Calculus of the first non-trivial 1-cocycle of the space of long knots
Abstract
For the space of long knots in R3, Vassiliev's theory defines the so called finite order cocycles. Zero degree cocycles are finite type knot invariants. The first non-trivial cocycle of positive dimension in the space of long knots has dimension one and order three. We apply Vassiliev's combinatorial formula, and find the value mod 2 of this cocycle on the 1-cycles that are obtained by dragging knots one along the other or by rotating around a fixed line.
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