nLab
braid group

Contents

Idea

The braid group Br nBr_n is the group whose elements are isotopy classes of nn 1-dimensional braids running vertically in 3-dimensional Cartesian space, the group operation being their concatenation.

Here a braid with nn strands is thought of as nn pieces of string joining nn points at the top of the diagram with nn-points at the bottom.

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(This is a picture of a 33-strand braid.)

We can transform / ‘isotope’ these braid diagrams just as we can transform knot diagrams, again using Reidemeister moves. The ‘isotopy’ classes of braid diagrams form a group in which the composition is obtained by putting one diagram above another.

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The identity consists of nn vertical strings, so the inverse is obtained by turning a diagram upside down:

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This is the inverse of the first 3-braid we saw.

There are useful group presentations of the braid groups. We will return later to the interpretation of the generators and relations in terms of diagrams.

Geometric definition

Let C n nC_n \hookrightarrow \mathbb{C}^n be the space of configurations of nn points in the complex plane, whose elements are those nn-tuples (z 1,,z n)(z_1, \ldots, z_n) such that z iz jz_i \neq z_j whenever iji \neq j. The symmetric group S nS_n acts on C nC_n by permuting coordinates. Let C n/S nC_n/S_n be the orbit space (the space of nn-element subsets of \mathbb{C} if one likes), and let [z 1,,z n][z_1, \ldots, z_n] be the image of (z 1,,z n)(z_1, \ldots, z_n) under the quotient π:C nC n/S n\pi: C_n \to C_n/S_n. We take p=(1,2,,n)p = (1, 2, \ldots, n) as basepoint for C nC_n, and [p]=[1,2,n][p] = [1, 2, \ldots n] as basepoint for C n/S nC_n/S_n.

Definition

The braid group Br nBr_n is the fundamental group π 1(C n/S n,[p])\pi_1(C_n/S_n, [p]). The pure braid group P nP_n is π 1(C n,p)\pi_1(C_n, p).

Evidently a braid β\beta is represented by a path α:IC n/S n\alpha: I \to C_n/S_n with α(0)=[p]=α(1)\alpha(0) = [p] = \alpha(1). Such a path may be uniquely lifted through the covering projection π:C nC n/S n\pi: C_n \to C_n/S_n to a path α˜\tilde{\alpha} such that α˜(0)=p\tilde{\alpha}(0) = p. The end of the path α˜(1)\tilde{\alpha}(1) has the same underlying subset as pp but with coordinates permuted: α˜(1)=(σ(1),σ(2),,σ(n))\tilde{\alpha}(1) = (\sigma(1), \sigma(2), \ldots, \sigma(n)). Thus the braid β\beta is exhibited by nn non-intersecting strands, each one connecting an ii to σ(i)\sigma(i), and we have a map βσ\beta \mapsto \sigma appearing as the quotient map of an exact sequence

1P nBr nS n11 \to P_n \to Br_n \to S_n \to 1

which is part of a long exact homotopy sequence corresponding to the fibration π:C nC n/S n\pi: C_n \to C_n/S_n.

Group-theoretic descriptions

Artin presentation of Br n+1Br_{n+1}

The Artin braid group, Br n+1Br_{n+1}, defined using n+1n+1 strands is a group given by

  • generators: y iy_i, i=1,,ni = 1, \ldots, n;

  • relations:

    • r i,jy iy jy i 1y j 1r_{i,j} \equiv y_i y_j y_i^{-1} y_j^{-1} for i+1<ji+1 \lt j

    • r i,i+1y iy i+1y iy i+1 1y i 1y i+1 1r_{i,i+1}\equiv y_i y_{i+1} y_i y_{i+1}^{-1} y_i^{-1} y_{i+1}^{-1} for 1i<n1 \leq i \lt n.

In terms of automorphisms on free groups

The braid group B nB_n may be alternatively described as the mapping class group of a 2-disk D 2D^2 with nn punctures (call it X nX_n). Meanwhile, the fundamental group π 1(X n)\pi_1(X_n) (with basepoint on the boundary) is a free group F nF_n on nn generators; the functoriality of π 1\pi_1 implies we have an induced homomorphism

Aut(X n)Aut(π 1(X n))=Aut(F n).Aut(X_n) \to Aut(\pi_1(X_n)) = Aut(F_n).

If an automorphism ϕ:X nX n\phi: X_n \to X_n is isotopic to the identity, then of course π 1(ϕ)\pi_1(\phi) is trivial, and so the homomorphism factors through the quotient MCG(X n)=Aut(X n)/Aut 0(X)MCG(X_n) = Aut(X_n)/Aut_0(X), so we get a homomorphism

B n=MCG(X n)Aut(F n)B_n = MCG(X_n) \to Aut(F_n)

and this turns out to be an injection.

Explicitly, the generator y iy_i used in the Artin presentation above is mapped to the automorphism σ i\sigma_i on the free group on nn generators x 1,,x nx_1, \ldots, x_n defined by

σ i(x i)=x i+1,σ i(x i+1)=x i+1 1x ix i+1,elseσ(x j)=x j.\sigma_i(x_i) = x_{i+1}, \sigma_i(x_{i+1}) = x_{i+1}^{-1} x_i x_{i+1}, \; \else\; \sigma(x_j) = x_j.

Examples

We will look at such groups for small values of nn.

The group Br 1Br_1

By default, Br 1Br_1 has no generators and no relations, so is trivial.

The group Br 2Br_2

By default, Br 2Br_2 has one generator and no relations, so is infinite cyclic.

The group Br 3Br_3

(We will simplify notation writing u=y 1u = y_1, v=y 2v = y_2.)

This then has presentation

𝒫=(u,v:ruvuv 1u 1v 1).\mathcal{P} = ( u,v : r \equiv u v u v^{-1} u^{-1} v^{-1}).

It is also the ‘trefoil group’, i.e., the fundamental group of the complement of a trefoil knot.

The group Br 4Br_4

Simplifying notation as before, we have generators u,v,wu,v,w and relations

  • r uvwvw 1v 1w 1r_u \equiv v w v w^{-1} v^{-1} w^{-1},
  • r vuwu 1w 1r_v \equiv u w u^{-1} w^{-1},
  • r wuvuv 1u 1v 1r_w \equiv u v u v^{-1} u^{-1} v^{-1}.

References

Classical references are

  • Joan S. Birman, Braids, links, and mapping class groups, Princeton Univ Press, 1974.
  • R. H. Fox, L. Neuwirth, The braid groups, Math. Scand. 10 (1962) 119-126, pdf, MR150755

and in addition see

category: knot theory

Revised on April 10, 2014 17:00:10 by Todd Trimble (67.81.95.215)