相关论文: Legendrian and Transversal Knots

We investigate the ramifications of the Legendrian satellite construction on the relation of Lagrangian cobordism between Legendrian knots. Under a simple hypothesis, we construct a Lagrangian concordance between two Legendrian satellites…

We prove that loose Legendrian knots in a rational homology contact 3-sphere, satisfying some additional hypothesis, are Legendrian isotopic if and only if they have the same classical invariants. The proof requires a result of Dymara on…

In this article, we introduce rack invariants of oriented Legendrian knots in the 3-dimensional Euclidean space endowed with the standard contact structure, which we call Legendrian racks. These invariants form a generalization of the…

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these…

An expository article on Turaev surfaces written for "A Concise Encyclopedia of Knot Theory," to appear.

We construct infinite families of non-simple isotopy classes of links in overtwisted contact structures on $S^1$-bundles over surfaces. These examples include: (1) a pair of Legendrian links that are not Legendrian isotopic, but which are…

We investigate properties of the odd Khovanov homology, compare and contrast them with those of the original (even) Khovanov homology, and discuss applications of the odd Khovanov homology to other areas of knot theory and low-dimensional…

This is a survey article to appear in the "Handbook on Teichmueller Theory".

We construct equisingular semiuniversal deformations of Legendrian curves.

We investigate the question of the existence of a Lagrangian concordance between two Legendrian knots in $\mathbb{R}^3$. In particular, we give obstructions to a concordance from an arbitrary knot to the standard Legendrian unknot, in terms…

In this paper we give necessary and sufficient conditions for a knot type to admit non-loose Legendrian and transverse representatives in some overtwisted contact structure, classify all non-loose rational unknots in lens spaces, and…

For a knot $K$ the cube number is a knot invariant defined to be the smallest $n$ for which there is a cube diagram of size $n$ for $K$. There is also a Legendrian version of this invariant called the \emph{Legendrian cube number}. We will…

The present paper is an addendum to "Spherical structures on torus knots and links", arXiv:1008.0312, and concerns more general case of torus knot and link cone-manifolds.

This article is solicited by C.\ Adams for a special issue of {\it Chaos, Solitons and Fractals\/} devoted to knot theory and its applications. We present some recent results about Dehn surgeries on arborescent knots and links.

In this essay dedicated to Yakov Eliashberg we survey the current state of the field of Lagrangian (un)knots, reviewing some constructions and obstructions along with a number of unsolved questions. The appendix by Georgios Dimitroglou…

For any knot T transverse to a given contact structure on a 3-manifold, we exhibit a Legendrian two-component link such that T equals the transverse push-off of one of the link components and contact (+1)-surgery on the link has the same…

This paper studies an algebraic invariant of virtual knots called the biquandle. The biquandle generalizes the fundamental group and the quandle of virtual knots. The approach taken in this paper to the biquandle emphasizes understanding…

Perverse schober defined by Kapranov--Schechtman is a categorification of the notion of perverse sheaf. In their definition, a key ingredient is certain purity property of perverse sheaves. In this short note, we attempt to describe a real…

In this note we show that $+1$-contact surgery on distinct Legendrian knots frequently produces contactomorphic manifolds. We also give examples where this happens for $-1$-contact surgery. As an amusing corollary we find overtwisted…

This paper is a very brief introduction to knot theory. It describes knot coloring by quandles, the fundamental group of a knot complement, and handle-decompositions of knot complements.