<s>
The	O
loop	B-Algorithm
braid	I-Algorithm
group	I-Algorithm
is	O
a	O
mathematical	O
group	O
structure	O
that	O
is	O
used	O
in	O
some	O
models	O
of	O
theoretical	O
physics	O
to	O
model	O
the	O
exchange	O
of	O
particles	O
with	O
loop-like	O
topologies	O
within	O
three	O
dimensions	O
of	O
space	O
and	O
time	O
.	O
</s>
<s>
The	O
basic	O
operations	O
which	O
generate	O
a	O
loop	B-Algorithm
braid	I-Algorithm
group	I-Algorithm
for	O
n	O
loops	O
are	O
exchanges	O
of	O
two	O
adjacent	O
loops	O
,	O
and	O
passing	O
one	O
adjacent	O
loop	O
through	O
another	O
.	O
</s>
<s>
To	O
be	O
precise	O
,	O
the	O
loop	B-Algorithm
braid	I-Algorithm
group	I-Algorithm
on	O
n	O
loops	O
is	O
defined	O
as	O
the	O
motion	O
group	O
of	O
n	O
disjoint	O
circles	O
embedded	O
in	O
a	O
compact	O
three-dimensional	O
"	O
box	O
"	O
diffeomorphic	O
to	O
the	O
three-dimensional	O
disk	O
.	O
</s>
<s>
One	O
may	O
also	O
show	O
that	O
the	O
loop	B-Algorithm
braid	I-Algorithm
group	I-Algorithm
is	O
isomorphic	O
to	O
the	O
welded	O
braid	B-Application
group	I-Application
,	O
as	O
is	O
done	O
for	O
example	O
in	O
a	O
paper	O
by	O
John	O
C	O
.	O
Baez	O
,	O
Derek	O
Wise	O
,	O
and	O
Alissa	O
Crans	O
,	O
which	O
also	O
gives	O
some	O
presentations	O
of	O
the	O
loop	B-Algorithm
braid	I-Algorithm
group	I-Algorithm
using	O
the	O
work	O
of	O
Xiao-Song	O
Lin	O
.	O
</s>
