Related papers: Tied links in the solid torus
We obtain the full list of Goeritz invariants of all torus knots and links.
This paper is a self-contained development of an invariant of graphs embedded in three-dimensional Euclidean space using the Jones polynomial and skein theory. Some examples of the invariant are computed. An unlinked embedded graph is one…
We introduce "book links" as a generalization of braids in open book decompositions; this new class of objects includes both braids and plats as special cases. We then prove a version of Markov's theorem in this general setting by extending…
We introduce and study strongly invertible Legendrian links in the standard contact three-dimensional space. We establish the equivariant analogs of basic results separately well-known for strongly invertible and Legendrian links, i.e. the…
Notes based on lessons given at {\sc Escuela " Fico Gonz\'alez Acu\~na" de Nudos y 3-variedades}, M\'erida Yucat\'an, M\'exico, 7--10 (2015) and {\sc Encuentro de nudos, trenzas y \'algebras}, Oaxaca--M\'exico, 3--10 October (2018).
We introduce a new method for computing triply graded link homology, which is particularly well-adapted to torus links. Our main application is to the (n,n)-torus links, for which we give an exact answer for all n. In several cases, our…
In 2014 Jones showed how to associate links in the $3$-sphere to elements of Thompson's group $F$. We provide an analogue of this program for annular links and Thompson's group $T$. The main result is that any edge-signed graph embedded in…
The purpose of this paper is to construct and study equivariant Khovanov homology - a version of Khovanov homology theory for periodic links. Since our construction works regardless of the characteristic of the coefficient ring it…
A group invariant for links in thickened closed orientable surfaces is studied. Associated polynomial invariants are defined. The group detects nontriviality of a virtual link and determines its virtual genus.
We introduce a new construction of a surface link in the 4-space. We construct a surface link as a branched covering over the standard torus, which we call a torus-covering link. We show that a certain torus-covering $T^2$-link is…
We define monotone links on a torus, obtained as projections of curves in the plane whose coordinates are monotone increasing. Using the work of Morton-Samuelson, to each monotone link we associate elements in the double affine Hecke…
We define a new notion of thin position for a graph in a 3-manifold which combines the ideas of thin position for manifolds first originated by Scharlemann and Thompson with the idea of thin position for knots first originated by Gabai.…
The notion of a braided chord diagram is introduced and studied. An equivalence relation is given which identifies all braidings of a fixed chord diagram. It is shown that finite-type invariants are stratified by braid index for knots which…
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…
The slope conjecture gives a precise relation between the degree of the colored Jones polynomial of a knot and the boundary slopes of essential surfaces in the knot complement. In this note we propose a generalization of the slope…
We extend the results of our previous paper from knots to links by using a formula for the Jones polynomial of a link derived recently by N. Reshetikhin. We illustrate this formula by an example of a torus link. A relation between the…
We introduce new polynomial isotopy invariants for closed braids. They are constructed as polynomial valued {\em Gauss diagram 1-cocycles} evaluated on the full rotation of the closed braid $\hat \beta$ around the core of the corresponding…
Following the approach to grid homology of links in $S^3$, we prove combinatorially that the grid homology of links in lens spaces defined by Baker, Grigsby, and Hedden is a link invariant. Further, using the sign assignment defined by…
We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S^3. The key role is played by the SL(2,Z) transformation, which generates a general torus knot from the unknot. Applying…
The Jones polynomial and the Kauffman bracket are constructed, and their relation with knot and link theory is described. The quantum groups and tangle functor formalisms for understanding these invariants and their descendents are given.…