# 6.01 Braids: Introduction

Below the video you will some pre-class questions and notes to accompany the video.

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**Braids: Artin action**. - Index of all lectures.

# Notes

## Definitions

*(0.00)*

Fix a collection of \(n\) points \(z_1,\ldots,z_n\) in
\(\mathbf{C}\). An

We can draw a picture of a braid as a collection of pairwise disjoint
paths \(\gamma_1,\ldots,\gamma_n\) in \(\mathbf{C}\times[0,1]\):
\[\gamma_k(t)=(F_k(t),t).\] In fact, since \([0,1]\) is compact and
the image of a compact set by a continuous map is compact, the images
of the paths \(F_k\) are contained in some compact set in the plane,
and we can always homotope everything (by a family of rescalings
depending on \(t\)) to assume that our braids are contained in
\(D\times[0,1]\), where \(D\) is the unit disc.
*\(n\)-strand braid*\(F\) is a collection of \(n\) continuous maps \(F_1,\ldots,F_n\colon[0,1]\to\mathbf{C}\) such that:- \(F_i(t)\neq F_j(t)\) if \(i\neq j\)
- \(F_i(0)=z_i\), \(F_i(1)=z_{s(i)}\) for some permutation
\(s\colon\{1,\ldots,n\}\to\{1,\ldots,n\}\).

## Equivalence of braids

*(3.23)*We say that two \(n\)-strand braids \(F\) and \(G\) are

*equivalent*if there is a collection of homotopies \(H_k(s,t)\), \(k=1,\ldots,n\), such that \(\{H_k(s,t)\}_{k=1}^n\) is a braid for each fixed value of \(s\) and such that \[H_k(0,t)=F_k(t),\ H_k(1,t)=G_k(t),\]

## Group law

*(5.20)*If \(F\) and \(G\) are two \(n\)-strand braids with associated permutations \(\sigma\) and \(\tau\) respectively then their product \(G\cdot F\) is the braid \[ (G\cdot F)_i(t) =\begin{cases} F_i(2t)&\mbox{ if }t\in[0,1/2]\\ G_{s(i)}(2t-1)&\mbox{ if }t\in[1/2,1]. \end{cases} \]

The set of equivalence classes of \(n\)-strand braids form a group \(B_n\)
under this stacking product.

This is an exercise.

Much of the proof of this theorem should look a little bit like the
proof that the fundamental group is a group. This is not a
coincidence: the \(n\)-strand braid group

**is**the fundamental group of a particular space, the

*unordered configuration space of \(n\) points in the disc*.

## Configuration space

*(8.20)*Let \(OC_n\) be the subset of \(\mathbf{C}^n\) defined by \[OC_n:=\{(x_1,\ldots,x_n)\in \mathbf{C}^n\ :\ x_i\neq x_j\mbox{ for }i\neq h\}.\] We call a point \((x_1,\ldots,x_n)\in OC_n\) an

*ordered configuration*of points in the disc and \(OC_n\) is called the

*ordered configuration space of \(n\) points in the plane*. There is an action of the permutation group \(S_n\) on \(OC_n\); a permutation \(s\) acts as \[(x_1,\ldots,x_n)s=(x_{s(1)},\ldots,x_{s(n)}).\] The quotient \(UC_n:=OC_n/S_n\) is called the space is called the

*unordered configuration space of \(n\) points in the plane*.

The fundamental group \(\pi_1(UC_n,[z_1,\ldots,z_n])\) is isomorphic
to the \(n\)-strand braid group.

This should be clear from the definition of a braid: a braid is a
collection of paths \(F_1(t),\ldots, F_n(t)\) with \(F_i(t)\neq
F_j(t)\) if \(i\neq j\) and such that \(F_i(0)=z_i\),
\(F_i(1)=z_{s(i)}\). Such a collection of paths defines a loop: \[
[F_1(t),\ldots,F_n(t)]\] in the unordered configuration space based
at \([z_1,\ldots,z_n]\) and conversely. A homotopy of braids gives a
homotopy of loops in the unordered configuration space (again, just
by definition). Stacking braids corresponds to concatenating loops.

## Presentation of the braid group

*(11.38)*We will assume that the points \(z_i\) are equally spaced along a line. For each \(i=1,2,\ldots,n-1\) there is an

*elementary braid*:

The braid group \(B_n\) is generated by the elementary braids
subject to the following relations:
\begin{gather*}
\sigma_i\sigma_j=\sigma_j\sigma_i\mbox{ if }|i-j|\geq 1\\
\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}.
\end{gather*}

Proof not included! It is an exercise to check that the braid
relations hold. Later, I will give \(\epsilon\) more explanation for
how one would go about checking that these relations suffice.

# Pre-class questions

- Why do braids form a group under stacking?

# Navigation

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**Braids: Artin action**. - Index of all lectures.