scholarly journals A BRAIDED SIMPLICIAL GROUP

2002 ◽  
Vol 84 (3) ◽  
pp. 645-662 ◽  
Author(s):  
JIE WU
Keyword(s):  
Group Action ◽  
Braid Group ◽  
Homotopy Group ◽  
Braid Groups ◽  
Subject Classification ◽  
Homotopy Groups ◽  
Simplicial Group ◽  
Pure Braid Group ◽  
Fixed Set ◽  
Braid Group Action

By studying the braid group action on Milnor's construction of the 1-sphere, we show that the general higher homotopy group of the 3-sphere is the fixed set of the pure braid group action on certain combinatorially described groups. This establishes a relation between the braid groups and the homotopy groups of the sphere.2000Mathematical Subject Classification: 20F36, 55P35, 55Q05, 55Q40, 55U10.

scholarly journals Planar pure braids on six strands

2020 ◽  
Vol 29 (01) ◽  
pp. 1950097
Author(s):  
Jacob Mostovoy ◽  
Christopher Roque-Márquez
Keyword(s):  
Abelian Group ◽  
Fundamental Group ◽  
Free Product ◽  
Free Group ◽  
Configuration Space ◽  
Braid Group ◽  
Free Abelian Group ◽  
Braid Groups ◽  
Pure Braid Group

The group of planar (or flat) pure braids on [Formula: see text] strands, also known as the pure twin group, is the fundamental group of the configuration space [Formula: see text] of [Formula: see text] labeled points in [Formula: see text] no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.

scholarly journals TRIPLY-GRADED LINK HOMOLOGY AND HOCHSCHILD HOMOLOGY OF SOERGEL BIMODULES

2007 ◽  
Vol 18 (08) ◽  
pp. 869-885 ◽  
Author(s):  
MIKHAIL KHOVANOV
Keyword(s):  
Group Action ◽  
Braid Group ◽  
Polynomial Algebra ◽  
Homology Theory ◽  
Hochschild Homology ◽  
Polynomial Algebras ◽  
Homology Groups ◽  
Link Homology ◽  
Soergel Bimodules ◽  
Braid Group Action

We consider a class of bimodules over polynomial algebras which were originally introduced by Soergel in relation to the Kazhdan–Lusztig theory, and which describe a direct summand of the category of Harish–Chandra modules for sl(n). Rouquier used Soergel bimodules to construct a braid group action on the homotopy category of complexes of modules over a polynomial algebra. We apply Hochschild homology to Rouquier's complexes and produce triply-graded homology groups associated to a braid. These groups turn out to be isomorphic to the groups previously defined by Lev Rozansky and the author, which depend, up to isomorphism and overall shift, only on the closure of the braid. Consequently, our construction produces a homology theory for links.

scholarly journals EXTENDING REPRESENTATIONS OF BRAID GROUPS TO THE AUTOMORPHISM GROUPS OF FREE GROUPS

2005 ◽  
Vol 14 (08) ◽  
pp. 1087-1098 ◽  
Author(s):  
VALERIJ G. BARDAKOV
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Automorphism Groups ◽  
Free Groups ◽  
Linear Representation ◽  
Braid Groups ◽  
Burau Representation ◽  
Pure Braid Group

We construct a linear representation of the group IA (Fn) of IA-automorphisms of a free group Fn, an extension of the Gassner representation of the pure braid group Pn. Although the problem of faithfulness of the Gassner representation is still open for n > 3, we prove that the restriction of our representation to the group of basis conjugating automorphisms Cbn contains a non-trivial kernel even if n = 2. We construct also an extension of the Burau representation to the group of conjugating automorphisms Cn. This representation is not faithful for n ≥ 2.

scholarly journals Braid Group Action and Canonical Bases

1996 ◽  
Vol 122 (2) ◽  
pp. 237-261 ◽  
Author(s):  
George Lusztig
Keyword(s):  
Group Action ◽  
Braid Group ◽  
Canonical Bases ◽  
Braid Group Action

Brunnian braids over the 2-sphere and Artin combed form

2021 ◽  
pp. 2150012
Author(s):  
Duzhin Fedor ◽  
Loh Sher En Jessica
Keyword(s):  
Exact Sequence ◽  
Word Problem ◽  
Open Problem ◽  
Homotopy Group ◽  
Mapping Class Groups ◽  
Braid Groups ◽  
Mapping Class ◽  
Homotopy Groups ◽  
Class Groups ◽  
Homotopy Groups Of Spheres

Finding homotopy group of spheres is an old open problem in topology. Berrick et al. derive in [A. J. Berrick, F. Cohen, Y. L. Wong and J. Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006)] an exact sequence that relates Brunnian braids to homotopy groups of spheres. We give an interpretation of this exact sequence based on the combed form for braids over the sphere developed in [R. Gillette and J. V. Buskirk, The word problem and consequences for the braid groups and mapping class groups of the two-sphere, Trans. Amer. Math. Soc. 131 (1968) 277–296] with the aim of helping one to visualize the sequence and to do calculations based on it.

scholarly journals Minimal Asymptotic Translation Lengths of Torelli Groups and Pure Braid Groups on the Curve Graph

2018 ◽  
Vol 2020 (24) ◽  
pp. 9974-9987
Author(s):  
Hyungryul Baik ◽  
Hyunshik Shin
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Braid Groups ◽  
Mapping Class ◽  
Torelli Group ◽  
Pure Braid Group ◽  
Curve Graph ◽  
Translation Length

Abstract In this paper, we show that the minimal asymptotic translation length of the Torelli group ${\mathcal{I}}_g$ of the surface $S_g$ of genus $g$ on the curve graph asymptotically behaves like $1/g$, contrary to the mapping class group ${\textrm{Mod}}(S_g)$, which behaves like $1/g^2$. We also show that the minimal asymptotic translation length of the pure braid group ${\textrm{PB}}_n$ on the curve graph asymptotically behaves like $1/n$, contrary to the braid group ${\textrm{B}}_n$, which behaves like $1/n^2$.

scholarly journals GENERALIZED LONG-MOODY REPRESENTATIONS OF BRAID GROUPS

2008 ◽  
Vol 10 (supp01) ◽  
pp. 1093-1102
Author(s):  
STEPHEN BIGELOW ◽  
JIANJUN PAUL TIAN
Keyword(s):  
Braid Group ◽  
Hecke Algebra ◽  
Braid Groups ◽  
Pure Braid Group ◽  
Special Case

Long and Moody give a method of constructing representations of the braid groups Bn. We discuss some ways to generalize their construction. One of these gives representations of subgroups of Bn, including the Gassner representation of the pure braid group as a special case. Another gives representations of the Hecke algebra.

Sign in / Sign up

Export Citation Format

Share Document