Let @math be a contact form on a manifold @math , and @math a closed Legendrian submanifold. I prove that @math intersects some characteristic for @math at least twice if all characteristics are closed and of the same period, and @math embeds nicely into the product of @math and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most @math . This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.
C. Abbas @cite_17 @cite_5 @cite_19 proved the strict chord property for certain Legendrian knots in tight closed contact 3-manifolds.
@cite_19 @cite_5 @cite_17
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Arnold conjectured that every Legendrian knot in the standard contact structure on the 3-sphere possesses a characteristic chord with respect to any contact form. I confirm this conjecture if the know has Thurston-Bennequin invariant −1. More generally, existence of chords is proved for a standard Legendrian unknot on the boundary of a subcritical Stein manifold of any dimension. There is also a multiplicity result which implies in some situations existence of infinitely many chords.¶The proof relies on the behaviour of symplectic homology under handle attaching. The main observation is that symplectic homology only changes in the presence of chords. Let (S, fλ) be the three sphere endowed with a contact form fλ, where λ is the standard tight contact form on S and f : S → R 0 is a smooth function. Moreover let L ⊂ S be a Legendrian knot. Then V.I. Arnold raised the conjecture in 1986 that there is a characteristic chord, i.e. an orbit x of the Reeb vectorfield that intersects the knot L at two different times. Characteristic chords are important in the theory of contact homology, see [6], [5]. Very little is known about their existence or non-existence . Using a deep result by H. Hofer, K. Wysocki and E. Zehnder about the existence of a global surface of section in strictly convex hypersurfaces in R, we observe that characteristic chords exist for certain Legendrian knots in strictly convex hypersurfaces in R. Let M be a closed three-dimensional manifold with contact form λ so that Ker λ is tight. In this paper, we will present a first application of the filling method by pseudoholomorphic curves recently developed by the author. We will show that Legendrian knots L ⊂ M satisfying suitable assumptions admit a Reeb Chord, that is, there is a trajectory x(t) of the Reeb vector field and T > 0 such that x(0),x(T) ∈ L and x(0) ≠ x(T). Given a knot K in the 3-sphere, let QK be its fundamental quan- dle as introduced by D.Joyce. Its first homology group is easily seen to be H1(QK) = Z. We prove that H2(QK) = 0 if and only if K is trivial, and H2(QK) = Z whenever K is non-trivial. An analogous result holds for links, thus characterizing trivial components. More detailed information can be derived from the conjugation quandle: let Q K be the conjugacy class of a meridian in the knot group 1(S 3 rK). We show that H2(Q K) = Z p , where p is the number of prime summands in a connected sum decomposition of K.