  In this manuscript we define Vassiliev measures of complexity for open curves
in 3-space. These are related to the coefficients of the enhanced Jones
polynomial of open curves in 3-space. These Vassiliev measures are continuous
functions of the curve coordinates and as the ends of the curve tend to
coincide, they converge to the corresponding Vassiliev invariants of the
resulting knot. We focus on the second Vassiliev measure from the enhanced
Jones polynomial for closed and open curves in 3-space. For closed curves, this
second Vassiliev measure can be computed by a Gauss code diagram and it has an
integral formulation, the double alternating self-linking integral. The double
alternating self-linking integral is a topological invariant for closed curves
and a continuous function of the curve coordinates for open curves in 3-space.
For polygonal curves, the double alternating self-linking integral obtains a
simpler expression in terms of geometric probabilities. For a polygonal curve
with 4 edges, the double alternating self-linking integral coincides with the
signed geometric probability of obtaining the knotoid k2.1 in a random
projection direction.

Geometric Topology;General Topology