John B. Etnyre

Legendrian and Transversal Knots

Publisher Summary Contact structures on manifolds and Legendrian and transversal knots in them are very natural objects, born over a century ago, in the work of Huygens, Hamilton, and Jacobi on geometric optics and work of Lie on partial differential equations. This chapter discusses Legendrian and transversal knots in dimension three where their theory is most fully developed and where they are most intimately tied to topology. In this dimension a predominately topological and combinatorial approach may be used to their study. The chapter then examines higher dimensional Legendrian knots. The chapter also explains the concept of Lagrangian protection. Analogous to the definition of finite type invariants of topological knots, finite type invariants of Legendrian knots can be defined. It is then possible to get a global contact isotopy if there is a fixed over twisted disk in the complement of both Legendrian knot. A complete understanding of Legendrian unknots in over twisted contact structures must wait for future work.

On the nonexistence of tight contact structures

We exhibit a 3-manifold which admits no tight contact structure.

Planar open book decompositions and contact structures

We observe that while all overtwisted contact structures on compact 3-manifolds are supported by planar open book decompositions, not all contact structures are. This has relevance to the Weinstein conjecture and invariants of contact structures.

ORIENTATIONS IN LEGENDRIAN CONTACT HOMOLOGY AND EXACT LAGRANGIAN IMMERSIONS

We show how to orient moduli spaces of holomorphic disks with boundary on an exact Lagrangian immersion of a spin manifold into complex n-space in a coherent manner. This allows us to lift the coefficients of the contact homology of Legendrian spin submanifolds of standard contact (2n + 1)-space from ℤ2 to ℤ. We demonstrate how the ℤ-lift provides a more refined invariant of Legendrian isotopy. We also apply contact homology to produce lower bounds on double points of certain exact Lagrangian immersions into ℂn and again including orientations strengthens the results. More precisely, we prove that the number of double points of an exact Lagrangian immersion of a closed manifold M whose associated Legendrian embedding has good DGA is at least half of the dimension of the homology of M with coefficients in an arbitrary field if M is spin and in ℤ2 otherwise.

On connected sums and Legendrian knots

Abstract We study the behavior of Legendrian and transverse knots under the operation of connected sums. As a consequence we show that there exist Legendrian knots that are not distinguished by any known invariant. Moreover, we classify Legendrian knots in some non-Legendrian-simple knot types.

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].

Invariants of contact structures from open books

In this note we define three invariants of contact structures in terms of open books supporting the contact structures. These invariants are the support genus (which is the minimal genus of a page of a supporting open book for the contact structure), the binding number (which is the minimal number of binding components of a supporting open book for the contact structure with minimal genus pages) and the norm (which is minus the maximal Euler characteristic of a page of a supporting open book).

Symplectic convexity in low-dimensional topology

Abstract In this paper we will survey the various forms of convexity in symplectic geometry, paying particular attention to applications of convexity in low-dimensional topology.

On symplectic fillings

In this note we make several observations concerning symplectic fillings. In particular we show that a (strongly or weakly) semi-fillable contact structure is fillable and any filling embeds as a symplectic domain in a closed symplectic manifold. We also relate properties of the open book decomposition of a contact manifold to its possible fillings. These results are also useful in showing the contact Heegaard Floer invariant of a fillable contact structure does not vanish [28] and property P for knots [18].

TIGHT CONTACT STRUCTURES ON LENS SPACES

In this paper we develop a method for studying tight contact structures on lens spaces. We then derive uniqueness and non-existence statements for tight contact structures with certain (half) Euler classes on lens spaces.