Related papers: Reducible braids and Garside theory
The main result of this paper is a negative answer to the question: are all transversal knot types transversally simple? An explicit infinite family of examples is given of closed 3-braids that define transversal knot types that are not…
We establish faithfulness of braid group actions generated by twists along an ADE configuration of $2$-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result…
Recently, there have been several progresses for the conjugacy search problem (CSP) in Garside groups, especially in braid groups. All known algorithms for solving this problem use a sort of exhaustive search in a particular finite set such…
We present a new algorithm to solve the conjugacy problem in Artin braid groups, which is faster than the one presented by Birman, Ko and Lee. This algorithm can be applied not only to braid groups, but to all Garside groups (which include…
In this note we solve the twisted conjugacy problem for braid groups, i.e. we propose an algorithm which, given two braids $u,v\in B_n$ and an automorphism $\phi \in Aut (B_n)$, decides whether $v=(\phi (x))^{-1}ux$ for some $x\in B_n$. As…
We study a subset of square free positive braids and we give a few algebraic characterizations of them and one geometric characterization: the set of positive braids whose closures are unlinks. We describe canonical forms of these braids…
Since the braid group was discovered by E. Artin, the question of its conjugacy problem has been solved by Garside and Birman, Ko and Lee. However, the solutions given thus far are difficult to compute with a computer, since the number of…
Given a system of equations in a "random" finitely generated subgroup of the braid group, we show how to find a small ordered list of elements in the subgroup, which contains a solution to the equations with a significant probability.…
Braid monodromy is an important tool for computing invariants of curves and surfaces. In this paper, the \emph{rectangular braid diagram (RBD)} method is proposed to compute the braid monodromy of a completely reducible $n$-gonal curve,…
There exists a multiplicative homomorphism from the braid group B to the Temperley-Lieb algebra TL. Moreover, the homomorphic images in TL of the simple elements form a basis for the vector space underlying TL. In analogy with the case of…
Derived braids have been used to classify categorical structures based on the braid underlying a braided monoidal category V. With four-strand braids underlying the composition morphisms of tensor products of categories enriched over V,…
Recent general results on Hamiltonian reductions under polar group actions are applied to study some reductions of the free particle governed by the Laplace-Beltrami operator of a compact, connected, simple Lie group. The reduced systems…
We give an explicit geometric argument that Artin's braid group $B_n$ is right-orderable. The construction is elementary, natural, and leads to a new, effectively computable, canonical form for braids which we call left-consistent canonical…
In this paper we prove a relative version of the classical Mumford-Newstead theorem for a family of smooth curves degenerating to a reducible curve with a simple node. We also prove a Torelli-type theorem by showing that certain moduli…
We associate to a braided 2-stack ${\cal C}$ a torsor, endowed with a symmetric cube structure (or $\Sigma$-structure), whose triviality is equivalent to the existence on ${\cal C}$ of a fully symmetric monoidal structure. In order to…
If a smooth, closed, and embedded curve is deformed along its normal vector field at a rate proportional to its curvature, it shrinks to a circular point. This curve evolution is called Euclidean curve shortening and the result is known as…
Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic from the braid group to the mapping class group. We prove here that this map is trivial in stable homology with…
Analysing statistical properties of the normal forms of random braids, we observe that, except for an initial and a final region whose lengths are uniformly bounded (that is, the bound is independent of the length of the braid), the…
The recent proof by Bigelow and Krammer that the braid groups are linear opens the possibility of applications to the study of knots and links. It was proved by the first author and Menasco that any closed braid representative of the unknot…
We study the notion of twisted conjugacy separability (essentially introduced in our previous paper for a proof of twisted version of Burnside-Frobenius theorem) and some related properties. We give examples of groups with and without this…