Hopf algebras and regular isotopy invariants for link diagrams
Mark A. Hennings · Mathematical Proceedings of the Cambridge Philosophical Society · 1991
In this paper, we shall consider the following method for obtaining regular isotopy invariants of link diagrams. Given any link diagramL, equip it with a Morse functionh, so that the diagram consists entirely of crossings, maxima, minima and vertical arcs. Introduce 2-valent graphical vertices to separate the various segments of the diagram. Given a finite index setI, a state σ forLhis an assignation of one element ofIto each graphical vertex. Each segment of the diagram now has a weight associated with it, given in terms of tensor coordinates indexed by the setIby the pictures and, for any state σ, [Li|σ] denotes the product of the various weights. We then define 〈Lh〉 to be the sum of [Lh|σ] over all possible states σ,