Related papers: Braids for pretzel links
We use a variation on the commutator collection process to characterize those pure braids which become trivial when any one strand is deleted, or, more generally, those pure braids which become trivial when all the strands in any one of a…
The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology of some sub-families of 3-strand pretzel knots, no general formula has been…
The virtual braid groups are generalizations of the classical braid groups. This paper gives an elementary proof that the classical braid group injects into the virtual braid group over the same number of strands.
We recall some abstract connectivity concepts, and apply them to special chains in partially ordered sets, called veins, that are defined as order-convex chains that are contained in every maximal chain they meet. Veins enable us to define…
After defining reduced minimum braid word and criteria for a braid family representative, different braid family representatives are derived, and a correspondence between them and families of knots and links given in Conway notation is…
We define braid presentation of edge-oriented spatial graphs as a natural generalization of braid presentation of oriented links. We show that every spatial graph has a braid presentation. For an oriented link it is known that the braid…
In this paper we give new presentations of the braid groups and the pure braid groups of a closed surface. We also give an algorithm to solve the word problem in these groups, using the given presentations.
An infinitary version of braid groups has been considered as a direct limit of n-braid groups. However, we can imagine more complicated braids with infinitely many strings. We invetisgate basic properties especially when the number of…
In the note, we give a proof, based on the Generalized Thom Conjecture, of Bennequin's Theorem on upper bound for the Euler number of a link which is considered as a closed braid. A lower bound for the Euler number of a link is also given.
In the paper we give a survey on braid groups and subjects connected with them. We start with the initial definition, then we give several interpretations as well as several presentations of these groups. Burau presentation for the pure…
A link in $S^{3}$ is a fully positive braid link if it is the closure of a positive braid that contains at least one full-twist. We show that a fully positive braid link is a satellite link if and only if it is the satellite of a fully…
Generalizing work of Marin [12], we construct in a unified way all the "braids and ties'' algebras available in literature and new ones.
The contents of this 6-page paper have been subsumed into the 13-page paper, "A note on closed 3-braids", arXiv:0802.1072 [math.GT]. This paper is correct, but contains less information than the new one. The topological classification of…
Braids can be represented geometrically as laminations of punctured disks. The geometric complexity of a braid is the minimal complexity of a lamination that represents it, and tight laminations are representatives of minimal complexity.…
We review a braid theoretic self-linking number formula and study its applications.
In previous papers, the author realized the following principle for many knot theories: if a knot diagram is complicated enough then it reproduces itself, i.e., is a subdiagram of any other diagram equivalent to it. This principle is…
Laundry surfaces for closed braid diagrams are presented. It is shown that braid diagrams are characterized by linking matrices obtained by lifting cycles from these surfaces. Oriented link types are then characterized by equivalence…
This paper explores the problem of unknotting closed braids and classical knots in mathematical knot theory. We apply evolutionary computation methods to learn sequences of moves that simplify knot diagrams, and show that this can be…
We introduce the notion of a ribbon-clasp surface-link, which is a generalization of a ribbon surface-link. We generalize the notion of a normal form on embedded surface-links to the case of immersed surface-links and prove that any…
To an oriented link in a solid torus we associate a trace graph in a thickened torus in such a way that links are isotopic if and only if their trace graphs can be related by moves of finitely many standard types. The key ingredient is a…