Instability of Legendrian knottedness, and non-regular Lagrangian concordances of knots

TL;DR

This paper explores the instability of Legendrian knottedness, constructs non-regular Lagrangian concordances between knots, and demonstrates that Lagrangian concordance does not form a partial order, revealing new flexibility in the field.

Contribution

It introduces the first examples of non-regular Lagrangian concordances and shows the non-antisymmetry of Lagrangian concordance relations.

Findings

Legendrian pretzel knots with same invariants have isotopic front-spuns

Constructed non-regular Lagrangian concordances between Legendrian knots

Lagrangian concordance relation is not antisymmetric

Abstract

We show that the family of smoothly non-isotopic Legendrian pretzel knots from the work of Cornwell-Ng-Sivek that all have the same Legendrian invariants as the standard unknot have front-spuns that are Legendrian isotopic to the front-spun of the unknot. Besides that, we construct the first examples of Lagrangian concordances between Legendrian knots that are not regular, and hence not decomposable. Finally, we show that the relation of Lagrangian concordance between Legendrian knots is not anti-symmetric, and hence does not define a partial order. The latter two results are based upon a new type of flexibility for Lagrangian concordances with stabilised Legendrian ends.