scholarly journals On the homology of the commutator subgroup of the pure braid group

2020 ◽  
pp. 1
Author(s):  
Andrea Bianchi
Keyword(s):  
Braid Group ◽  
Commutator Subgroup ◽  
Pure Braid Group

scholarly journals A geometric interpretation of the normal closure of the braid group Bn in the braid group of the torus Bn(T)

2019 ◽  
Vol 28 (05) ◽  
pp. 1950038
Author(s):  
Liming Pang
Keyword(s):  
Braid Group ◽  
Commutator Subgroup ◽  
Geometric Interpretation ◽  
Braid Groups ◽  
Normal Closure ◽  
Geometric Description ◽  
Pure Braid Group

It had been proved by Birman and Goldberg that the normal closure of the pure braid group [Formula: see text] in the pure braid group of the torus [Formula: see text] is the commutator subgroup [Formula: see text]. In this paper, we are going to study the case of full braid groups: i.e. the normal closure of [Formula: see text] in [Formula: see text], which turns out to have an interesting geometric description.

scholarly journals Representation Stability of Power Sets and Square Free Polynomials

2015 ◽  
Vol 67 (5) ◽  
pp. 1024-1045
Author(s):  
Samia Ashraf ◽  
Haniya Azam ◽  
Barbu Berceanu
Keyword(s):  
Symmetric Group ◽  
Braid Group ◽  
Point Of View ◽  
Pure Braid Group ◽  
Representation Stability ◽  
Stability Point ◽  
Power Set ◽  
The Stability

AbstractThe symmetric group 𝓢n acts on the power set 𝓟(n) and also on the set of square free polynomials in n variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.

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 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 Non-realizability of the pure braid group as area-preserving homeomorphisms

2020 ◽  
pp. 1-12
Author(s):  
LEI CHEN
Keyword(s):  
Braid Group ◽  
Natural Projection ◽  
Realization Problem ◽  
Pure Braid Group ◽  
Rotation Numbers ◽  
Marked Points ◽  
Area Preserving ◽  
Nielsen Realization

Let $\operatorname{Homeo}_{+}(D_{n}^{2})$ be the group of orientation-preserving homeomorphisms of $D^{2}$ fixing the boundary pointwise and $n$ marked points as a set. The Nielsen realization problem for the braid group asks whether the natural projection $p_{n}:\operatorname{Homeo}_{+}(D_{n}^{2})\rightarrow B_{n}:=\unicode[STIX]{x1D70B}_{0}(\operatorname{Homeo}_{+}(D_{n}^{2}))$ has a section over subgroups of $B_{n}$ . All of the previous methods use either torsion or Thurston stability, which do not apply to the pure braid group $PB_{n}$ , the subgroup of $B_{n}$ that fixes $n$ marked points pointwise. In this paper, we show that the pure braid group has no realization inside the area-preserving homeomorphisms using rotation numbers.

scholarly journals Classifying links under fused isotopy

2016 ◽  
Vol 25 (07) ◽  
pp. 1650042
Author(s):  
Andrew Fish ◽  
Ebru Keyman
Keyword(s):  
Braid Group ◽  
Algebraic Proof ◽  
Pure Braid Group ◽  
Linking Numbers

All knots have been shown to be isotopic to the unknot using a process known as virtualization. We extend and adapt this process to show that, up to fused isotopy, classical links are classified by their linking numbers. We provide an algebraic proof, utilizing Alexander’s Theorem and some simple results about the pure braid group.

Quantization of Bending Deformations of Polygons In , Hypergeometric Integrals and the Gassner Representation

2001 ◽  
Vol 44 (1) ◽  
pp. 36-60 ◽  
Author(s):  
Michael Kapovich ◽  
John J. Millson
Keyword(s):  
Singular Point ◽  
Lie Algebra ◽  
Moduli Space ◽  
Braid Group ◽  
Vector Fields ◽  
Flat Connection ◽  
Hamiltonian Action ◽  
Pure Braid Group ◽  
Hypergeometric Integrals

AbstractThe Hamiltonian potentials of the bending deformations of n-gons instudied in [KM] and [Kly] give rise to a Hamiltonian action of the Malcev Lie algebra𝓟nof the pure braid groupPnon the moduli spaceMrofn-gon linkages with the side-lengthsr = (r1, … , rn)in. Ife∈Mris a singular point wemay linearize the vector fields in𝓟nate. This linearization yields a flat connection ∇ on the spaceof n distinct points on. We show that the monodromy of ∇ is the dual of a quotient of a specialized reduced Gassner representation.

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