Invariants of Legendrian and transverse knots in monopole knot homology

TL;DR

This paper introduces a new Legendrian knot invariant in monopole knot homology that is preserved under negative stabilization, enabling the definition of a transverse knot invariant and advancing the understanding of Lagrangian concordances.

Contribution

The authors develop an improved Legendrian invariant in monopole knot homology that remains invariant under negative stabilization, facilitating the study of transverse knots and Lagrangian concordances.

Findings

The new invariant is preserved by negative stabilization.

It enables defining a transverse knot invariant in KHM.

The paper provides the first infinite family of nonreversible Lagrangian concordances.

Abstract

We use the contact invariant defined in [2] to construct a new invariant of Legendrian knots in Kronheimer and Mrowka's monopole knot homology theory (KHM), following a prescription of Stipsicz and V\'ertesi. Our Legendrian invariant improves upon an earlier Legendrian invariant in KHM defined by the second author in several important respects. Most notably, ours is preserved by negative stabilization. This fact enables us to define a transverse knot invariant in KHM via Legendrian approximation. It also makes our invariant a more likely candidate for the monopole Floer analogue of the "LOSS" invariant in knot Floer homology. Like its predecessor, our Legendrian invariant behaves functorially with respect to Lagrangian concordance. We show how this fact can be used to compute our invariant in several examples. As a byproduct of our investigations, we provide the first infinite family of nonreversible Lagrangian concordances between prime knots.