A BRACKET POLYNOMIAL FOR GRAPHS, II: LINKS, EULER CIRCUITS AND MARKED GRAPHS

Abstract

Let D be an oriented classical or virtual link diagram with directed universe [Formula: see text]. Let C denote a set of directed Euler circuits, one in each connected component of U. There is then an associated looped interlacement graph [Formula: see text] whose construction involves very little geometric information about the way D is drawn in the plane; consequently [Formula: see text] is different from other combinatorial structures associated with classical link diagrams, like the checkerboard graph, which can be difficult to extend to arbitrary virtual links. [Formula: see text] is determined by three things: the structure of [Formula: see text] as a 2-in, 2-out digraph, the distinction between crossings that make a positive contribution to the writhe and those that make a negative contribution, and the relationship between C and the directed circuits in [Formula: see text] arising from the link components; this relationship is indicated by marking the vertices where C does not follow the incident link component(s). We introduce a bracket polynomial for arbitrary marked graphs, defined using either a formula involving matrix nullities or a recursion involving the local complement and pivot operations; the marked-graph bracket of [Formula: see text] is the same as the Kauffman bracket of D. This provides a unified combinatorial description of the Jones polynomial that applies seamlessly to both classical and non-classical virtual links.