<s>
A	O
braided	B-Algorithm
monoidal	I-Algorithm
category	I-Algorithm
is	O
a	O
monoidal	O
category	O
equipped	O
with	O
a	O
braiding	O
—	O
that	O
is	O
,	O
a	O
commutativity	O
constraint	O
that	O
satisfies	O
axioms	O
including	O
the	O
hexagon	O
identities	O
defined	O
below	O
.	O
</s>
<s>
The	O
term	O
braided	O
references	O
the	O
fact	O
that	O
the	O
braid	B-Application
group	I-Application
plays	O
an	O
important	O
role	O
in	O
the	O
theory	O
of	O
braided	B-Algorithm
monoidal	I-Algorithm
categories	I-Algorithm
.	O
</s>
<s>
Partly	O
for	O
this	O
reason	O
,	O
braided	B-Algorithm
monoidal	I-Algorithm
categories	I-Algorithm
and	O
other	O
topics	O
are	O
related	O
in	O
the	O
theory	O
of	O
knot	O
invariants	O
.	O
</s>
<s>
Alternatively	O
,	O
a	O
braided	B-Algorithm
monoidal	I-Algorithm
category	I-Algorithm
can	O
be	O
seen	O
as	O
a	O
tricategory	O
with	O
one	O
0-cell	O
and	O
one	O
1-cell	O
.	O
</s>
<s>
Braided	B-Algorithm
monoidal	I-Algorithm
categories	I-Algorithm
were	O
introduced	O
by	O
André	O
Joyal	O
and	O
Ross	O
Street	O
in	O
a	O
1986	O
preprint	O
.	O
</s>
<s>
For	O
along	O
with	O
the	O
commutativity	O
constraint	O
to	O
be	O
called	O
a	O
braided	B-Algorithm
monoidal	I-Algorithm
category	I-Algorithm
,	O
the	O
following	O
hexagonal	O
diagrams	O
must	O
commute	O
for	O
all	O
objects	O
.	O
</s>
<s>
The	O
action	O
of	O
on	O
an	O
-fold	O
tensor	O
product	O
factors	O
through	O
the	O
braid	B-Application
group	I-Application
.	O
</s>
<s>
There	O
are	O
several	O
variants	O
of	O
braided	B-Algorithm
monoidal	I-Algorithm
categories	I-Algorithm
that	O
are	O
used	O
in	O
various	O
contexts	O
.	O
</s>
<s>
A	O
braided	B-Algorithm
monoidal	I-Algorithm
category	I-Algorithm
is	O
called	O
symmetric	O
if	O
also	O
satisfies	O
for	O
all	O
pairs	O
of	O
objects	O
and	O
.	O
</s>
<s>
In	O
this	O
case	O
the	O
action	O
of	O
on	O
an	O
-fold	O
tensor	O
product	O
factors	O
through	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
A	O
braided	B-Algorithm
monoidal	I-Algorithm
category	I-Algorithm
is	O
a	O
ribbon	O
category	O
if	O
it	O
is	O
rigid	O
,	O
and	O
it	O
may	O
preserve	O
quantum	O
trace	O
and	O
co-quantum	O
trace	O
.	O
</s>
<s>
The	O
first	O
property	O
shows	O
us	O
that	O
,	O
thus	O
allowing	O
us	O
to	O
omit	O
the	O
analog	O
to	O
the	O
second	O
defining	O
diagram	O
of	O
a	O
braided	B-Algorithm
monoidal	I-Algorithm
category	I-Algorithm
and	O
ignore	O
the	O
associator	O
maps	O
as	O
implied	O
.	O
</s>
<s>
The	O
category	O
of	O
representations	O
of	O
a	O
quantized	O
universal	O
enveloping	O
algebra	O
is	O
a	O
braided	B-Algorithm
monoidal	I-Algorithm
category	I-Algorithm
,	O
where	O
is	O
constructed	O
using	O
the	O
universal	O
R-matrix	O
.	O
</s>
