<s>
In	O
mathematics	O
,	O
the	O
braid	B-Application
group	I-Application
on	O
strands	O
(	O
denoted	O
)	O
,	O
also	O
known	O
as	O
the	O
Artin	B-Application
braid	I-Application
group	I-Application
,	O
is	O
the	O
group	O
whose	O
elements	O
are	O
equivalence	O
classes	O
of	O
-braids	O
(	O
e.g.	O
</s>
<s>
Example	O
applications	O
of	O
braid	B-Application
groups	I-Application
include	O
knot	O
theory	O
,	O
where	O
any	O
knot	O
may	O
be	O
represented	O
as	O
the	O
closure	O
of	O
certain	O
braids	O
(	O
a	O
result	O
known	O
as	O
Alexander	O
's	O
theorem	O
)	O
;	O
in	O
mathematical	O
physics	O
where	O
Artin	O
's	O
canonical	O
presentation	O
of	O
the	O
braid	B-Application
group	I-Application
corresponds	O
to	O
the	O
Yang	O
–	O
Baxter	O
equation	O
(	O
see	O
)	O
;	O
and	O
in	O
monodromy	O
invariants	O
of	O
algebraic	O
geometry	O
.	O
</s>
<s>
Braid	B-Algorithm
theory	I-Algorithm
has	O
recently	O
been	O
applied	O
to	O
fluid	O
mechanics	O
,	O
specifically	O
to	O
the	O
field	O
of	O
chaotic	O
mixing	O
in	O
fluid	O
flows	O
.	O
</s>
<s>
Another	O
field	O
of	O
intense	O
investigation	O
involving	O
braid	B-Application
groups	I-Application
and	O
related	O
topological	O
concepts	O
in	O
the	O
context	O
of	O
quantum	O
physics	O
is	O
in	O
the	O
theory	O
and	O
(	O
conjectured	O
)	O
experimental	O
implementation	O
of	O
so-called	O
anyons	O
.	O
</s>
<s>
These	O
may	O
well	O
end	O
up	O
forming	O
the	O
basis	O
for	O
error-corrected	O
quantum	B-Architecture
computing	I-Architecture
and	O
so	O
their	O
abstract	O
study	O
is	O
currently	O
of	O
fundamental	O
importance	O
in	O
quantum	O
information	O
.	O
</s>
<s>
To	O
put	O
the	O
above	O
informal	O
discussion	O
of	O
braid	B-Application
groups	I-Application
on	O
firm	O
ground	O
,	O
one	O
needs	O
to	O
use	O
the	O
homotopy	O
concept	O
of	O
algebraic	O
topology	O
,	O
defining	O
braid	B-Application
groups	I-Application
as	O
fundamental	O
groups	O
of	O
a	O
configuration	O
space	O
.	O
</s>
<s>
Alternatively	O
,	O
one	O
can	O
define	O
the	O
braid	B-Application
group	I-Application
purely	O
algebraically	O
via	O
the	O
braid	B-Application
relations	I-Application
,	O
keeping	O
the	O
pictures	O
in	O
mind	O
only	O
to	O
guide	O
the	O
intuition	O
.	O
</s>
<s>
To	O
explain	O
how	O
to	O
reduce	O
a	O
braid	B-Application
group	I-Application
in	O
the	O
sense	O
of	O
Artin	O
to	O
a	O
fundamental	O
group	O
,	O
we	O
consider	O
a	O
connected	O
manifold	B-Architecture
of	O
dimension	O
at	O
least	O
2	O
.	O
</s>
<s>
The	O
symmetric	O
product	O
of	O
copies	O
of	O
means	O
the	O
quotient	O
of	O
,	O
the	O
-fold	O
Cartesian	O
product	O
of	O
by	O
the	O
permutation	B-Algorithm
action	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
strands	O
operating	O
on	O
the	O
indices	O
of	O
coordinates	O
.	O
</s>
<s>
This	O
is	O
invariant	O
under	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
and	O
is	O
the	O
quotient	O
by	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
the	O
non-excluded	O
-tuples	O
.	O
</s>
<s>
With	O
this	O
definition	O
,	O
then	O
,	O
we	O
can	O
call	O
the	O
braid	B-Application
group	I-Application
of	O
with	O
strings	O
the	O
fundamental	O
group	O
of	O
(	O
for	O
any	O
choice	O
of	O
base	O
point	O
this	O
is	O
well-defined	O
up	O
to	O
isomorphism	O
)	O
.	O
</s>
<s>
The	O
number	O
of	O
components	O
of	O
the	O
link	O
can	O
be	O
anything	O
from	O
1	O
to	O
n	O
,	O
depending	O
on	O
the	O
permutation	B-Algorithm
of	O
strands	O
determined	O
by	O
the	O
link	O
.	O
</s>
<s>
Braid	B-Application
groups	I-Application
were	O
introduced	O
explicitly	O
by	O
Emil	O
Artin	O
in	O
1925	O
,	O
although	O
(	O
as	O
Wilhelm	O
Magnus	O
pointed	O
out	O
in	O
1974	O
)	O
they	O
were	O
already	O
implicit	O
in	O
Adolf	O
Hurwitz	O
's	O
work	O
on	O
monodromy	O
from	O
1891	O
.	O
</s>
<s>
Braid	B-Application
groups	I-Application
may	O
be	O
described	O
by	O
explicit	O
presentations	O
,	O
as	O
was	O
shown	O
by	O
Emil	O
Artin	O
in	O
1947	O
.	O
</s>
<s>
Braid	B-Application
groups	I-Application
are	O
also	O
understood	O
by	O
a	O
deeper	O
mathematical	O
interpretation	O
:	O
as	O
the	O
fundamental	O
group	O
of	O
certain	O
configuration	O
spaces	O
.	O
</s>
<s>
As	O
Magnus	O
says	O
,	O
Hurwitz	O
gave	O
the	O
interpretation	O
of	O
a	O
braid	B-Application
group	I-Application
as	O
the	O
fundamental	O
group	O
of	O
a	O
configuration	O
space	O
(	O
cf	O
.	O
</s>
<s>
braid	B-Algorithm
theory	I-Algorithm
)	O
,	O
an	O
interpretation	O
that	O
was	O
lost	O
from	O
view	O
until	O
it	O
was	O
rediscovered	O
by	O
Ralph	O
Fox	O
and	O
Lee	O
Neuwirth	O
in	O
1962	O
.	O
</s>
<s>
This	O
presentation	O
leads	O
to	O
generalisations	O
of	O
braid	B-Application
groups	I-Application
called	O
Artin	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
The	O
cubic	O
relations	O
,	O
known	O
as	O
the	O
braid	B-Application
relations	I-Application
,	O
play	O
an	O
important	O
role	O
in	O
the	O
theory	O
of	O
Yang	O
–	O
Baxter	O
equations	O
.	O
</s>
<s>
The	O
braid	B-Application
group	I-Application
is	O
trivial	O
,	O
is	O
the	O
infinite	O
cyclic	O
group	O
,	O
and	O
is	O
isomorphic	O
to	O
the	O
knot	O
group	O
of	O
the	O
trefoil	O
knot	O
–	O
in	O
particular	O
,	O
it	O
is	O
an	O
infinite	O
non-abelian	O
group	O
.	O
</s>
<s>
The	O
-strand	O
braid	B-Application
group	I-Application
embeds	O
as	O
a	O
subgroup	O
into	O
the	O
-strand	O
braid	B-Application
group	I-Application
by	O
adding	O
an	O
extra	O
strand	O
that	O
does	O
not	O
cross	O
any	O
of	O
the	O
first	O
strands	O
.	O
</s>
<s>
The	O
increasing	O
union	O
of	O
the	O
braid	B-Application
groups	I-Application
with	O
all	O
is	O
the	O
infinite	O
braid	B-Application
group	I-Application
.	O
</s>
<s>
There	O
is	O
a	O
left-invariant	O
linear	B-Algorithm
order	O
on	O
called	O
the	O
Dehornoy	O
order	O
.	O
</s>
<s>
This	O
map	O
corresponds	O
to	O
the	O
abelianization	O
of	O
the	O
braid	B-Application
group	I-Application
.	O
</s>
<s>
By	O
forgetting	O
how	O
the	O
strands	O
twist	O
and	O
cross	O
,	O
every	O
braid	O
on	O
strands	O
determines	O
a	O
permutation	B-Algorithm
on	O
elements	O
.	O
</s>
<s>
This	O
assignment	O
is	O
onto	O
and	O
compatible	O
with	O
composition	O
,	O
and	O
therefore	O
becomes	O
a	O
surjective	B-Algorithm
group	O
homomorphism	O
from	O
the	O
braid	B-Application
group	I-Application
onto	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
These	O
transpositions	O
generate	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
satisfy	O
the	O
braid	B-Application
group	I-Application
relations	O
,	O
and	O
have	O
order	O
2	O
.	O
</s>
<s>
This	O
transforms	O
the	O
Artin	O
presentation	O
of	O
the	O
braid	B-Application
group	I-Application
into	O
the	O
Coxeter	B-Algorithm
presentation	I-Algorithm
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
:	O
</s>
<s>
The	O
kernel	O
of	O
the	O
homomorphism	O
is	O
the	O
subgroup	O
of	O
called	O
the	O
pure	B-Application
braid	I-Application
group	O
on	O
strands	O
and	O
denoted	O
.	O
</s>
<s>
In	O
a	O
pure	B-Application
braid	I-Application
,	O
the	O
beginning	O
and	O
the	O
end	O
of	O
each	O
strand	O
are	O
in	O
the	O
same	O
position	O
.	O
</s>
<s>
This	O
sequence	O
splits	O
and	O
therefore	O
pure	B-Application
braid	I-Application
groups	O
are	O
realized	O
as	O
iterated	O
semi-direct	O
products	O
of	O
free	O
groups	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
modular	O
group	O
has	O
trivial	O
center	O
,	O
and	O
thus	O
the	O
modular	O
group	O
is	O
isomorphic	O
to	O
the	O
quotient	O
group	O
of	O
modulo	O
its	O
center	O
,	O
and	O
equivalently	O
,	O
to	O
the	O
group	O
of	O
inner	B-Algorithm
automorphisms	I-Algorithm
of	O
.	O
</s>
<s>
From	O
the	O
braid	B-Application
relations	I-Application
it	O
follows	O
that	O
.	O
</s>
<s>
where	O
and	O
are	O
the	O
standard	O
left	O
and	O
right	O
moves	O
on	O
the	O
Stern	B-Data_Structure
–	I-Data_Structure
Brocot	I-Data_Structure
tree	I-Data_Structure
;	O
it	O
is	O
well	O
known	O
that	O
these	O
moves	O
generate	O
the	O
modular	O
group	O
.	O
</s>
<s>
Mapping	O
to	O
and	O
to	O
yields	O
a	O
surjective	B-Algorithm
group	O
homomorphism	O
.	O
</s>
<s>
The	O
center	O
of	O
is	O
equal	O
to	O
,	O
a	O
consequence	O
of	O
the	O
facts	O
that	O
is	O
in	O
the	O
center	O
,	O
the	O
modular	O
group	O
has	O
trivial	O
center	O
,	O
and	O
the	O
above	O
surjective	B-Algorithm
homomorphism	O
has	O
kernel	O
.	O
</s>
<s>
The	O
braid	B-Application
group	I-Application
can	O
be	O
shown	O
to	O
be	O
isomorphic	O
to	O
the	O
mapping	O
class	O
group	O
of	O
a	O
punctured	O
disk	O
with	O
punctures	O
.	O
</s>
<s>
Alexander	O
's	O
theorem	O
in	O
braid	B-Algorithm
theory	I-Algorithm
states	O
that	O
the	O
converse	O
is	O
true	O
as	O
well	O
:	O
every	O
knot	O
and	O
every	O
link	O
arises	O
in	O
this	O
fashion	O
from	O
at	O
least	O
one	O
braid	O
;	O
such	O
a	O
braid	O
can	O
be	O
obtained	O
by	O
cutting	O
the	O
link	O
.	O
</s>
<s>
The	O
word	O
problem	O
for	O
the	O
braid	B-Application
relations	I-Application
is	O
efficiently	O
solvable	O
and	O
there	O
exists	O
a	O
normal	O
form	O
for	O
elements	O
of	O
in	O
terms	O
of	O
the	O
generators	O
.	O
</s>
<s>
The	O
free	O
GAP	B-General_Concept
computer	I-General_Concept
algebra	I-General_Concept
system	I-General_Concept
can	O
carry	O
out	O
computations	O
in	O
if	O
the	O
elements	O
are	O
given	O
in	O
terms	O
of	O
these	O
generators	O
.	O
</s>
<s>
There	O
is	O
also	O
a	O
package	O
called	O
CHEVIE	O
for	O
GAP3	O
with	O
special	O
support	O
for	O
braid	B-Application
groups	I-Application
.	O
</s>
<s>
The	O
word	O
problem	O
is	O
also	O
efficiently	O
solved	O
via	O
the	O
Lawrence	B-Algorithm
–	I-Algorithm
Krammer	I-Algorithm
representation	I-Algorithm
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
word	O
problem	O
,	O
there	O
are	O
several	O
known	O
hard	O
computational	O
problems	O
that	O
could	O
implement	O
braid	B-Application
groups	I-Application
,	O
applications	O
in	O
cryptography	O
have	O
been	O
suggested	O
.	O
</s>
<s>
In	O
analogy	O
with	O
the	O
action	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
by	O
permutations	B-Algorithm
,	O
in	O
various	O
mathematical	O
settings	O
there	O
exists	O
a	O
natural	O
action	O
of	O
the	O
braid	B-Application
group	I-Application
on	O
-tuples	O
of	O
objects	O
or	O
on	O
the	O
-folded	O
tensor	O
product	O
that	O
involves	O
some	O
"	O
twists	O
"	O
.	O
</s>
<s>
Thus	O
the	O
elements	O
and	O
exchange	O
places	O
and	O
,	O
in	O
addition	O
,	O
is	O
twisted	O
by	O
the	O
inner	B-Algorithm
automorphism	I-Algorithm
corresponding	O
to	O
—	O
this	O
ensures	O
that	O
the	O
product	O
of	O
the	O
components	O
of	O
remains	O
the	O
identity	O
element	O
.	O
</s>
<s>
It	O
may	O
be	O
checked	O
that	O
the	O
braid	B-Application
group	I-Application
relations	O
are	O
satisfied	O
and	O
this	O
formula	O
indeed	O
defines	O
a	O
group	O
action	O
of	O
on	O
.	O
</s>
<s>
As	O
another	O
example	O
,	O
a	O
braided	B-Algorithm
monoidal	I-Algorithm
category	I-Algorithm
is	O
a	O
monoidal	O
category	O
with	O
a	O
braid	B-Application
group	I-Application
action	O
.	O
</s>
<s>
Elements	O
of	O
the	O
braid	B-Application
group	I-Application
can	O
be	O
represented	O
more	O
concretely	O
by	O
matrices	O
.	O
</s>
<s>
One	O
classical	O
such	O
representation	O
is	O
Burau	B-Algorithm
representation	I-Algorithm
,	O
where	O
the	O
matrix	O
entries	O
are	O
single	O
variable	O
Laurent	O
polynomials	O
.	O
</s>
<s>
It	O
had	O
been	O
a	O
long-standing	O
question	O
whether	O
Burau	B-Algorithm
representation	I-Algorithm
was	O
faithful	O
,	O
but	O
the	O
answer	O
turned	O
out	O
to	O
be	O
negative	O
for	O
.	O
</s>
<s>
More	O
generally	O
,	O
it	O
was	O
a	O
major	O
open	O
problem	O
whether	O
braid	B-Application
groups	I-Application
were	O
linear	B-Algorithm
.	O
</s>
<s>
In	O
1996	O
,	O
Chetan	O
Nayak	O
and	O
Frank	O
Wilczek	O
posited	O
that	O
in	O
analogy	O
to	O
projective	O
representations	O
of	O
,	O
the	O
projective	O
representations	O
of	O
the	O
braid	B-Application
group	I-Application
have	O
a	O
physical	O
meaning	O
for	O
certain	O
quasiparticles	O
in	O
the	O
fractional	O
quantum	O
hall	O
effect	O
.	O
</s>
<s>
Around	O
2001	O
Stephen	O
Bigelow	O
and	O
Daan	O
Krammer	O
independently	O
proved	O
that	O
all	O
braid	B-Application
groups	I-Application
are	O
linear	B-Algorithm
.	O
</s>
<s>
Their	O
work	O
used	O
the	O
Lawrence	B-Algorithm
–	I-Algorithm
Krammer	I-Algorithm
representation	I-Algorithm
of	O
dimension	O
depending	O
on	O
the	O
variables	O
and	O
.	O
</s>
<s>
By	O
suitably	O
specializing	O
these	O
variables	O
,	O
the	O
braid	B-Application
group	I-Application
may	O
be	O
realized	O
as	O
a	O
subgroup	O
of	O
the	O
general	O
linear	B-Algorithm
group	I-Algorithm
over	O
the	O
complex	O
numbers	O
.	O
</s>
<s>
The	O
simplest	O
way	O
is	O
to	O
take	O
the	O
direct	O
limit	O
of	O
braid	B-Application
groups	I-Application
,	O
where	O
the	O
attaching	O
maps	O
send	O
the	O
generators	O
of	O
to	O
the	O
first	O
generators	O
of	O
(	O
i.e.	O
,	O
by	O
attaching	O
a	O
trivial	O
strand	O
)	O
.	O
</s>
<s>
The	O
second	O
group	O
can	O
be	O
thought	O
of	O
the	O
same	O
as	O
with	O
finite	O
braid	B-Application
groups	I-Application
.	O
</s>
<s>
Place	O
a	O
strand	O
at	O
each	O
of	O
the	O
points	O
and	O
the	O
set	O
of	O
all	O
braidswhere	O
a	O
braid	O
is	O
defined	O
to	O
be	O
a	O
collection	O
of	O
paths	O
from	O
the	O
points	O
to	O
the	O
points	O
so	O
that	O
the	O
function	O
yields	O
a	O
permutation	B-Algorithm
on	O
endpointsis	O
isomorphic	O
to	O
this	O
wilder	O
group	O
.	O
</s>
<s>
A	O
classifying	O
space	O
for	O
the	O
braid	B-Application
group	I-Application
is	O
the	O
th	O
unordered	O
configuration	O
space	O
of	O
,	O
that	O
is	O
,	O
the	O
set	O
of	O
distinct	O
unordered	O
points	O
in	O
the	O
plane	O
:	O
</s>
<s>
Similarly	O
,	O
a	O
classifying	O
space	O
for	O
the	O
pure	B-Application
braid	I-Application
group	O
is	O
,	O
the	O
th	O
ordered	O
configuration	O
space	O
of	O
.	O
</s>
