The decomposition of linking number of the edges of a ribbon into the sum of twist and writhe is well known in the applied knot theory of closed polymer chains and DNA. A straightforward proof follows from each projection of the ribbon link: the crossings of the resulting link diagram are naturally partitioned into "ribbon crossings" (equalling 2 $\times$ planar writhe of an edge) and ribbon twists; averaging these over all projection directions gives the familiar writhe and twist respectively. This can be further visualized as a certain decomposition of the secant manifold of the curve, Gauss mapped to the direction sphere, closed with a framing manifold equivalent to the ribbon framing. Vassiliev invariants are knot (and link) invariants which can be considered as $n$-point generalizations of the linking number with certain combinatorial constraints. Indeed, the first Vassiliev invariant for links is the linking number. In this talk, I will discuss work in progress in understanding the standard '$X+Y$' decomposition of the second-order Vassiliev invariant as a sum of a certain 2-point generalized writhe and twist, and interpret this in terms of the knot’s secant manifold under the Gauss map.

This is joint work with Alexander Taylor. This research was supported by a Leverhulme Research Programme Grant: RP2013-K-009, "SPOCK: Scientific Properties of Complex Knots".