Related papers: Markov Theorem For Free Links

In this work, we find a closed form formula for the braid index of an $n$-bridge braid, a class of positive braid knots which simultaneously generalizes torus knots, 1-bridge braids, and twisted torus knots. Our proof is elementary,…

We define a homology theory of virtual links built out of the direct sum of the standard Khovanov complex with itself, motivating the name doubled Khovanov homology. We demonstrate that it can be used to show that some virtual links are…

There are many results asserting the existence of tree-decompositions of minimal width which still represent local connectivity properties of the underlying graph, perhaps the best-known being Thomas' theorem that proves for every graph $G$…

We show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links…

This is an expository article on diagrammatic representations of knots and links in various settings via braids.

We define new notions of groups of virtual and welded knots (or links) and we study their relations with other invariants, in particular the Kauffman group of a virtual knot.

Using a probabilistic interpretation of the Burau representation of the braid group offered by Vaughan Jones, we generalize the Burau representation to a representation of the semigroup of string links. This representation is determined by…

In a recent paper, McMullen showed an inequality between the Thurston norm and the Alexander norm of a 3-manifold. This generalizes the well-known fact that twice the genus of a knot is bounded from below by the degree of the Alexander…

This paper defines a new sequence of finite dimensional algebras as quotients of the group algebras of the braid groups. This sequence depends on three homogeneous parameters and has a one-parameter family of Markov traces, and so gives a…

New presentations of a link and a virtual link are introduced and algebraic systems on links and virtual links are constructed respectively. Based on the algebraic systems, Reduction Crossing Algorithms for them are proposed which are used…

We use the famous knot-theoretic consequence of Freedman's disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a…

We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index $\ge 5$. In order to have the necessity part, graphs are organized into small classes so that one of homological…

It is proved that every disconnected surface-link with meridian-based free fundamental group is a trivial (i.e., an unknotted-unlinked) surface-link. This result is a surface-link version of the author's recent announcement result on smooth…

Free-minor closed classes [2] and free-planar graphs [3] are considered. Versions of Kuratowski-like theorem for free-planar graphs and Kuratowski theorem for planar graphs are considered.

For classical links Ohyama proved an inequality involving the minimal crossing number and the braid index, then motivated from this Takeda showed an analogous inequality for virtual links. In this paper, we are interested in studying…

We define half grid diagrams and prove every link is half grid presentable by constructing a canonical half grid pair (which gives rise to a grid diagram of some special type) associated with an element in the oriented Thompson group. We…

In this paper we show how generalized quaternions, including 2X2 matrices, can be used to find solutions of a non-commuting equation intimately connected with braid groups. These solutions can then be used to find polynomial invariants of…

We propose two definitions of configuration Lie groupoids and in both the cases we prove a Fadell-Neuwirth type fibration theorem for a class of Lie groupoids. We show that this is the best possible extension, in the sense that, for the…

We give a general procedure that provides, given any particular pretzel link, a braid whose closure is the pretzel link. Moreover, we manage to give a specific braid word in terms of the entries of the pretzel link.

The goal of this paper is to address A. Shumakovitch's conjecture about the existence of $\Z_2$-torsion in Khovanov link homology. We analyze torsion in Khovanov homology of semi-adequate links via chromatic cohomology for graphs which…