Some Problems on Knots, Braids, and Automorphism Groups (original) (raw)

The pure braid groups and their relatives

In this talk, I discuss the resonance varieties, the lower central series ranks, the Chen ranks, and the formality properties of several families of braid-like groups: the pure braid groups P_n, the welded pure braid groups wP_n, the virtual pure braid groups vP_n, as well as their `upper' variants, wP_n^+ and vP_n^+. I will also discuss several natural homomorphisms between these groups, and various ways to distinguish among the pure braid groups and their relatives. This is joint work with He Wang.

An acyclic extension of the braid group

Commentarii Mathematici Helvetici, 1991

We relate Artin's braid group B~ = lim ~ B n to a certain group F' of p/-homeomorphisms of the interval. Namely, there exists a short exact sequence 1~ B~-,A-* F'~ 1, where HkA = 0, k ~ 1.

On braid groups and right-angled Artin groups

Geometriae Dedicata, 2014

In this article we prove a special case of a conjecture of A. Abrams and R. Ghrist about fundamental groups of certain aspherical spaces. Specifically, we show that the n−point braid group of a linear tree is a right angled Artin group for each n.

On some groups related to the Braid Groups of type A

Annals of the University of Craiova - Mathematics and Computer Science Series, 2010

We prove that a family of groups R(n) forms the algebraic structure of an operad and that they admit a presentation similar to that of the Braid groups of type A. This result provides a new proof that the Braid Groups form an operad, a topic emphasized in ~\cite{16}~\cite{ulrike}. These groups proved to be useful in several problems which belong to different areas of Mathematics. Representations of R(n) came from a system of mixed Yang-Baxter type equations. We define the Hopf equation in braided monoidal categories and we prove that representations for our groups came from any braided Hopf algebra with invertible antipode. Using this result, we prove that there is a morphism from R(n) to the mapping class group Gamman,1\Gamma_{n,1}Gamman,1, using some results from 3-dimensional topology.

Braids, links, cobordisms and formal groups

Cornell University - arXiv, 2022

V.V. Sharko in his papers and books has investigated functions on manifolds and cobordism. Braids intimately connect with functions on manifolds. These connections are represented by mapping class groups of corresponding discs, by fundamental groups of corresponding punctured discs, and by some other topological or algebraic structures. This paper presents selected algebraic methods and results of braids, links, cobordism connect with investigations by V.V. Sharko. These includes group theoretic results on braids and links, infinitesimal braid group relations and connections as well as connections on coherent sheaves on smooth schemes, a sketch of our algorithm for constructing of Lazard's one dimensional universal commutative formal group and selected results on applications of commutative formal groups to cobordism theory. as with the theory of cobordism, took place in 1970-1971. during his internship with academician A.A. Markov. (A.A. Markov was the head of the laboratory at the Computing Center of the USSR Academy of Sciences, and, at the same time, the head of the department of mathematical logic at Moscow State University). A.A. Markov gave the description of the set of isotopy classes of oriented links in R 3 in terms of braids. For manifolds of the dimension grater than 3 A.A. Markov has proved the undecidability of the problem of homeomorphy.

A New Approach to the Word and Conjugacy Problems in the Braid Groups

Advances in Mathematics, 1998

A new approach to the word and conjugacy problems in the braid groups Abstract A new presentation of the n-string braid group B n is studied. Using it, a new solution to the word problem in B n is obtained which retains most of the desirable features of the Garside-Thurston solution, and at the same time makes possible certain computational improvements. We also give a related solution to the conjugacy problem, but the improvements in its complexity are not clear at this writing. * Partially supported by NSF Grant 94-02988. This paper was completed during a visit by the first author to MSRI. She thanks MSRI for its hospitality and partial support, and thanks Barnard College for its support under a Senior Faculty Research Leave. † This work was initiated during the second author's sabbatical visit to Columbia University in 1995-6. He thanks Columbia University for its hospitality during that visit.