8
votes
0answers
128 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 ...
6
votes
0answers
107 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 $I^3$, ...
8
votes
1answer
264 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 ...
14
votes
4answers
788 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 ...
6
votes
1answer
313 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 ...
3
votes
0answers
158 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 ...
2
votes
1answer
232 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 ...
4
votes
1answer
342 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 ...
3
votes
0answers
334 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 ...
0
votes
0answers
283 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 ...
1
vote
1answer
540 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 ...
1
vote
1answer
216 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 ...
0
votes
1answer
220 views

Are braid links proper links?

Are braid links proper links? Or are the concepts involved unrelated?
3
votes
1answer
223 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 $$ \displaystyle \coprod\_n \mathcal B_n\longrightarrow \{\text{ oriented links }\} $$ ...
12
votes
5answers
1k 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 ...