Question about terminology, and reference request related to the braid operad
Let $\Delta_n$ stand for the Garside element of the braid group $B_n$. It turns out that the family of all Garside elements have the following ``operadic'' property:$$\Delta_n\left[...
View ArticleHow 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...
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Orbifold fundamental group and configuration space
I'm not very familiar with (even simple examples of) orbifolds, so my first question is:Let $C_2$ be $\mathbb{C}$ with one cone singularity at 0 of index 2. What is the fundamental group of $C_2$ minus...
View ArticleSymmetries 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...
View ArticlePerfect 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.$ B_3...
View ArticleAction 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...
View ArticleSeveral 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 of...
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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}$....
View ArticleLink invariants from Hecke relations of higher order
Alexander theorem says oriented links in $\mathbb{R}^3$ can berepresented by closures of braids. Markov theorem says thatbraids related by Markov moves produce isotopic braid closures,and vice versa....
View ArticleStochastic 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 "braid" a...
View ArticleSoftware for finding conjugates in the braid group
The conjugacy problem for the braid group was solved by Garside, and gives an algorithm for determining whether two braids are conjugate. Since this algorithm is rather tedious, I was wondering if...
View ArticleNormal 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...
View ArticleWhat 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 giving...
View ArticleGiven a word $w$ in the braid group $B_n$, representing a pure braid, find...
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...
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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| \geq...
View ArticleNielsen–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...
View ArticleLoop 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 "loops" $i...
View ArticleHomotopy 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 & Tillmann.The...
View ArticleGroup 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...
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What is known/expected on the co-growth series of the braid group?
The co-growth series of finitely generated group with respect to generating set $S$ is generating function for the number of words of length $n$ which are equal to 1 in the group.Its studies originates...
View ArticleWhy does the definition of a braided monoidal category not mention the braid...
Let $\mathcal{M}$ be a braided monoidal category (BMC) with braiding $\gamma$. In the definition of a BMC $\gamma$ is required to satisfy the two hexagon identities. However since "braided" appears in...
View ArticleThe word problem in braid groups
I have read a statement from Sossinsky and Prasolov' s book "Knots, Lİnks, Braids and 3-Manifold", it says that two reduced word represent isotopic braids if and only if they have the same reduced...
View ArticleInsights on non-commutative operator families on rational functions...
I am studying the article "Symmetrization operators in polynomial rings" by A. Lascoux and M.-P. Schützenberger (MSN). Specifically, I am trying to prove the following claim involving operators defined...
View ArticleWhat does $\mathrm{Conf}_n(M)^{h S_n}$ look like?
$\DeclareMathOperator\Conf{Conf}$Let $M$ be a manifold, and $\Conf_n M$ the ordered configuration space of n points on $M$. The symmetric group $S_n$ acts by permuting the points.Is there a simple...
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Segal's electrostatic map given in "Configuration Spaces and Iterated Loop...
In Segal's "Configuration Spaces and Iterated Loop Spaces", I'm not understanding the map $C_n(X) \rightarrow \Omega^n S^n X$ given in the first page, Section 1 (the picturesque electrostatic map) as...
View ArticleExtension between the abelianization of the pure braid group and the...
The braid group $B_n$ and the pure braid group $P_n$ sits in a short exact sequence$$1\to P_n\to B_n\to S_n\to 1.$$The pure braid group $P_n$ has abelianization $\mathbb Z^{n\choose 2}$, with...
View ArticleExample from Segal's "Configuration Spaces and Iterated Loop Spaces
After Theorem 3 in Segal's Configuration Spaces and Iterated Loop Spaces, he gives some special cases.I do not understand $n = 2$ case, i.e. how is $B(\coprod_{k\geq 0} B(Br_k)) \simeq \Omega...
View ArticleMarkov theorem for braid partial closures
The classical Markov theorem tells us that the closures of two braids are isotopic links if and only if the braids are related by a sequence of Markov moves(MI) $b \sim aba^{-1} $(MII) $b \sim b...
View ArticleWhat do tangles teach us about braids?
A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\dotsc,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\dotsc,n\} \times \{1\}$....
View ArticleMapping class group of $n$-punctured annulus
I am looking for an explicit presentation of the mapping class group of the annulus $\mathbb{A}^2$, after equipping it with $n$ interior punctures/marked points $\{x_1, \cdots, x_n\} \hookrightarrow...
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