Having a good length function on a group is the main ingredient in a recent combined memory/length approach to solving random equations in that group. Currently, the groups of greatest interest in this respect are Artin’s braid groups. We demonstrate, by a series of experiments, that a length function based on the Birman-Ko-Lee presentation of the braid group is substantially better than the co...

Let G be a finite group. By Riemann’s Existence Theorem, braid orbits of generating systems of G with product 1 correspond to irreducible families of covers of the Riemann sphere with monodromy group G. Thus many problems on algebraic curves require the computation of braid orbits. In this paper we describe an implementation of this computation. We discuss several applications, including the cl...

In this paper, we explore algebraic structures and low dimensional topology underlying quantum information and computation. We revisit quantum teleportation from the perspective of the braid group, the symmetric group and the virtual braid group, and propose the braid teleportation, the teleportation swapping and the virtual braid teleportation, respectively. Besides, we present a physical inte...

Enhanced gauge symmetry appears in Type II string theory (as well as Fand M-theory) compactified on Calabi–Yau manifolds containing exceptional divisors meeting in Dynkin configurations. It is shown that in many such cases, at enhanced symmetry points in moduli a braid group acts on the derived category of sheaves of the variety. This braid group covers the Weyl group of the enhanced symmetry a...

Governed by locality, we explore a connection between unitary braid group representations associated to a unitary R-matrix and to a simple object in a unitary braided fusion category. Unitary R-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is...

which is Artin’s braid relation. In general we will refer to (1) as the braid relation satisfied by si and sj. In order for W to be a Coxeter group it is required that the given set of relations between elements of I give a presentation of W . Informally, this means that any relation between generators in I can be deduced from the fact that the s ∈ Σ have order 2 and the braid relations. Formal...

In this paper we prove that braid monodromy of an affine plane curve determines the topology of a related projective plane curve. Introduction Our purpose in this paper is to relate the topological embedding of algebraic curves to a refinement of a well-known invariant of curves such as braid monodromy. Roughly speaking, braid monodromy is defined for a triple (C , L , P), where C ⊂ P2 is a cur...

In his initial paper on braids E. Artin gave a presentation with two generators for an arbitrary braid group. We give analogues of this Artin’s presentation for various generalizations of braids. The diverse aspects of presentations of braid groups and their generalizations continue to attract attention [12], [5], [15]. The canonical presentation of the braid group Brn was given by E. Artin [1]...