Calculating the First Nontrivial 1-Cocycle in the Space of Long Knots

For spaces of knots in $\mathbb{R}^3$, the Vassiliev theory defines the so-called cocycles of finite order. The zero-dimensional cocycles are the finite order invariants. The first nontrivial cocycle of positive dimension in the space of long knots is one-dimensional and is of order 3. We apply the combinatorial formula given by Vassiliev in his paper [1] and find the value $\bmod\, 2$ of this cocycle on 1-cycles obtained by dragging knots one through another or by rotating a knot around a given line.