<s>
In	O
mathematics	O
the	O
Burau	B-Algorithm
representation	I-Algorithm
is	O
a	O
representation	O
of	O
the	O
braid	B-Application
groups	I-Application
,	O
named	O
after	O
and	O
originally	O
studied	O
by	O
the	O
German	O
mathematician	O
Werner	O
Burau	O
during	O
the	O
1930s	O
.	O
</s>
<s>
The	O
Burau	B-Algorithm
representation	I-Algorithm
has	O
two	O
common	O
and	O
near-equivalent	O
formulations	O
,	O
the	O
reduced	O
and	O
unreduced	O
Burau	B-Algorithm
representations	I-Algorithm
.	O
</s>
<s>
Consider	O
the	O
braid	B-Application
group	I-Application
to	O
be	O
the	O
mapping	O
class	O
group	O
of	O
a	O
disc	O
with	O
marked	O
points	O
.	O
</s>
<s>
By	O
the	O
basic	O
theory	O
of	O
covering	O
spaces	O
,	O
acts	O
on	O
,	O
and	O
this	O
representation	O
is	O
called	O
the	O
reduced	O
Burau	B-Algorithm
representation	I-Algorithm
.	O
</s>
<s>
The	O
unreduced	O
Burau	B-Algorithm
representation	I-Algorithm
has	O
a	O
similar	O
definition	O
,	O
namely	O
one	O
replaces	O
with	O
its	O
(	O
real	O
,	O
oriented	O
)	O
blow-up	O
at	O
the	O
marked	O
points	O
.	O
</s>
<s>
Let	O
denote	O
the	O
standard	O
generators	O
of	O
the	O
braid	B-Application
group	I-Application
.	O
</s>
<s>
Vaughan	O
Jones	O
gave	O
the	O
following	O
interpretation	O
of	O
the	O
unreduced	O
Burau	B-Algorithm
representation	I-Algorithm
of	O
positive	O
braids	O
for	O
in	O
i.e.	O
</s>
<s>
for	O
braids	O
that	O
are	O
words	O
in	O
the	O
standard	O
braid	B-Application
group	I-Application
generators	O
containing	O
no	O
inverses	O
which	O
follows	O
immediately	O
from	O
the	O
above	O
explicit	O
description	O
:	O
</s>
<s>
Then	O
the	O
'	O
th	O
entry	O
of	O
the	O
unreduced	O
Burau	B-Algorithm
representation	I-Algorithm
of	O
is	O
the	O
probability	O
that	O
a	O
ball	O
thrown	O
into	O
the	O
'	O
th	O
lane	O
ends	O
up	O
in	O
the	O
'	O
th	O
lane	O
.	O
</s>
<s>
where	O
is	O
the	O
reduced	O
Burau	B-Algorithm
representation	I-Algorithm
of	O
the	O
braid	O
.	O
</s>
<s>
The	O
first	O
nonfaithful	O
Burau	B-Algorithm
representations	I-Algorithm
were	O
found	O
by	O
John	O
A	O
.	O
Moody	O
without	O
the	O
use	O
of	O
computer	O
,	O
using	O
a	O
notion	O
of	O
winding	O
number	O
or	O
contour	O
integration	O
.	O
</s>
<s>
Stephen	O
Bigelow	O
combined	O
computer	O
techniques	O
and	O
the	O
Long	O
–	O
Paton	O
theorem	O
to	O
show	O
that	O
the	O
Burau	B-Algorithm
representation	I-Algorithm
is	O
not	O
faithful	O
for	O
.	O
</s>
<s>
The	O
Burau	B-Algorithm
representation	I-Algorithm
for	O
has	O
been	O
known	O
to	O
be	O
faithful	O
for	O
some	O
time	O
.	O
</s>
<s>
The	O
faithfulness	O
of	O
the	O
Burau	B-Algorithm
representation	I-Algorithm
when	O
is	O
an	O
open	O
problem	O
.	O
</s>
<s>
The	O
Burau	B-Algorithm
representation	I-Algorithm
appears	O
as	O
a	O
summand	O
of	O
the	O
Jones	O
representation	O
,	O
and	O
for	O
,	O
the	O
faithfulness	O
of	O
the	O
Burau	B-Algorithm
representation	I-Algorithm
is	O
equivalent	O
to	O
that	O
of	O
the	O
Jones	O
representation	O
,	O
which	O
on	O
the	O
other	O
hand	O
is	O
related	O
to	O
the	O
question	O
of	O
whether	O
or	O
not	O
the	O
Jones	O
polynomial	O
is	O
an	O
unknot	O
detector	O
.	O
</s>
<s>
Craig	O
Squier	O
showed	O
that	O
the	O
Burau	B-Algorithm
representation	I-Algorithm
preserves	O
a	O
sesquilinear	B-Algorithm
form	I-Algorithm
.	O
</s>
<s>
Thus	O
the	O
Burau	B-Algorithm
representation	I-Algorithm
of	O
the	O
braid	B-Application
group	I-Application
can	O
be	O
thought	O
of	O
as	O
a	O
map	O
into	O
the	O
unitary	O
group	O
U(n )	O
.	O
</s>
