Monopole Floer homology and Legendrian knots

Abstract

We use monopole Floer homology for sutured manifolds to construct invariants of unoriented Legendrian knots in a contact $3$–manifold. These invariants assign to a knot $K\subset Y$ elements of the monopole knot homology $KHM\left(-Y,K\right)$, and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We prove several vanishing results, investigate their behavior under contact surgeries, and use this to construct many examples of nonloose knots in overtwisted $3$–manifolds. We also show that these invariants are functorial with respect to Lagrangian concordance.