arXiv research

Let $(X^{m+1}, g)$ be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of $\mathbb R^m$. The Legendrian Low conjecture formulated by Natário and Tod says that two events $x,y\inß$ are causally related if and only if the Legendrian link of spheres $\mathfrak S_x, \mathfrak S_y$ whose p…

The set N of all null geodesics of a globally hyperbolic (d+1)-dimensional spacetime (M,g) is naturally a smooth (2d-1)-dimensional contact manifold. The sky of an event is the subset of N defined by all null geodesics through that event, and is an embedded Legendrian submanifold of N diffeomorphic to a (d-1)-dimension…

The study generalizes causality and linking conjectures in spacetimes.

New algebraic structures help distinguish Legendrian knots.

We construct, somewhat non-standard, Legendrian surgery diagrams for some Stein fillable contact structures on some plumbing trees of circle bundles over spheres. We then show how to put such a surgery diagram on the pages of an open book for $S^3,$ with relatively low genus. Thus we produce open books with low genus p…

Legendrian products can be twist spuns under certain conditions.

Innovative rack theory applied to Legendrian links.

The study finds non-simple isotopy classes of links in 3-manifolds, including Legendrian and pseudo-Legendrian examples.

An elementary stabilization of a Legendrian link $L$ in the spherical cotangent bundle $ST^*M$ of a surface $M$ is a surgery that results in attaching a handle to $M$ along two discs away from the image in $M$ of the projection of the link $L$. A virtual Legendrian isotopy is a composition of stabilizations, destabiliz…

We investigate Legendrian graphs in $(\R^3, ξ_{std})$. We extend the classical invariants, Thurston-Bennequin number and rotation number to Legendrian graphs. We prove that a graph can be Legendrian realized with all its cycles Legendrian unknots with $tb=-1$ and $rot=0$ if and only if it does not contain $K_4$ as a mi…

Classifies Legendrian torus and cable links, revealing symmetries and invariants.

4-Legendrian permutation racks can't distinguish knots but recover classical invariants.

A reconstruction theorem in terms of the topology and geometrical structures on the spaces of light rays and skies of a given space-time is discussed. This result can be seen as part of Penrose and Low's programme intending to describe the causal structure of a space-time $M$ in terms of the topological and geometrical…

Generalizes ruling polynomials to Legendrian tangles and proves their composition property.

The abstract discusses conjectures about virtual Legendrian knots and their relation to causality.

Legendrian Lavrentiev links are shown to be equivalent to smooth links.

Loose Legendrian knots in rational homology spheres are Legendrian isotopic if they have the same classical invariants.

Classifies exceptional Legendrian realizations of Hopf link connected sums.

Introduces a Cost function to measure Legendrian knot obstructions.

In this paper we study Legendrian knots in the knot types of satellite knots. In particular, we classify Legendrian Whitehead patterns and learn a great deal about Legendrian braided patterns. We also show how the classification of Legendrian patterns can lead to a classification of the associated satellite knots if th…

The study of the Vassiliev invariants of Legendrian knots was started by D. Fuchs and S. Tabachnikov who showed that the groups of complex-valued Vassiliev invariants of Legendrian and of framed knots in the standard contact $R^3$ are canonically isomorphic. Recently we constructed the first examples where Vassiliev in…

Study of cable links of uniformly thick knots, revealing new isotopy phenomena.

The Gronwall conjecture states that a planar 3-web of foliations which admits more than one distinct linearizations is locally equivalent to an algebraic web. We propose an analogue of the Gronwall conjecture for the 3-web of foliations by Legendrian curves in a contact three manifold. The Legendrian Gronwall conjectur…

New distances defined on Legendrian spaces without positive loops.

The study constructs and shows isotopy of high-dimensional Legendrian spheres.

Projection maps virtual Legendrian knots to classical ones.

A new algebra for Legendrian graphs defined combinatorially.

Characterizes Legendrian knots in lens spaces.

Study of strongly invertible Legendrian links in contact 3-space.

Persistent Legendrian contact homology distinguishes knots using height functional.

The problem of classification of Legendrian knots (links) up to isotopy in the class of Legendrian embeddings (Legendrian isotopy) naturally leads to the following two subproblems. The first of them is: which combinations of the three classical invariants can be realized by a Legendrian knot? (It is well-known that eac…

In this paper Legendrian graphs in $(\mathbb{R}^3,ξ_{\mathrm{st}})$ are considered modulo Legendrian isotopy and edge contraction. To a Legendrian graph we associate a (generalized) rectangular diagram --- a purely combinatorial object. Moves of rectangular diagrams are introduced so that equivalence classes of Legendr…

The paper classifies Willmore Legendrian surfaces in S^5 and studies their properties.

Minimal Legendrian surfaces found in 5D sphere.

Study the spaces of Legendrian knots and links with maximal Thurston-Bennequin invariant.

New methods link Legendrian satellites to Lagrangian cobordisms.

Classifies Legendrian Hopf links in 3-sphere.

New algebra structure for Legendrian knots preserves contact homology invariants.

The surgery unknotting number of a Legendrian link is defined as the minimal number of particular oriented surgeries that are required to convert the link into a Legendrian unknot. Lower bounds for the surgery unknotting number are given in terms of classical invariants of the Legendrian link. The surgery unknotting nu…

Characterizes Legendrian knots with exact Lagrangian fillings.

Computes bounds on mosaic number of Legendrian knots.

We give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in PxR, where P is a punctured Riemann surface. As an application we show that for any integer k and any homology class h in H_1(PxR) there are k Legendrian knots all representing h which are pairwise smoot…

Study on exact Lagrangian submanifolds in unit ball with Legendrian boundary.

A rigidity theorem for smooth Legendrian self-shrinkers is proven.

Study Legendrian surfaces using N-graphs and flag moduli.

Study of Legendrian links using Floer theory and cluster varieties.

New method confirms conjectures about specific Legendrian knots.

New integer-valued functions for Legendrian knots.