Elements that satisfy an equation in an affine Artin group of type...
Consider the affine Artin group of type $\widetilde{A}_2$, that is:$$A[\widetilde{A}_2]:=\langle a,b,c\ |\ aba=bab,\ bcb=cbc,\ aca=cac\rangle,$$and let $\varphi:A[\widetilde{A}_2]\longrightarrow...
View ArticleDiagonalizable Yang-Baxter solutions
Let $V$ be a finite-dimensional vector space and $R: V \otimes V \to V \otimes V$ a solution of the Yang-Baxter equation$$(R \otimes \mathrm{id}) (\mathrm{id} \otimes R) (R \otimes \mathrm{id}) =...
View ArticleCohomology of a semidirect product of the pure braid group via the Burau...
Let $P_n$ denote the pure braid group on $n$ strands, and consider its (reduced) Burau representation$$\rho : P_n \to \mathrm{GL}_{n-1}(R),$$where $ R = \mathbb{Z}[t,t^{-1}] $.Fix a module $ M $...
View ArticleCurrent knowledge on the cohomology of surface (pure) braid groups
Let $\Sigma_{g,n}$ denote a compact orientable surface of genus $g$ with $n$ punctures, and let $P_n(\Sigma_g)$ denote the corresponding pure braid group on $n$ strands on $\Sigma_g$. I am interested...
View ArticleSet-theoretic solutions to the Yang-Baxter equations and racks
A (finite) set-theoretic solution to the Yang-Baxter equation is a finite set $X$ with a bijection $r\colon X \times X \to X\times X$ such that $(id \times r)(r\times id)(id \times r)=(r\times id)(id...
View ArticleThe Alexander polynomial via the reduced Burau representation
Let $L$ be an oriented link and let $\varphi$ be an $m$-braid whose closure equals $L$.Let $\beta\colon B_m\to \operatorname{GL}(m-1,\Bbb{Z}[t^{\pm 1}])$ be the reduced Burau representation. It is...
View ArticleIs there a q-analog to the braid group?
The braid group $B_n$ on $n$ strands fits into a short exact sequence of groups:$$ 1 \longrightarrow P_n \longrightarrow B_n \longrightarrow S_n \longrightarrow 1,$$where $S_n$ is the symmetric group...
View ArticlePresentation of the mixed Braid group
This question concerns the mixed Braid groups$B_{n,P}$. Suppose $P$ is a partition of $\{1, \ldots, n\}$. Consider the subgroup $S_{n,P}$ of the symmetric group $S_n$ consisting of the permutations...
View ArticleCurves fixed by periodic elements of mapping class groups of punctured spheres
Let $\overline{B}_{2n+1}$ be the mapping class group of a $(2n+1)$-punctured sphere fixing one distinguished puncture. Let $f\in \overline{B}_{2n+1}$ be a periodic element. Since there is a concrete...
View ArticleWhen do two positive braids represent the same link?
Let $B_n$ be the braid group on $n$ strands, with the usual generators: $s_1, \ldots, s_{n-1}$ and their inverses, where $s_i$ is a positive half-twist interchanging the strands labelled $i$ and $i+1$....
View ArticleSecond homology of braid group via Hopf's formula
The Hopf's formula says that if $1 \to R \to F \to G \to 1$ is a presentation of the group $G$ (i.e. the sequence is exact, $R, F$ are free) then $H_2(G) = \frac{R \cap [F, F]}{[F, R]}$. As far as I...
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How to split discrete torus braids of size $k$ resulting from symmetric group...
Consider the symmetric group $S_{2n}$ and $[2n]:=\lbrace 1,..,2n\rbrace$. All notations regarding the symmetric group come from its action on the set $[2n]$. We define discrete torus braids of size $k$...
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...
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 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 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 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 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 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 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...
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