Quasipositive surfaces and decomposable Lagrangians

TL;DR

This paper demonstrates how quasipositive surfaces influence knot Floer homology and uses this to obstruct certain Lagrangian cobordisms, advancing understanding in contact and symplectic topology.

Contribution

It introduces a new map induced by quasipositive surfaces on knot Floer homology that preserves the transverse invariant, with applications to Lagrangian cobordisms.

Findings

The induced map preserves the transverse invariant.

The invariant obstructs decomposable Lagrangian cobordisms.

Recovery of naturality statements under contact +1 surgery.

Abstract

We show that a quasipositive surface with disconnected boundary induces a map between the knot Floer homology groups of its boundary components preserving the transverse invariant. As an application, we show that this invariant can be used to obstruct decomposable Lagrangian cobordisms of arbitrary genus within Weinstein cobordisms. The construction of our maps rely on the comultiplicativity of the transverse invariant. Along the way, we also recover various naturality statements for the invariant under contact +1 surgery.