Non-loose Legendrian spheres with trivial contact homology DGA

Abstract

Loose Legendrian n-submanifolds, n 2, were introduced by Murphy ('Loose Legendrian embeddings in high dimensional contact manifolds ', Preprint, 2012, arXiv:1201.2245) and proved to be flexible in the h-principle sense: any two loose Legendrian submanifolds that are formally Legendrian isotopic are also actually Legendrian isotopic. Legendrian contact homology is a Floer theoretic invariant that associates a differential graded algebra (DGA) to a Legendrian submanifold. The DGA of a loose Legendrian submanifold is trivial. We show that the converse is not true by constructing non-loose Legendrian n-spheres in standard contact (2n + 1)-space, n 2, with trivial DGA.

Symplectic Fillings, Contact Surgeries, and Lagrangian Disks

“…For example, [15] shows that the maximal Thurston-Bennequin unknot essentially has one Lagrangian filling disk (at least if a suitably large portion of the symplectization of S 3 is added to B 4 ) and it is also known that contact (+1)-surgery on the maximal Thurston-Bennequin unknot has a unique Stein filling up to symplectomorphism. In contrast, Ekholm [12] proved that the knot 9 46 has two Lagrangian disk fillings which are not Hamiltonian isotopic. It is known that the complements of neighborhoods of these two ribbon disks are diffeomorphic 4-manifolds, but one naturally wonders whether or not these disks give rise to non-symplectomorphic fillings of contact (+1)-surgery on the Legendrian knot 9 46 .…”

Nearby Lagrangian fibers and Whitney sphere links

“…For sufficiently large we are then far from and can cap the fiber sphere off with a standard half of a Whitney sphere in over , see §§3.1 and 4.1. We point out that there is a rich class of non-standard Lagrangian disks (with Legendrian knotted), see [CNS16] for examples when , and [Ekh16, §2.4] for constructing analogous disks in higher dimensions.…”

A note on coherent orientations for exact Lagrangian cobordisms

“…For other examples of applications of such a signed lift which allows Φ L to be defined over the integers, see e.g. [CDRGG15], [CDRGG], [Ekh16], [EL]. Note that the existence of such a signed lift is indicated but not proved in these papers.…”