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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 ArticleOpen neighborhood of a geometric knot in an orientable 3-dimensional manifold
I was reading "Braid Groups" by Christian Kassel and Vladimir Turaev where I found the following question: Prove that an arbitrary geometric knot $L$ in an orientable 3-dimensional manifold has an open...
View ArticleNormal subgroups of braid groups
There is a lot of normal subgroups in braid groups (for example there is an action of $B_n$ on unitriangular bilinear forms on $R^n$ over arbitrary commutative ring $R$: $b_i\colon e_j\mapsto e_j$,...
View ArticleAlexander 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...
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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