Invariants of Legendrian and transverse knots in monopole knot homology

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 Vértesi. 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.

28 pages, 7 figures; this paper was originally part of arXiv:1403.1930

Cited by in corpus (5)

• Khovanov homology detects the trefoils
• Naturality in sutured monopole and instanton homology
• On the equivalence of contact invariants in sutured Floer homology theories
• Functoriality of the EH class and the LOSS invariant under Lagrangian concordances
• Upper bounds for the Lagrangian cobordism relation on Legendrian links