%% somewhere in here, mention the construction of the D4 subfactor as a crossed product.

If you've spent some time with people who enjoy their category theory,
you'll have picked up the idea that ``links invariants come from braided
tensor categories''. While this isn't always the case --- there are
plenty of link invariants which don't seem to arise in this way --- there
are many pretty examples. In particular, all of the link invariants
defined using quantum groups, via $R$-matrices, fit into this pattern:
the category of representations of a quantum group $U_q(\mathfrak{g})$ is
a braided tensor category. Sadly, the invariant we've given a recipe for
above doesn't quite fit the mould. On the other hand, happily, it doesn't
fall too far outside it. Below, we'll describe quite explicitly something
not-too-far from a braided tensor category, which is still sufficient to
produce a link invariant, and explain how we can sometime produce
examples from a subfactor, and in particular from the $D_4$ subfactor.

\input{text/bioidal}
\input{text/translation}
