Legendrian Satellites and Decomposable Concordances

TL;DR

This paper explores how Legendrian satellite constructions affect Lagrangian cobordisms between Legendrian knots, providing new methods to construct and obstruct such cobordisms, and identifying conditions for decomposability.

Contribution

It introduces a construction for Lagrangian concordances between Legendrian satellites and establishes obstructions based on Thurston-Bennequin numbers.

Findings

Constructed Lagrangian concordances via elementary cobordisms.

Identified new families of decomposable Lagrangian slice knots.

Showed Thurston-Bennequin number bounds obstruct certain satellites.

Abstract

We investigate the ramifications of the Legendrian satellite construction on the relation of Lagrangian cobordism between Legendrian knots. Under a simple hypothesis, we construct a Lagrangian concordance between two Legendrian satellites by stacking up a sequence of elementary cobordisms. This construction narrows the search for "non-decomposable" Lagrangian cobordisms and yields new families of decomposable Lagrangian slice knots. Finally, we show that the maximum Thurston-Bennequin number of a smoothly slice knot provides an obstruction to any Legendrian satellite of that knot being Lagrangian slice.