We prove that the braid group on 4 or more strands modulo its center is co-Hopfian. We then show that any injective endomorphism of these braid groups is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from the braid group on n strands to the braid group on n + 1 strands is geometric (n ≥ 7). Additionally, we obtain related r...

In this paper we prove a Markov Theorem for the virtual braid group and for some analogs of this structure. The virtual braid group is the natural companion to the category of virtual knots, just as the Artin braid group is the natural companion to classical knots and links. In classical knot theory the braid group gives a fundamental algebraic structure associated with knots. The Alexander The...

In the last decade, a number of public key cryptosystems based on combinatorial group theoretic problems in braid groups have been proposed. Our tutorial is aimed at presenting these cryptosystems and some known attacks on them. We start with some basic facts on braid groups and on the Garside normal form of its elements. We then present some known algorithms for solving the word problem in the...

We study some quadratic algebras which are appeared in the low–dimensional topology and Schubert calculus. We introduce the Jucys–Murphy elements in the braid algebra and in the pure braid group, as well as the Dunkl elements in the extended affine braid group. Relationships between the Dunkl elements, Dunkl operators and Jucys–Murphy elements are described.

To a plane algebraic curve of degree n, Moishezon associated a braid monodromy homomorphism from a finitely generated free group to Artin’s braid group Bn. Using Hansen’s polynomial covering space theory, we give a new interpretation of this construction. Next, we provide an explicit description of the braid monodromy of an arrangement of complex affine hyperplanes, by means of an associated “b...

We found Gröbner-Shirshov basis for the braid semigroup B n+1. It gives a new algorithm for the solution of the word problem for the braid semigroup and so for the braid group.

We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged “convexly”, which is why we describe them as geometric presentations. Motivated by a presentation for the full braid group that we call the “rotation presentation”, we introduce...

We introduce the idea of a geometric categorical Lie algebra action on derived categories of coherent sheaves. The main result is that such an action induces an action of the braid group associated to the Lie algebra. The same proof shows that strong categorical actions in the sense of Khovanov-Lauda and Rouquier also lead to braid group actions. As an example, we construct an action of Artin’s...

We consider Krammer’s representation of the pure braid group on three strings: P3 → GL 3, Z t±1, q±1 , where t and q are indeterminates. As it was done in the case of the braid group, B3, we specialize the indeterminates t and q to nonzero complex numbers. Then we present our main theorem that gives us a necessary and sufficient condition that guarantees the irreducibility of the complex specia...

We discuss some consequences of the braid group action on a categorified quantum group. Results include a description of reflection functors for quiver Hecke algebras and a theory of restricting categorical representations along a face.