Monopole Floer homology and Legendrian knots

TL;DR

This paper develops monopole Floer homology invariants for Legendrian knots in contact 3-manifolds, demonstrating their properties and applications to non-loose knots and Lagrangian concordance.

Contribution

It introduces new monopole Floer homology invariants for Legendrian knots and explores their behavior under contact surgeries and Lagrangian concordance.

Findings

Invariants resemble knot Floer homology invariants.

Proves vanishing results for these invariants.

Constructs examples of non-loose knots in overtwisted manifolds.

Abstract

We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold. These invariants assign to a knot K in Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsv\'ath, Stipsicz, and Szab\'o. We prove several vanishing results, investigate their behavior under contact surgeries, and use this to construct many examples of non-loose knots in overtwisted 3-manifolds. We also show that these invariants are functorial with respect to Lagrangian concordance.