Doubly slice knots and obstruction to Lagrangian concordance

TL;DR

This paper shows how the doubly slice genus can obstruct certain Legendrian knots from being concordant to the trivial knot, providing new obstructions where Legendrian contact homology fails.

Contribution

It introduces a novel obstruction method for Legendrian knot concordance using doubly slice genus, expanding the toolkit beyond Legendrian contact homology.

Findings

Obstructed concordances from P(3,-3,-m) to the unknot for m > 3

Doubly slice genus as an effective obstruction tool

Limitations of Legendrian contact homology in these cases

Abstract

In this short note we observe that a result of Eliashberg and Polterovitch allows to use the doubly slice genus as an obstruction for a Legendrian knot to be a slice of a concordance from the trivial Legendrian knot with maximal Thurston-Bennequin invariant to itself. This allows to obstruct concordances from the Pretzel knot P(3,-3,-m) when m > 3 to the unknot. Those examples are of interest because the Legendrian contact homology algebra cannot be used to obstruct such a concordance.