Related papers: Morse structures on open books
We study open book foliations on surfaces in 3-manifolds, and give applications to contact geometry of dimension 3. We prove a braid-theoretic formula of the self-linking number of transverse links, which reveals an unexpected link to the…
We extend Turaev's definition of torsion invariants of 3-dimensional manifolds equipped with non-singular vector fields, by allowing (suitable) tangency circles to the boundary, and manifolds with non-zero Euler characteristic. We show that…
We use contact fiber sums of open book decompositions to define an infinite hierarchy of filling obstructions for contact 3-manifolds, called planar k-torsion for nonnegative integers k, all of which cause the contact invariant in Embedded…
We use an estimate on the Thurston--Bennequin invariant of a Legendrian link in terms of its Kauffman-polynomial to show that links of topological unknots, e.g. the Borromean rings or the Whithead link, may not be represented by Legendrian…
In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that…
We study the relation of an embedded Lagrangian cobordism between two closed, orientable Legendrian submanifolds of R^{2n+1}. More precisely, we investigate the behavior of the Thurston-Bennequin number and (linearized) Legendrian contact…
Final revision. To appear in the Journal of Differential Geometry. This paper studies knots that are transversal to the standard contact structure in $\reals^3$, bringing techniques from topological knot theory to bear on their transversal…
Sivek proves a "van Kampen" decomposition theorem for the combinatorial Legendrian contact algebra (also known as the Chekanov-Eliashberg algebra) of knots in standard contact $\R^3$ . We prove an analogous result for the holomorphic curve…
We define relative versions of the classical invariants of Legendrian and transverse knots in contact 3-manifolds for knots that are homologous to a fixed reference knot. We show these invariants are well-defined and give some basic…
In this survey, we give an overview of Brieskorn manifolds and varieties, and their role in contact topology. We discuss open books, fillings and invariants such as contact and symplectic homology. We also present some new results involving…
A theorem of Ding and Geiges states that every closed, connected contact $3$-manifold can be obtained from the standard tight contact $3$-sphere by contact $(\pm1)$-surgery along a Legendrian link. The literature also contains some examples…
We show that on any closed contact manifold of dimension greater than 1 a contact structure with vanishing contact homology can be constructed. The basic idea for the construction comes from Giroux. We use a special open book decomposition…
It is shown that Legendrian (resp. transverse) cable links in the 3-sphere with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the…
Given an open book decomposition of a contact three man-ifold (M, $\xi$) with pseudo-Anosov monodromy and fractional Dehn twist coefficient c = k n, we construct a Legendrian knot $\Lambda$ close to the stable foliation of a page, together…
We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K…
The paper deals with topologically trivial Legendrian knots in tight and overtwisted contact 3-manifolds. The first part contains a thorough exposition of the proof of the classification of topologically trivial Legendrian knots (i.e.…
Fractional Dehn twists give a measure of the difference between the relative isotopy class of a homeomorphism of a bordered surface and the Thurston representative of its free isotopy class. We show how to estimate and compute these…
We construct new knot polynomials. Let $V$ be the standard solid torus in 3-space and let $pr$ be its standard projection onto an annulus. Let $M$ be the space of all smooth oriented knots in $V$ such that the restriction of $pr$ is an…
We provide a translation between Chekanov's combinatorial theory for invariants of Legendrian knots in the standard contact R^3 and a relative version of Eliashberg and Hofer's Contact Homology. We use this translation to transport the idea…
We study Legendrian embeddings of a compact Legendrian submanifold $L$ sitting in a closed contact manifold $(M,\xi)$ whose contact structure is supported by a (contact) open book $\mathcal{OB}$ on $M$. We prove that if $\mathcal{OB}$ has…