and establishes that their Legendrian isotopy classes are equivalent to smooth isotopy classes.

Findings

Legendrian Lavrentiev links are characterized by contact form integrals.

Legendrian isotopies preserve the equivalence classes of these links.

The classes of Legendrian Lavrentiev links coincide with smooth classes.

Abstract

Lavrentiev curves form a special class of rectifiable curves which includes cusp-free piecewise smooth curves. We call a Lavrentiev curve Legendrian if the integral of the contact form equals zero on any its subarc. We define Legendrian isotopies of such curves and prove that the equivalence classes of Legendrian Lavrentiev links with respect to Legendrian isotopies coincide with smooth classes.