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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...
View ArticleFinite quotients of the braid group that remember the winding number modulo $d$
Let $B_n$ be the braid group on $n$ points, and let $S_n$ be the symmetric group on $n$ letters. There is a homomorphism $B_n\rightarrow S_n$, given by forgetting the paths of the strings and only...
View ArticlePresentation of the pure Artin groups
Let $W$ be a Coxeter group attached to a Coxeter matrix with entries $m_{ij}$ . The presentation of $W$ is given by $$W= < T_1, \dots, T_n | T_i^2=1, T_iT_jT_i \ldots = T_jT_iT_j \ldots, i \neq...
View ArticleReference request on Braid groups as almost actions
I have heard recently that one can define the usual braid groups $B_n$ using almost actions (in the sense of Yves Cornulier, as in https://arxiv.org/pdf/1901.05065) using the symmetric groups $S_n$. I...
View ArticleBounds 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...
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