Questions tagged [braid-groups]
The braid-groups tag has no usage guidance.
81
questions with no upvoted or accepted answers
19
votes
0
answers
420
views
Are braid groups known to not be linear over $\mathbb{Z}$?
$\DeclareMathOperator\GL{GL}$It is known that every braid group $B_n$ embeds as a subgroup of $\GL_m(\mathbb{Z}[q^{\pm 1},t^{\pm 1}])$, where $m=n(n-1)/2$ (see Krammer - Braid groups are linear). This ...
- 3,363
18
votes
0
answers
467
views
What do tangles teach us about braids?
A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times \{1\}$....
- 21.9k
17
votes
0
answers
983
views
Symmetries 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 ...
- 22.2k
15
votes
0
answers
555
views
Relation Between Truncated Braid Groups and Regular Tilings of the Complex and Hyperbolic Plane
This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question on math.SE.
There exists a rather remarkable ...
- 715
14
votes
0
answers
784
views
Splitting of homomorphism from cactus group to permutation group
We all learned in kindergarten that the category of finite-dimensional (type I, say) $U_q(\mathfrak{g})$-modules is braided monoidal for $\mathfrak{g}$ a complex semisimple Lie algebra. This gives an ...
- 8,409
12
votes
0
answers
382
views
The Markov trace via Bott-Samelson fibers?
Let $H_n$ be the Hecke algebra of GL(n), i.e., the algebra over $\mathbb{Q}(q)$ with generators $T_1,
\ldots, T_{n-1}$ which satisfy the braid relations and also $T^2 = (q-1) T + q$.
Recall the ...
- 8,643
11
votes
0
answers
210
views
On 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 ...
- 40.2k
11
votes
0
answers
240
views
Cohomology of a configuration space of points on $\mathbb C^\times$ with an additional restriction
Let $Conf_{1,n}^3$ be the configuration space of collections of $n$ distinct numbered points on the annulus $\mathbb C^\times$ with an imposed restriction: for any $r\in \mathbb R^+$ the circle $\...
- 401
10
votes
0
answers
335
views
Surjective homomorphisms between braid groups
There cannot be a surjective homomorphism $B_2 \to B_n$ for any $n > 2$ because $B_2$ is commutative and $B_n$ is not. It seems plausible that if $m < n$, there cannot be a surjective ...
- 323
10
votes
0
answers
343
views
Finite quotients of surface braid groups
Let $\Sigma_b$ be a closed orientable surface of genus $b \geq 2$, and denote by $\mathsf{P}_2(\Sigma_b)$ the pure braid group with two strands on $\Sigma_b$.
There is a braid $A_{12} \in \Sigma_b$ ...
- 65.1k
10
votes
0
answers
241
views
What is the preimage of a braid in a covering space branched over the braid?
For a knot $K\subset \mathbb{S}^3$, one can construct the covering space branched over that knot by assigning elements of the symmetric group $S_n$ to each arc of the knot. You can find the knot group ...
- 145
9
votes
0
answers
167
views
Configuration space of 4 points as an orbifold
Setup: Consider the braid group $B_n$. One way to define this is as the fundamental group of the unordered configuration space $UC_n(\mathbb{C}) = \{\{z_1,\dotsc,z_n\}\subset \mathbb{C} \mid z_i \not= ...
- 1,267
9
votes
0
answers
184
views
Interactions between pseudoline arrangements and braid groups?
It is common to represent
pseudoline arrangements
as wiring diagrams:
Fig. from: "Hamiltonicity and colorings of arrangement ...
- 149k
9
votes
0
answers
368
views
Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$
This is related to my previous MathOverflow question Fundamental group of $\mathrm{Sym}^2(C_g)$ minus the diagonal.
Let $C_2$ be a smooth curve of genus $2$ and $X:=\mathrm{Sym}^2(C_2)$ its second ...
- 65.1k
8
votes
0
answers
162
views
Is the Lawrence–Krammer representation faithful, reduced modulo p?
It is well-known that the braid group $B_n$ is linear for every $n$ by the Lawrence–Krammer (or LKB) representation. It embeds $B_n$ faithfully into $\mathrm{GL}\left(\frac{n(n-1)}{2},\mathbb{Z}[q^{\...
- 113
8
votes
0
answers
245
views
Monotone invariants of braid forcing
Let $\phi$ be a diffeomorphism of the unit disk $D^2$, fixed on the boundary, and suppose that $Q$ is a finite subset of the interior permuted by $\phi$. The isotopy class of $\phi$ relative to $Q$ ...
- 2,562
7
votes
0
answers
97
views
Normal 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 ...
- 3,363
7
votes
0
answers
248
views
Generating cycles inside Tits' graph of words for a positive braid
Let $Br_n$ be the braid group and consider words in its generators (not in the inverses). Two such words define the same "positive" braid if one can be obtained from the other by commuting ...
- 27.7k
7
votes
0
answers
400
views
Very frustrated reading a proof of the faithfulness of Artin's representation of braid groups
I am reading BRAID GROUPS, FREE GROUPS, AND THE LOOP SPACE OF THE 2-SPHERE by F.R. Cohen and J. Wu and here is an extract of the paper:
(The proof is not finished yet but I am very confused by now.)
...
- 1,108
7
votes
0
answers
218
views
Does the braid group act faithfully on the quantized enveloping algebra?
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$, where $...
- 8,409
7
votes
0
answers
382
views
Braid group action on canonical basis
This is a question about Lusztig's theory of based modules. This theory is elementary but far from easy and is developed in Chapter 27 of "Introduction to quantum groups".
Let $V$ be a highest weight ...
- 9,052
6
votes
0
answers
135
views
Is there a natural, purely group-theoretic definition of the virtual braid group?
The braid group $B_n$ has the well-known presentation $$\left<\sigma_i,i=1\ldots n-1\, \left| \begin{cases}\sigma_i\sigma_j=\sigma_j\sigma_i & |i-j|>1\\\sigma_i\sigma_j\sigma_i=\sigma_j\...
- 1,005
6
votes
0
answers
105
views
Permutative Yang-Baxter monoids
Suppose that $f,g:X^{2}\rightarrow X,T:X^{2}\rightarrow X^{2}$ are mappings such that $T(x,y)=(f(x,y),g(x,y))$. An element
$1\in X$ is said to be an identity if $T(1,x)=(x,1),T(x,1)=(1,x)$. The ...
- 27.8k
6
votes
0
answers
126
views
Braid groups representations on infinite dimensional vector spaces
Let $V$ be an infinite dimensional complex vector space. Let $R:V\otimes V\to V\otimes V$ be a solution to the quantum Yang Baxter Equation. In other words: $R$ is invertible and satisfies the ...
- 4,979
6
votes
0
answers
375
views
Presentation of Homotopy Pure Braid Group?
Let $\tilde{B_n}$ be the homotopy braid group; namely, in the deformation of braids, a braid string is allowed to self-intersect itself. Similarly let $\tilde{P_n}$ be the homotopy pure braid group.
...
- 1,108
5
votes
0
answers
146
views
What 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 ...
- 765
5
votes
0
answers
229
views
Coxeter's braid group quotients
Coxeter's result is that if the generators of the braid group $B_n$ on $n$ strands fulfill a relation $\forall_i\sigma_i^k=1$, then $1/n+1/k>1/2$ must hold to get a finite quotient of $B_n$. In ...
- 4,699
5
votes
0
answers
588
views
Do the ternary braid groups arise in algebraic topology?
Let $TB_{n}$ be the group defined by the presentation with generators $t_{1},...,t_{n-2}$ and relations $t_{i}t_{i+1}t_{i+2}t_{i}=t_{i+2}t_{i}t_{i+1}t_{i+2}$
and $t_{i}t_{j}=t_{j}t_{i}$ whenever $|i-j|...
- 27.8k
5
votes
0
answers
168
views
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[ \Delta_{k_1},\...
5
votes
0
answers
120
views
How long does it take for the action of the braid monoids on Laver tables to become trivial?
Let $A_{n}$ denote the classical Laver table of cardinality $2^{n}$.
Let $B_{n}^{*}$ denote the positive (including the identity) braid monoid on $n$ elements generated by $\sigma_{1},...,\sigma_{n}$....
- 27.8k
5
votes
0
answers
260
views
What's the story with the Hopf fibration and the Jacobi identity?
I like the Hopf fibration of the 3-sphere $S^3$ enough that I found a nice way to make a physical model of it. All you need is to combine a bunch of key rings in such a way that (ii) every pair of ...
- 1,267
4
votes
0
answers
101
views
Interplay 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 ...
- 6,484
4
votes
0
answers
140
views
Does the Hurwitz action of the braid group on rank-into-rank embeddings tend to increase the critical points?
An algebraic structure $(X,*)$ is said to be self-distributive if it satisfies the identity $x*(y*z)=(x*y)*(x*z)$.
Suppose that $X$ is a self-distributive algebra. Then the positive braid monoid $B_{...
- 27.8k
4
votes
0
answers
113
views
semisimplicity of maps in braided vector spaces
Let $V$ be a finite dimensional braided vector space over $\mathbb{C}$.
This means that we have a map $$c_{V,V}:V\otimes V\to V\otimes V$$ which gives us an action of the braid group $B_n$ on $V^{\...
- 4,979
4
votes
0
answers
105
views
Do the normal forms for braid groups really conceal information about better than randomly applying the braid relations?
In braid based cryptography, one typically wants to conceal the way a certain braid $b$ has been obtained. One therefore puts $b$ into some normal form. Since every braid has a unique normal form, the ...
- 27.8k
4
votes
0
answers
205
views
How do we see the rank of the braid group?
The only presentation of the braid group that most people ever see is the standard Artin presentation
$$B_n=\langle σ_1,\cdots,σ_{n−1}|\ σ_iσ_j=σ_jσ_i\ \ (|i−j|>1),\ σ_iσ_{i+1}σ_i=σ_{i+1}σ_i σ_{i+...
- 1,691
4
votes
0
answers
230
views
The Alexander-Conway polynomial: from knots to braids?
The Alexander-Conway polynomial was the first knot invariant to be discovered, as far back as 1923 according to this link. Given that knots can be expressed in terms of quasi-toric braid closures, it ...
HeloLobo
4
votes
0
answers
423
views
Generalized pure braid groups
Generalized braid groups are by definition obtained from finite Coxeter groups after removing some of its relations. For example the Coxeter group $H_3=< a_1, a_2, a_3 | a_i^2=1, a_1a_3=a_3a_1, ...
- 49
4
votes
0
answers
353
views
Constructing Markov traces simply
Short version: I wondering how to simply check if a proposed Markov trace, $\phi$ had the correct property using techniques similar to those from the Akutsu-Wadati 1987 paper `Exactly solvable models ...
- 41
4
votes
0
answers
607
views
Positivity of braid monodromy of curve singularities
I recall the notion of braid monodromy. Let $C \subset \mathbb{C}^2$ be an algebraic curve, and choose a projection $\pi: \mathbb{C}^2 \to \mathbb{C}$ such that the restriction $\pi: C \to \mathbb{C}...
- 8,643
4
votes
0
answers
195
views
restricting "dances of minimal cost" (optimization on braids/permutations?)
Consider applying the permutation (1,3,0,5,2,7,4,6) to the integers (0,1,2,3,4,5,6,7) three times.
I call this a "dance of minimal cost" because all unordered pairs in {0..7} meet each other, and the
...
- 121
3
votes
0
answers
126
views
Stochastic 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 "...
- 1,944
3
votes
0
answers
191
views
Braids of fibered knots
There are some theorems saying that the closure of a braid of a specific form is fibered. For instance, a theorem of Stallings says that the closure of a homogeneous braid is a fibered knot.
I am ...
- 1,410
3
votes
0
answers
587
views
Braided lobsters
If $(X,m)$ is a median algebra, then for each $x\in X$, define an operation $\wedge_{x}$ by letting $y\wedge_{x}z=m(x,y,z)$. Then $(X,\wedge_{x})$ is a meet-semilattice with least element $x$. Define ...
- 27.8k
3
votes
0
answers
96
views
What are the composition series for these series of groups?
A rack is an algebra $(X,*,*^{-1})$ that satisfies the identities
$x*(y*z)=(x*y)*(x*z)$ and
$x*(x*^{-1}y)=x*^{-1}(x*y)=y$.
If $X$ is a rack then define a homomorphism $\phi_{n,X}:B_{n}\rightarrow \...
- 27.8k
3
votes
0
answers
103
views
Is the variety of algebras that satisfy the Yang-Baxter equation generated by its finite members?
Suppose that $f,g:X^{2}\rightarrow X$, and $T:X^{2}\rightarrow X^{2}$ is the function where $T(x,y)=(f(x,y),g(x,y))$. Then $(X,f,g)$ is said to satisfy the Yang-Baxter equation if $(T\times 1_{X})\...
- 27.8k
3
votes
0
answers
128
views
Does the notion of a critical point extend from set theory to Braid groups?
Let $B_{\infty}$ denote the infinite strand braid group. Let $\text{sh}:B_{\infty}\rightarrow B_{\infty}$ be the homomorphism defined by $\text{sh}(\sigma_{i})=\sigma_{i+1}$ whenever $i\geq 1$. Give $...
- 27.8k
3
votes
0
answers
72
views
Partially permutative matrices
Let $V$ be a finite dimensional vector space over a field $K$. Then a map
$L:V\otimes V\rightarrow V\otimes V$ is said to satisfy the Yang-Baxter equation if $(L\otimes I)(I\otimes L)(L\otimes I)=(I\...
- 27.8k
3
votes
0
answers
703
views
the braid groups on the sphere
The braid group $B_n(S^2)$ on the sphere possesses a finite element of order which generates a cyclic group $M$ in $B_n(S^2)$. My question is this:
What is the index of $H$ in $B_n (S^2)$?
Same ...
3
votes
0
answers
254
views
Braids with an infinite number of strings
Has anyone developed a theory for braids with an infinite number of strings?
- 1,161