Alain Bruguieres
Double Braidings, Twists and Tangle Invariants
Preprint series: ESI preprints

MSC:

57M25 Knots and links in $S^3$, {For higher dimensions, See

18D10 Monoidal categories (= multiplicative categories), See also {19D23}

Abstract: A tortile (or ribbon) category defines invariants of ribbon (framed)
links and tangles. We observe that these invariants, when
restricted to links, string links, and more general tangles which
we call \emph{turbans}, do not actually depend on the braiding of
the tortile category. Besides duality, the only pertinent data for
such tangles are the double braiding and twist. We introduce the
general notions of twine, which is meant to play the rôle of the
double braiding (in the absence of a braiding), and the
corresponding notion of twist. We show that the category of
(ribbon) pure braids is the free category with a twine (a twist).
We show that a category with duals and a self-dual twist defines
invariants of stringlinks. We introduce the notion of
\emph{turban} category, so that the category of turban tangles is
the free turban category. Lastly we give a few examples and a
tannaka dictionary for twines and twists.
Keywords: braided categories, double braiding, twists, pure braids, string links, tangles