All Questions
Tagged with knot-theory braid-groups
44 questions
6
votes
1
answer
311
views
Loop 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 ...
- 647
6
votes
1
answer
444
views
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| \...
- 156k
4
votes
1
answer
258
views
Bounds 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 ...
- 9,114
2
votes
0
answers
80
views
Composition 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 ...
- 73
2
votes
1
answer
164
views
Are 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 ...
- 73
0
votes
1
answer
205
views
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 ...
- 73
4
votes
1
answer
186
views
Alexander 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 $\sigma_\kappa:=\sigma_1^{k_1}\...
- 685
7
votes
0
answers
265
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 ...
- 27.8k
2
votes
1
answer
244
views
Determine if a closed braid is a link/unlink
I am relatively new to the world of braids/knots so really sorry if this question is simple. However, I am not able to find if there is any theorem/procedure that determines if a closed braid, given ...
- 73
8
votes
0
answers
185
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
5
votes
0
answers
237
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 ...
- 4,829
4
votes
1
answer
336
views
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}$....
- 309
3
votes
0
answers
209
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 ...
- 1,430
3
votes
1
answer
166
views
Knots with a braid presentation with only positive or negative crossings on each fixed position
I am interested in the following class of knots $K$:
$\{$$K$ has a braid presentation such that for any fixed position $k$, either only positive or negative powers of $\sigma_k$ appear in the braid ...
- 1,430
20
votes
2
answers
2k
views
Several 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 ...
- 2,099
5
votes
1
answer
269
views
Do there exist any variational principles on the space of braids (or knots)?
This is very speculative question and I do not know where to start looking up the literature, or if what I am looking for is even mathematically possible/meaningful.
Q: I am interested in finding out ...
- 2,281
3
votes
1
answer
211
views
Basis for Annular Skein Algebra
Background/Notation:
Given the Iwahori-Hecke algebra $H_{n}$ (over some ring commutative ring $R$ with identity) with generators $\{T_{1},\ldots T_{n-1}\}$, we know it has basis
$\{T_{w}\}_{w\in S_{...
- 318
9
votes
0
answers
186
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 ...
- 151k
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
3
votes
1
answer
280
views
The singularity type of a non-torus link
It is a well-know result that singular surfaces like
$$x^p+y^q=0$$
(for complex $x$ and $y$) can be associated with $(p,q)$-torus links by considering the intersection of this surface with a small ...
- 145
6
votes
1
answer
255
views
How to braid a ribbon knot
Is there any algorithm known for braiding ribbon knots? More specifically I need to braid a generic ribbon knot presented as boundary of a ribbon surface= union of two 0-handles and one 1-handle. (...
- 143
2
votes
1
answer
285
views
Plat representations of unlinks
Suppose that $\beta$ is a $2n$-strand braid with plat closure $L$. We can multiply $\beta$ on either side by a member of the Hilden subgroup to get a new braid whose plat closure is still $L$. Or we ...
- 268
2
votes
1
answer
184
views
A three-rope tangle that is equal to its mirror tangle
In the same way that a figure-eight knot is equal (after a suitable rotation) to its mirror knot, I am looking for simple tangles made of three ropes that are equal to their mirror tangle.
The ...
5
votes
0
answers
122
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 $\sigma_{1},...,\sigma_{n}$....
4
votes
1
answer
824
views
Are Markov traces matrix traces?
When starting this question I was very hesitant - literature on the subject is vast and I thought most likely the answer is already there somewhere.
Then when the list "Questions that may already ...
- 18.4k
2
votes
2
answers
583
views
How can i change 8_19 to (3,4)-torus knot K(3,4)?
In the knot table, it is well-known that 8_19 is (3,4)-torus knot. But, it is not clear to me. How can i change 8_19 to (3,4)-torus knot K(3,4)? Moreover, it is well-known that braids of two ...
- 21
4
votes
1
answer
300
views
Computable link invariants
I am interested in the following situation: given a braid $B$, it induces a link $L$ in a pretty straightforward way ("glue" the endpoints, like here). For a braid $B$, we know how to represent it in ...
- 647
18
votes
0
answers
502
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 $f(k,1) \in \{1,2,\ldots,n\} \times \{1\}$....
- 22.1k
12
votes
0
answers
386
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 ...
- 8,723
9
votes
1
answer
288
views
Does the shortest path between two braids pass through string links?
One of the fundamental facts underlying the application of braid theory to knot theory is that braids inject into string links.
This means that braids $B_1$ and $B_2$, considered inside a cube $I^3$, ...
- 22.1k
11
votes
3
answers
2k
views
When 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 $i+1$....
- 8,723
20
votes
4
answers
1k
views
Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group?
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 ...
- 715
7
votes
1
answer
522
views
Resemblance between Birman-Murakami-Wenzl algebra representations and the Lawrence-Krammer representations
At the end of Stephen Bigelow's paper "Braid Groups are Linear", he mentioned that there is a striking resemblance between the matrices of the Lawrence-Krammer (LK) representations and "those of a ...
- 2,297
4
votes
0
answers
408
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
2
votes
1
answer
403
views
Computing an Invariant for Knots via Braid Words?
I've been reading up on Knot Theory (which is not my area of expertise) and am stuck in the following bit of logic:
Statement 1: Every knot can be represented as a braid.
Statement 2: There's a ...
- 2,235
4
votes
1
answer
697
views
How does one relate the monodromy of the KZ equations with the WRT representation of the braid group?
The KZ equations on the configuration space of $n$ distinct points in $\mathbb C$ give rise to a representation of $B_n$ on $V^{\otimes n}$, where $V$ is any given representation of $SL(2)$ (we'll ...
- 18.7k
4
votes
0
answers
610
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}...
- 8,723
21
votes
3
answers
3k
views
Is 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 $\sigma_1,\...
- 29k
0
votes
0
answers
435
views
Properties of the Jones polynomial
Let $V(L)$ be the Jones polynomial of the oriented
link $L$. For $\alpha \in B_n$, we write $V(\alpha)$ for
$V(\hat{\alpha})$, where $\hat{\alpha}$ is the closed braid
associated to $\alpha$. The ...
- 63
2
votes
1
answer
2k
views
Markov Trace and Markov Property
Hey guys,
I'm a computer science student attempting to understand a quantum algorithm that uses braid theory - something I'm completely unfamiliar. I've getting through the algorithm but I can't ...
- 187
1
vote
1
answer
245
views
Simplified Jones trace invariant for links
Jones (1985) defines a simplified trace invariant for knots by $W_K(t)=\frac{1-V_K(t)}{(1-t^3)(1-t)}$. Then, e.g., the Arf invariant for $K$ is $Arf(K)=W_K(i)$. Does this work for oriented links as ...
- 63
0
votes
1
answer
305
views
Are braid links proper links?
Are braid links proper links? Or are the concepts involved unrelated?
- 63
3
votes
1
answer
382
views
Is a positive link the closure of a positive braid?
Alexander's Theorem guarantees that every oriented link is the closure of some braid. In other words, the map
$$ \displaystyle \coprod\_n \mathcal B_n\longrightarrow \{\text{ oriented links }\} $$
...
- 1,756
15
votes
5
answers
3k
views
Is the pure braid group on three strands generated as a normal subgroup of the braid group by the six-crossing braid?
Artin's presentation of braid group on three strands is:
$$ B_3 = \langle l,r : lrl = rlr \rangle $$
where you should think of "$l$" as the positive crossing between the left and middle strands and "$...
- 54.6k