Related papers: Simple Braids

We determine the Zariski closure of the representations of the braid groups that factorize through the Birman-Wenzl-Murakami algebra, for generic values of the parameters $\alpha,s$. For $\alpha,s$ of modulus 1 and close to 1, we prove that…

This article is an exposition of certain connections between the braid groups, classical homotopy groups of the 2-sphere, as well as Lie algebras attached to the descending central series of pure braid groups arising as Vassiliev invariants…

We show that the difference between the Seifert genus and the topological 4-genus of a prime positive braid knot is bounded from below by an affine function of the minimal number of strands among positive braid representatives of the knot.…

The role of quantum groups and braid groups in the description of Standard Model particles is discussed. Some recent results on the use of the quantum group $SU_q(3)$ as a flavour symmetry are reviewed and a connection between two…

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 and study extensions of Artin's representation and braid monodromy representation to the case of topological and algebraical generalisations of braid groups. In particular we provide faithful representations of braid groups of…

These are Lecture Notes of a course given by the author at the French-Spanish School "Tresses in Pau", held in Pau (France) in October 2009. It is basically an introduction to distinct approaches and techniques that can be used to show…

The knot group is the fundamental group of a knot or link complement. A necessary and sufficient conditions for a group to be realized as the knot group of some link was provided. This result was shown using the closed braid method.…

Finite type invariants (also known as Vassiliev invariants) of pure braids are considered from a group-theoretic point of view. New results include a construction of a universal invariant with integer coefficients based on the Magnus…

The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally…

We combine the newly discovered technique, which computes explicit formulas for the image of an algebraic curve under rational transformation, with techniques that enable to compute braid monodromies of such curves. We use this combination…

We survey the theory of totally symmetric sets, with applications to homomorphisms of symmetric groups, braid groups, linear groups, and mapping class groups.

The main result of this paper is an explicit construction of the free commutative skew brace -- that is, a skew brace whose circle group is commutative -- on an arbitrary generating set $X$. We embed this object into a set of rational…

We study simplicity of Lie skew braces from both global and infinitesimal perspectives. After reviewing the correspondence between connected Lie skew braces, simply transitive affine actions, and post-Lie algebras, we investigate ideals and…

A group is boundedly simple if, for some constant N, every nontrivial conjugacy class generates the whole group in N steps. For a large class of trees, Tits proved simplicity of a canonical subgroup of the automorphism group, which is…

The Rickard complex of a braid with strands colored by positive integers is a chain complex of singular Soergel bimodules. The complex determines the colored triply-graded homology and colored sl(N) homology of the braid closure, when…

For a positive braid $\beta \in \mathrm{Br}^{+}_{k}$, we consider the braid variety $X(\beta)$. We define a family of open sets $\mathcal{U}_{r, w}$ in $X(\beta)$, where $w \in S_k$ is a permutation and $r$ is a positive integer no greater…

We show that for every conjugacy class O in a connected semisimple algebraic group G over a field of characteristic good for G one can find a special transversal slice S to the set of conjugacy classes in G such that O intersects S and dim…

Let $G$ be a connected reductive group over an algebraically closed field with Weyl group $W$. The analogy between Lusztig varieties and Deligne-Lusztig varieties associated to minimal length elements in elliptic conjugacy classes of $W$…

We generalize the concept of a cycle from graphs to simplicial complexes. We show that a simplicial cycle is either a sequence of facets connected in the shape of a circle, or is a cone over such a structure. We show that a simplicial tree…