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.

Obstructions to Lagrangian concordance

“…The second author and Rutherford [35,Theorem 5.4] showed that if the ruling polynomial R Λ (z) has positive degree then Λ is A (2) -compatible. 1 The following result is similar but allows for R Λ (z) to be a constant as well.…”

“…There are infinitely many Legendrian knots Λ such that U ≺ Λ but Λ ≺ U . 1 The convention in [35] is that a ruling ρ with s switches and c right cusps contributes z s−c to the ruling polynomial, so their condition is deg RΛ(z) ≥ 0. We instead use the convention z s−c+1 of [40], so that the 2-graded and ungraded ruling polynomials match the appropriate coefficients of the HOMFLY-PT and Kauffman polynomials respectively.…”

“…In symplectic and contact topology, there has been a great deal of recent interest in the subject of Lagrangian cobordisms between Legendrian submanifolds; see for example [1,3,5,6,8,9,14,13,25,26,42]. A key motivation is that one can construct a category whose objects are Legendrian submanifolds and whose morphisms are exact Lagrangian cobordisms, and this category fits nicely into Symplectic Field Theory [15].…”

“…We also use this result in Theorem 2.10 to construct an infinite family of Legendrian knots Λ with U ≺ Λ and Λ ≺ U . It should be noted that Baldwin and the third author [1] previously presented a different infinite family of non-reversible concordances, involving stabilized unknots rather than U .…”

Nonorientable Lagrangian cobordisms between Legendrian knots

“…Given the existence of exact, non-orientable Lagrangian endocobordisms for a stabilized Legendrian, it is natural to ask: What Legendrian knots can appear as a "slice" of such an endocobordism? The parallel question for orientable Lagrangian endocobordisms has been studied in [9,4,12]. The non-orientable version of this question is closely tied to the question of whether non-orientable Lagrangian cobordisms define an equivalence relation on the set of Legendrian knots.…”