Related papers: Legendrian helix and cable links

We study algebraic aspects of generalized Legendrian racks, which are nonassociative structures based on the Legendrian Reidemeister moves. We answer an open question characterizing the group of GL-structures on a given rack. As…

For every link $L$ we construct a complex algebraic plane curve that intersects $S^3$ transversally in a link $\tilde{L}$ that contains $L$ as a sublink. This construction proves that every link $L$ is the sublink of a quasipositive link…

A new classification theorem for links by the authors and Roger Fenn leads to computable link invariants. As an illustration we distinguish the left and right trefoils and recover the result of Carter et al that the 2-twist-spun trefoil is…

We investigate families of Legendrian submanifolds of 1-jet spaces by developing and applying a theory of families of generating family homologies. This theory allows us to detect an infinite family of loops of Legendrian n-spheres embedded…

We prove some classification results for tight contact structure in the 3-space, -ball and -sphere that are invariant with respect to some arbitrary involution, that is conjugated to the standard rotation around the x-axis. Unlike the…

The characteristic varieties of a space are the jump loci for homology of rank 1 local systems. The way in which the geometry of these varieties may vary with the characteristic of the ground field is reflected in the homology of finite…

We prove that a version of the Thurston-Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere $(M,\xi)$, whenever $\xi$ is tight. More specifically, we show that the self-linking number of a…

In a recent paper we constructed a family of foliated 2-complexes of thin type whose typical leaves have two topological ends. Here we present simpler examples of such complexes that are, in addition, symmetric with respect to an involution…

Let $\lambda$ be a Legendrian link in standard contact $\mathbb{R}^3$, such that $L_1$, $L_2$ are two exact fillings of $\lambda$ and $\varphi$ is a Legendrian loop of $\lambda$. We study fillability and isotopy characterizations of…

We compare two associative algebras which encode the "quantum topology" of Legendrian curves in contact threefolds of product type $S\times\mathbb R$. The first is the skein algebra of graded Legendrian links and the second is the Hall…

We apply a recently proposed definition of a linear connection in non commutative geometry based on the natural bimodule structure of the algebra of differential forms to the case of the two-parameter quantum plane. We find that there…

Given a holomorphic line bundle $L$ on a compact complex torus $A$, there are two naturally associated holomorphic $\Omega_A$--torsors over $A$: one is constructed from the Atiyah exact sequence for $L$, and the other is constructed using…

In this paper, we study contact surgeries along Legendrian links in the standard contact 3-sphere. On one hand, we use algebraic methods to prove the vanishing of the contact Ozsv\'{a}th-Szab\'{o} invariant for contact $(+1)$-surgery along…

We provide an explicit example of a non trivial Legendrian knot $\Lambda$ such that there exists a Lagrangian concordance from $\Lambda_0$ to $\Lambda$ where $\Lambda_0$ is the trivial Legendrian knot. We then use the map induced in…

It has been observed that certain classical chains admit topologically protected zero-energy modes that are localized on the boundaries. The static features of such localized modes are captured by linearized equations of motion, but the…

We give new examples of 2-component links with linking number one and unknotted components that are topologically concordant to the positive Hopf link, but not smoothly so - in fact they are not smoothly concordant to the positive Hopf link…

We consider all compatible topologies of an arbitrary finite-dimensional vector space over a non-trivial valuation field whose metric completion is a locally compact space. We construct the canonical lattice isomorphism between the lattice…

We investigate the geometry of Legendrian complex projective manifolds $X\subset\PP V$. By definition, this means $V$ is a complex vector space of dimension $2n+2$, endowed with a symplectic form, and the affine tangent space to $X$ at each…

In an earlier paper we introduced rectangular diagrams of surfaces and showed that any isotopy class of a surface in the three-sphere can be presented by a rectangular diagram. Here we study transformations of those diagrams and introduce…

Let $Z \to Y^{2n+1}$ be the bundle of Legendrian $n$-planes over a contact manifold $Y$. We consider a foliation of $Z$ by canonical lifts of Legendrian submanifolds, called \emph{Legendrian submanifold path geometry}, whose flat model is…