Abstract:
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.
Keywords:long knot, Vassiliev invariant, finite order cocycle, Casson's invariant.
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