On knots in overtwisted contact structures

TL;DR

This paper demonstrates that overtwisted contact structures contain infinitely many distinct transverse knots sharing the same self-linking number, classifies some of these knots, and explores implications for Legendrian knots and classical invariants.

Contribution

It proves the existence of infinitely many transverse knots with identical invariants in overtwisted structures and classifies certain cases, advancing understanding of knot types in contact topology.

Findings

Infinitely many transverse knots with same self-linking number in overtwisted structures.

Classification results for some transverse and Legendrian knots.

Discussion of open problems in Legendrian knot 'geography' and 'botany' in overtwisted contact structures.

Abstract

We prove that each overtwisted contact structure has knot types that are represented by infinitely many distinct transverse knots all with the same self-linking number. In some cases, we can even classify all such knots. We also show similar results for Legendrian knots and prove a "folk" result concerning loose transverse and Legendrian knots (that is knots with overtwisted complements) which says that such knots are determined by their classical invariants (up to contactomorphism). Finally we discuss how these results partially fill in our understanding of the "geography" and "botany"' problems for Legendrian knots in overtwisted contact structures, as well as many open questions regarding these problems.