<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>
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
explain	O
how	O
to	O
reduce	O
a	O
braid	O
group	O
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>
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-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	O
groups	O
called	O
Artin	B-Algorithm
groups	I-Algorithm
.	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>
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	O
group	O
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	O
group	O
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	O
group	O
into	O
the	O
Coxeter	B-Algorithm
presentation	I-Algorithm
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
:	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>
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>
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
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>
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
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	O
group	O
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>
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	O
group	O
action	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	O
groups	O
were	O
linear	B-Algorithm
.	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	O
groups	O
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	O
group	O
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>
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>
