Obstructions to Lagrangian concordance

TL;DR

This paper explores conditions under which Lagrangian concordances exist between Legendrian knots, providing obstructions, restrictions, and classifications for knots with certain properties, especially focusing on the unknot and knots with up to 14 crossings.

Contribution

It introduces new obstructions to Lagrangian concordance based on normal rulings and classifies knots with Lagrangian slice representatives up to 14 crossings.

Findings

Obstructions to concordance from arbitrary knots to the unknot.

Restrictions on knots with concordances both to and from the unknot.

Complete classification of knots with up to 14 crossings that are Lagrangian slice.

Abstract

We investigate the question of the existence of a Lagrangian concordance between two Legendrian knots in $\mathbb{R}^3$. In particular, we give obstructions to a concordance from an arbitrary knot to the standard Legendrian unknot, in terms of normal rulings. We also place strong restrictions on knots that have concordances both to and from the unknot and construct an infinite family of knots with non-reversible concordances from the unknot. Finally, we use our obstructions to present a complete list of knots with up to 14 crossings that have Legendrian representatives that are Lagrangian slice.