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Group completion of a monoid (braid groups)

Let $B_n$ be the braid group on $n$ strands, $B_{\infty}$ the direct limit of braid groups. For a discrete group $G$, we let $BG$ to be the classifying space of $G$. After reading this question, I was ...
May's user avatar
  • 110
4 votes
1 answer
168 views

Homotopy equivalence between certain loop spaces

I've been reading some papers carefully, with their proofs (Notations are given at the end). The following comes from "Braids, mapping class groups and categorical delooping" by Song & ...
wind's user avatar
  • 164
6 votes
1 answer
287 views

Loop manipulation subgroup of the braid group

Recently, I came across a subgroup of the braid group $B_{2n}$ that I'm calling the "loop manipulation" group $H_n$. The idea is that we treat pairs of adjacent strands in the braid group as ...
pgadey's user avatar
  • 647
2 votes
0 answers
58 views

Nielsen–Thurston classification and configuration spaces

Viewing the $n$-strand braid group as the mapping class group of an $n$-punctured disk, braids can be classified as periodic, reducible, or pseudo-Anosov. The same group is also the fundamental group ...
Ben Knudsen's user avatar
6 votes
1 answer
418 views

Relations between relations in the positive braid monoid

The positive braid monoid on $n$ strands is the monoid with generators $s_1$, $s_2$, ..., $s_{n-1}$ and relations $$s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1} \qquad s_i s_j = s_j s_i \text{for}\ |i-j| \...
David E Speyer's user avatar
4 votes
1 answer
107 views

Given a word $w$ in the braid group $B_n$, representing a pure braid, find the image of $w$ in the abelianization of $P_n$

Suppose I have a word $w$ in the standard generators $\sigma_1,\dots,\sigma_{n-1}$ of the braid group $B_n$ representing an element which we know belongs to the pure braid group $P_n$, is there an ...
Harry Reed's user avatar
5 votes
0 answers
150 views

What are the finite quotients of the braid group?

Are all known finite quotients of the braid group given by reducing the Burau or Lawrence-Krammer representations mod $p$ and evaluating at some element in $\mathbb{F}_p$? I recently saw a paper ...
Harry Reed's user avatar
7 votes
0 answers
99 views

Normal subgroups of pure braid groups stable under strand bifurcation

$\DeclareMathOperator\PB{PB}\DeclareMathOperator\B{B}$Let $\PB_n$ be the $n$-strand pure braid group. For each $1\le k\le n$, let $\kappa_k^n \colon \PB_n \to \PB_{n+1}$ be the monomorphism that takes ...
Matt Zaremsky's user avatar
3 votes
0 answers
128 views

Stochastic braids

I am definitely not a probability guy, but I'd like to have a heuristic answer to the following question: do $n$ independently moving points in an open, connected, bounded region $R$ tend to "...
Andrea Marino's user avatar
1 vote
2 answers
228 views

Link invariants from Hecke relations of higher order

Alexander theorem says oriented links in $\mathbb{R}^3$ can be represented by closures of braids. Markov theorem says that braids related by Markov moves produce isotopic braid closures, and vice ...
Student's user avatar
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1 vote
0 answers
68 views

Mapping class group interpretation of braid closure

Given a braid (diagram) $\beta\in B_n$, the associated closed braid is the knot/link formed by attaching the ends on which the strings lie. We can also, however, think of $\beta$ as being an element ...
user2357's user avatar
4 votes
1 answer
352 views

Action of braid groups on regular trees

Question: Are there any well known actions of braid groups on trees? For example is there some action of a braid group $ B_n $ on a $ p $ regular tree for some $ p $ such that the action is transitive ...
Ian Gershon Teixeira's user avatar
6 votes
2 answers
559 views

Perfect quotients of braid groups

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSp{PSp}$The braid groups $ B_1=1 $ and $ B_2\cong \mathbb{Z} $ have no perfect quotients. $ ...
Ian Gershon Teixeira's user avatar
17 votes
0 answers
989 views

Symmetries of local systems on the punctured sphere

Let $X=S^2\setminus D$, for $D\subset S^2$ some finite set of points, say with $|D|=n\geq 1$. The category of locally constant sheaves of $\mathbb{C}$-vector spaces on $X$ (equivalently, complex ...
Daniel Litt's user avatar
  • 22.2k
2 votes
1 answer
245 views

How to get a presentation of the mapping class group of the $n$-punctured sphere

$\DeclareMathOperator\Mod{Mod}$I would like to compute the mapping class group (homeomorphism preserving orientation modulo those isotopic to the identity) of the sphere $S^2$ minus $n$ points $p_1,\...
Federico Fallucca's user avatar
4 votes
1 answer
232 views

Bounds for the crossing number in terms of the braid index?

Is there a lower bound on the crossing number of a knot (resp., link) with braid index $b$? For knots, I believe the smallest crossing number for braid index 2 is 3, the smallest crossing number for ...
Charles's user avatar
  • 9,004
3 votes
1 answer
182 views

Squier's conjecture on Burau at roots of unity

In Squier's short, yet influential, paper about the Burau representation, he made two conjectures that might have provided a proof for the faithfulness of the Burau representation (which we now know ...
Ethan Dlugie's user avatar
  • 1,267
2 votes
0 answers
75 views

Composition of 3-braids to obtain braids with trivial closure

Given a 3-braid $b=\sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_1$ (which has non-trivial closure), can we find a 3-braid $c$, which has trivial closure (closure results in any trivial knot or ...
Muqing Cao's user avatar
2 votes
1 answer
148 views

Are there infinite number of 3-braids with trivial closure?

Not counting equivalent braids, are there finite or infinite numbers of 3-braids whose closures are trivial knot or links? If the answer is infinite, are there some patterns in those infinite numbers ...
Muqing Cao's user avatar
0 votes
1 answer
203 views

Space-time trajectory that cannot be straightened and its braid form

Considering we have the space-time trajectory of multiple particles (or any objects) in the X-Y-Time coordination system. Given a projection direction, we can obtain the braid form of the space-time ...
Muqing Cao's user avatar
4 votes
0 answers
103 views

Interplay beween simplicial and Weyl algebra identities

Recall that the (first) Weyl algebra is the algebra generated by $x,y$ with the relation $xy-yx=1$. It can be realized as the algebra of differential operators on $k[x]$, where one generator acts as ...
Arshak Aivazian's user avatar
4 votes
1 answer
171 views

Alexander polynomials for a certain family of closed braids

Let $n\geq 3$ be a positive integer and $\kappa=(k_1, \dots, k_n)\in \mathbb{Z}^n$. Denote by $B_n$ the braid group on $n$ strings. Consider the braid on $n+1$ strings $\sigma_\kappa:=\sigma_1^{k_1}\...
Anwesh Ray's user avatar
11 votes
0 answers
211 views

On an Artin (?) subgroup of braid groups

While working on something apparently unrelated I encountered a "braid-like" group, which is a relatively geometric subgroup of a braid group and seems to be itself an Artin group. It seems ...
Simon Henry's user avatar
  • 40.4k
5 votes
1 answer
191 views

Why is the category of strong braided functors from the braid category to a braided monoidal $M$ equivalent to the subcategory of *strict* functors?

This is my first, and probably my last, (for a while) posting on MO. I am very much a student and I don't claim to be a research mathematician, at all, but I have seen that sometimes "regular&...
FShrike's user avatar
  • 831
7 votes
0 answers
253 views

Generating cycles inside Tits' graph of words for a positive braid

Let $Br_n$ be the braid group and consider words in its generators (not in the inverses). Two such words define the same "positive" braid if one can be obtained from the other by commuting ...
Allen Knutson's user avatar
2 votes
1 answer
204 views

Determine if a closed braid is a link/unlink

I am relatively new to the world of braids/knots so really sorry if this question is simple. However, I am not able to find if there is any theorem/procedure that determines if a closed braid, given ...
Muqing Cao's user avatar
8 votes
0 answers
166 views

Is the Lawrence–Krammer representation faithful, reduced modulo p?

It is well-known that the braid group $B_n$ is linear for every $n$ by the Lawrence–Krammer (or LKB) representation. It embeds $B_n$ faithfully into $\mathrm{GL}\left(\frac{n(n-1)}{2},\mathbb{Z}[q^{\...
Adel M's user avatar
  • 113
9 votes
2 answers
410 views

Coherence theorem in braided monoidal categories

In MacLane's Categories for the working mathematician, the author shows that the evaluation at 1 gives an equivalence of categories $\mathrm{hom}_{\mathrm{BMC}}(B,M)\simeq M_0$ where $B$ is the braid ...
QGM's user avatar
  • 201
19 votes
0 answers
424 views

Are braid groups known to not be linear over $\mathbb{Z}$?

$\DeclareMathOperator\GL{GL}$It is known that every braid group $B_n$ embeds as a subgroup of $\GL_m(\mathbb{Z}[q^{\pm 1},t^{\pm 1}])$, where $m=n(n-1)/2$ (see Krammer - Braid groups are linear). This ...
Matt Zaremsky's user avatar
5 votes
0 answers
229 views

Coxeter's braid group quotients

Coxeter's result is that if the generators of the braid group $B_n$ on $n$ strands fulfill a relation $\forall_i\sigma_i^k=1$, then $1/n+1/k>1/2$ must hold to get a finite quotient of $B_n$. In ...
Hauke Reddmann's user avatar
3 votes
1 answer
351 views

Dehornoy's proof that the application of two elementary embeddings is an elementary embedding

What is meant by the statement and the proof of Lemma 3.2 in Chapter XII of Dehornoy's book Braids and Self-Distributivity? That lemma states "Assume that $j_1$ and $j_2$ are elementary ...
kdog's user avatar
  • 245
2 votes
0 answers
101 views

Virtual cohomological dimension of mapping class group and braid group of punctured surfaces

$\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\vcd{vcd}\DeclareMathOperator\cd{cd}$Let $B_{k}(S_{g}),$ $\MCG(S_{g};k)$ and $\MCG(S_{g}))$ denote the braid group, the mapping class group (relative ...
King Khan's user avatar
  • 173
5 votes
1 answer
347 views

Representation stability for systems of braid group representations

In general, if I understand correctly, the representation theory of the braid groups is quite complicated, and there's no classification of the irreducibles. However, the braid groups form a sort of ...
Jonathan Beardsley's user avatar
4 votes
1 answer
266 views

Good algorithmic properties for quotients of braid groups

I'm trying to understand some things about quotients of braid groups, and particularly I'd like to solve the word problem for some elements of these quotients. I'm using MAGMA to try to access this, ...
Ethan Dlugie's user avatar
  • 1,267
9 votes
0 answers
167 views

Configuration space of 4 points as an orbifold

Setup: Consider the braid group $B_n$. One way to define this is as the fundamental group of the unordered configuration space $UC_n(\mathbb{C}) = \{\{z_1,\dotsc,z_n\}\subset \mathbb{C} \mid z_i \not= ...
Ethan Dlugie's user avatar
  • 1,267
3 votes
1 answer
79 views

Arc complex characterization of braids with trivial closure

A braid $\beta\in B_n,$ the braid group with $n$ strands, viewed as the mapping class group $\mathrm{Mod}(\mathbb{D}_n)$ of the disk with $n$ punctures is trivial if and only if $\beta$ acts trivially ...
Renaud Detcherry's user avatar
2 votes
0 answers
215 views

Braid group and the fundamental group of coefficients of polynomials

I am studying Norbert A'Campo's article "Tresses, monodromie et le groupe symplectique" and I am trying to understand why $\rho(t_i)=T_i$. I am stating this just for the sake of reference, ...
user avatar
2 votes
0 answers
43 views

Finiteness of almost maximal submonoids in positive braids

A numerical semigroup of genus $g$ is a submonoid $S=S+S$ of $\mathbb N$ missing $g$ elements (gaps) of $\mathbb N$. It is not hard to show that there are only finitely many numerical semigroups of ...
Roland Bacher's user avatar
2 votes
0 answers
128 views

Image of the pure braid group under the Artin presentation into the automorphism group of the nilpotent quotient of a free group?

As I know, it is unknown that the image of the mapping class group of the surface and its Johnson filtration under the higher Johnson homomorphisms. There are a relationship between the mapping class ...
qkqh's user avatar
  • 347
2 votes
0 answers
134 views

Braid 2-groups, symmetric 2-groups

Is there an object which can be called a "braid 2-group"? Or a "symmetric 2-group"? (Note: not a braided 2-group) I am ignorant about 2-categories but I hope that a good candidate ...
Alex Ogg's user avatar
  • 159
5 votes
1 answer
170 views

Composite knots and their braid words

Given a composite knot K = K_1 # K_2, I wonder how the braid word looks like. Is it possible to see from the word that the knot is composite? I am not aware of a statement such as "the closure of ...
Zen's user avatar
  • 71
8 votes
1 answer
248 views

Geometric intuition behind Garside's paper?

I apologize in advance for a somewhat wishy-washy question. I just read the paper "The Braid Group and Other Groups" by F. A. Garside in which he solves the conjugacy problem for the braid ...
user101010's user avatar
  • 5,319
4 votes
1 answer
306 views

From braid representations to link invariants

If one has a $\mathbb{C}$-linear representation of the braid algebra into e.g. the Temperley-Lieb algebra i.e. $\rho:\mathbb{C}[B_{n}]\to TL_{n}(\delta)$, we can deduce a skein relation $\mathcal{S}$....
Meths's user avatar
  • 309
0 votes
1 answer
105 views

Fundamental group to groupoid : bijection between homotopy classes?

I'm looking at the fundamental group $\pi_{1}(M)$ of the $n^{th}$ unordered configuration space $M$ of $\mathbb{R}^{d}$. In particular, it's well-known that $\pi_{1}(M)\cong S_{n}$ (symmetric group) ...
Meths's user avatar
  • 309
3 votes
0 answers
195 views

Braids of fibered knots

There are some theorems saying that the closure of a braid of a specific form is fibered. For instance, a theorem of Stallings says that the closure of a homogeneous braid is a fibered knot. I am ...
Henry's user avatar
  • 1,420
8 votes
2 answers
638 views

Braid groups and Kazhdan's property (T)

In Nica's dissertation Group actions on median spaces, we can read the following assertion: Braid groups do not contain infinite subgroups satisfying Kazhdan's property (T). This is used in order to ...
AGenevois's user avatar
  • 7,761
12 votes
1 answer
274 views

Artin groups of type $D_n$ as mapping class groups?

According to Allcock (Braid Pictures for Artin groups, https://arxiv.org/abs/math/9907194), the Artin group $A(D_n$) of type $D_n$ may be realized as an index 2 subgroup of the orbifold fundamental ...
Thomas Haettel's user avatar
1 vote
0 answers
53 views

How to get the braid word of the tubes in a reducible braid?

If a braid is reducible, it means that the strands can be separated into groups of strands (less groups than number of strands), such that each group is running inside one tube. The tubes themselves ...
Zen's user avatar
  • 71
3 votes
1 answer
126 views

Knots with a braid presentation with only positive or negative crossings on each fixed position

I am interested in the following class of knots $K$: $\{$$K$ has a braid presentation such that for any fixed position $k$, either only positive or negative powers of $\sigma_k$ appear in the braid ...
Henry's user avatar
  • 1,420
20 votes
2 answers
1k views

Several questions about Gauss's mathematical conception of braids

I'm trying to figure out several things about Gauss's thoughts concerning a certain four-strand braid. The reference my questions are based on is mainly Moritz Epple's excellent article "orbits ...
user2554's user avatar
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