Lagrangian Zigzag Cobordisms

TL;DR

This paper introduces and explores Lagrangian zigzag cobordisms as an equivalence relation among Legendrian knots, analyzing its properties, relations to other concordance notions, and implications for Legendrian knot invariants.

Contribution

It defines Lagrangian zigzag cobordisms, compares them to existing concordance relations, and studies the algebraic structure of their classes, including torsion and satellite operations.

Findings

Lagrangian zigzag cobordism forms a monoid structure.

Structural results on torsion and satellite operators are established.

Connections to non-classical Legendrian invariants are discussed.

Abstract

We investigate an equivalence relation on Legendrian knots in the standard contact three-space defined by the existence of an interpolating zigzag of Lagrangian cobordisms. We compare this relation, restricted to genus-$0$ surfaces, to smooth concordance and Lagrangian concordance. We then study the metric monoid formed by the set of Lagrangian zigzag concordance classes, which parallels the metric group formed by the set of smooth concordance classes, proving structural results on torsion and satellite operators. Finally, we discuss the relation of Lagrangian zigzag cobordism to non-classical invariants of Legendrian knots.