Why is the category of strong braided functors from the braid category to a...
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" MSE...
View ArticleOn 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 like...
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 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 ArticleInterplay 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...
View ArticleIs anything known about this braid group quotient?
Let $B_n$ be the braid group on $n$ strands. As is well known, if $\sigma_i$ is the operation of crossing the string in position $i$ over the string in position $i+1$, then the elements...
View ArticleAre 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...
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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...
View ArticleComposition 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...
View ArticleSquier'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 to...
View ArticleCan the n-string sphere braid group embed in to the (n+1)-string sphere braid...
This question was originally posted on math.SE by myself nearly a year ago. I've been thinking again about the problem after it recently received a little attention, but little progress was made in...
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 ArticleQuestion 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 ArticleMapping 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 of...
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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