Ribbon Hopf algebra

Summary

A ribbon Hopf algebra is a quasitriangular Hopf algebra with a central invertible element, the ribbon element, that behaves like a full twist. Its finite-dimensional modules form a ribbon category: a braided monoidal category with duals and compatible twists. In such a category every framed knot or link diagram coloured by modules gives a well-defined morphism. So each ribbon Hopf algebra gives invariants of knots and links. Suitable versions of the quantum enveloping algebras are ribbon, and the Jones polynomial arises this way from $U_q(\mathfrak{sl}_2)$.