Double braidings, twists and tangle invariants

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 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 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.