Narrow Results

Let k: S¹ → S² be a generic immersion with n double points. We present an algorithm that assigns to k a partitioned n × n matrix over Z/2Z, and show that k gives rise to an orthogonal decomposition of (Z/2Z)ⁿ. We discuss a connection between this decomposition and the trip matrix of an alternating knot diagram produced from k.

A knot diagram has an associated looped interlacement graph, obtained from the intersection graph of the Gauss diagram by attaching loops to the vertices that correspond to negative crossings. This construction suggests an extension of the Kauffman bracket to an invariant of looped graphs, and an extension of Reidemeister equivalence to an equivalence relation on looped graphs. The graph bracket polynomial can be defined recursively using the same pivot and local complementation operations used to define the interlace polynomial, and it gives rise to a graph Jones polynomial V_G(t) that is invariant under the graph Reidemeister moves.

Let D be an oriented classical or virtual link diagram with directed universe . Let C denote a set of directed Euler circuits, one in each connected component of U. There is then an associated looped interlacement graph whose construction involves very little geometric information about the way D is drawn in the plane; consequently 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. is determined by three things: the structure of 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 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 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.

In this note we show that the rank of the trip matrix of a positive knot diagram is exactly twice the genus of the associated positive knot. From this, we give a quick proof of the following result of Murasugi: The term of lowest degree in the Jones polynomial of a positive knot is 1 · t^g, where g is the genus of the knot.