braid group

braid group

Let

C

n

be the space of unordered

n

-tuples of distinct points in the complex plane. The

braid group

B

n

is the

fundamental group

of

C

n

.

A

closed path

γ

on this space is a set of

n

paths

γ

i

:

[

0

,

1

]

→

ℂ

with

γ

i

⁢

(

t

)

≠

γ

j

⁢

(

t

)

, and

γ

i

⁢

(

1

)

=

γ

σ

⁢

(

i

)

⁢

(

0

)

, where

σ

is some

permutation

of

{

1

,

…

,

n

}

. Drawing the graphs of all these paths in 3 space, what we see is

n

strands between the

z

=

0

and

z

=

1

planes, possibly tangled, with

composition

given by stacking these braids on top of each other.

Homotopy

corresponds to

isotopy

of the braid, homotopies of the strands such that none of them cross. This is the origin of the name “braid group”

The braid group determines a

homomorphism

ϕ

:

B

n

→

S

n

, where

S

n

is the

symmetric group

on

n

letters. For

γ

∈

B

n

, we get an element of

S

n

from map sending

i

↦

γ

i

⁢

(

1

)

. This works because of our requirement on the points that the braids start and end, and since our homotopies fix

basepoints

. The kernel of

ϕ

consists of the braids that bring each strand to its original order. This kernel gives us the

pure braid group on n strands

, and is denoted by

P

n

. Hence, we have a

short exact sequence

1

→

P

n

→

B

n

→

S

n

→

1

.

We can also describe braid groups in more generality. Let

M

be a

manifold

. The configuration space of

n

ordered points on

M

is defined to be

F

n

⁢

(

M

)

=

{

(

a

1

,

…

,

a

n

)

∈

M

n

∣

a

i

≠

a

j

⁢

for

⁢

i

≠

j

}

. The group

S

n

acts on

F

n

⁢

(

M

)

by permuting coordinates, and the corresponding quotient space

C

n

⁢

(

M

)

=

F

n

⁢

(

M

)

/

S

n

is called the configuration space of

n

unordered points on

M

. In the case that

M

=

ℂ

, we obtain the

regular

and pure braid groups as

π

1

⁢

(

C

n

⁢

(

M

)

)

and

π

1

⁢

(

F

n

⁢

(

M

)

)

respectively.

The group

B

n

can be given the following

presentation

. The presentation was given in Artin’s first paper

[

1

]

on the braid group. Label the braids

1

through

n

as before. Let

σ

i

be the braid that twists strands

i

and

i

+

1

, with

i

passing beneath

i

+

1

. Then the

σ

i

generate

B

n

, and the only

relations

needed are

σ

i

⁢

σ

j

=

σ

j

⁢

σ

i

for

⁢

|

i

-

j

|

≥

2

,

1

≤

i

,

j

≤

n

-

1

σ

i

⁢

σ

i

+

1

⁢

σ

i

=

σ

i

+

1

⁢

σ

i

⁢

σ

i

+

1

for

⁢

1

≤

i

≤

n

-

2

The pure braid group has a presentation with

generators

⁢

a

i

⁢

j

=

σ

j

-

1

⁢

σ

j

-

2

⁢

⋯

⁢

σ

i

+

1

⁢

σ

i

2

⁢

σ

i

+

1

-

1

⁢

⋯

⁢

σ

j

-

2

-

1

⁢

σ

j

-

1

-

1

⁢

for

⁢

1

≤

i

<

j

≤

n

that is,

a

i

⁢

j

wraps the ith strand around the jth strand, and

defining relations

a

r

⁢

s

-

1

⁢

a

i

⁢

j

⁢

a

r

⁢

s

=

{

a

i

⁢

j

if

i

<

r

<

s

<

j

⁢

or

⁢

r

<

s

<

i

<

j

a

r

⁢

j

⁢

a

i

⁢

j

⁢

a

r

⁢

j

-

1

if

r

<

i

=

s

<

j

a

r

⁢

j

⁢

a

s

⁢

j

⁢

a

i

⁢

j

⁢

a

s

⁢

j

-

1

⁢

a

r

⁢

j

-

1

if

i

=

r

<

s

<

j

a

r

⁢

j

⁢

a

s

⁢

j

⁢

a

r

⁢

j

-

1

⁢

a

s

⁢

j

-

1

⁢

a

i

⁢

j

⁢

a

s

⁢

j

⁢

a

r

⁢

j

⁢

a

s

⁢

j

-

1

⁢

a

r

⁢

j

-

1

if

r

<

i

<

s

<

j

References

1

E. Artin

Theorie der Zöpfe

. Abh. Math. Sem. Univ. Hamburg 4(1925), 42-72.

2

V.L. Hansen

Braids and

Coverings

. London Mathematical Society Student Texts 18. Cambridge University Press. 1989.

Title

braid group

Canonical name

BraidGroup

Date of creation

2013-03-22 13:51:51

Last modified on

2013-03-22 13:51:51

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

15

Author

bwebste (988)

Entry type

Topic

Classification

msc 20F36

Synonym

Artin’s braid group

Related topic

Tangle

Defines

pure braid group

Defines

braid

Defines

configuration space

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