Tobias Ekholm

A duality exact sequence for legendrian contact homology

We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact homology CH_* which maps to singular homology. In particular, the sequence implies a duality between the kernel of the map (CH_*\to H_*) and the cokernel of the map (H_* \to CH^*). Furthermore, this duality is compatible with Poincare duality in L in the following sense: the Poincare dual of a singular class which is the image of a in CH_* maps to a class \alpha in CH^* such that \alpha(a)=1. The exact sequence generalizes the duality for Legendrian knots in Euclidean 3-space [24] and leads to a refinement of the Arnold Conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [6].

Rational SFT, Linearized Legendrian Contact Homology, and Lagrangian Floer Cohomology

We relate the version of rational symplectic field theory for exact Lagrangian cobordisms introduced in [6] to linearized Legendrian contact homology. More precisely, if L ⊂ Xis an exact Lagrangian submanifold of an exact symplectic manifold with convex end Λ ⊂ Y, where Yis a contact manifold and Λis a Legendrian submanifold, and if Lhas empty concave end, then the linearized Legendrian contact cohomology of Λ, linearized with respect to the augmentation induced by L, equals the rational SFT of (X,L). Following ideas of Seidel [15], this equality in combination with a version of Lagrangian Floer cohomology of Lleads us to a conjectural exact sequence that in particular implies that if \(X = {\mathbb{C}}^{n}\), then the linearized Legendrian contact cohomology of Λ ⊂ S2n − 1is isomorphic to the singular homology of L. We outline a proof of the conjecture and show how to interpret the duality exact sequence for linearized contact homology of [7] in terms of the resulting isomorphism.

Morse flow trees and Legendrian contact homology in 1-jet spaces

Let L ⊂ J 1 (M) be a Legendrian submanifold of the 1-jet space of a Riemannian n-manifold M. A correspondence is established between rigid flow trees in M determined by L and boundary punctured rigid pseudo-holomorphic disks in TM, with boundary on the projection of L and asymptotic to the double points of this projection at punctures, provided n ≤ 2, or provided n > 2 and the front of L has only cusp edge singularities. This result, in particular, shows how to compute the Legendrian contact homology of L in terms of Morse theory.

Filtrations on the knot contact homology of transverse knots

We construct a new invariant of transverse links in the standard contact structure on

Algorithm for generating a Brownian motion on a sphere

{\mathbb R }^3.

Brownian dynamics simulations on a hypersphere in 4-space

This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link, see (Ekholm et al., Knot contact homology, Arxiv:1109.1542, 2011; Ng, Duke Math J 141(2):365–406, 2008). Here the knot contact homology of a link in

Symplectic homology product via Legendrian surgery.

{\mathbb R }^3