Legendrian Satellites and Decomposable Cobordisms

TL;DR

This paper explores how Legendrian satellite constructions interact with Lagrangian cobordisms, showing that certain cobordisms between knots induce cobordisms between their satellites, with added twists to account for genus differences.

Contribution

It introduces a method to construct Lagrangian cobordisms between Legendrian satellites from existing cobordisms, including decomposable cases, with adjustments for genus via extra twists.

Findings

Constructs Lagrangian cobordisms between satellites from original cobordisms.

Shows decomposable cobordisms between knots induce decomposable cobordisms between satellites.

Provides a framework for understanding satellite interactions in Legendrian knot theory.

Abstract

We investigate the interactions between the Legendrian satellite construction and the existence of exact, orientable Lagrangian cobordisms between Legendrian knots. Given Lagrangian cobordisms between two Legendrian knots and between two Legendrian tangles, we construct a Lagrangian cobordism between Legendrian satellites of the knots by the closures of the tangles, with extra twists on both the top and the bottom satellite to compensate for the genus of the cobordism. If the original cobordisms were decomposable, then a decomposable cobordism between satellites exists as well, again with extra twists.