Calculus of the first non-trivial 1-cocycle of the space of long knots

Details Calculus of the first non-trivial 1-cocycle of the space of long knots Calculus of the first non-trivial 1-cocycle of the space of long knots

2005-02-24

arxiv.org/abs/math/0502518v2

Mathematics

Algebraic Topology

15 pages, 2 figures In the second version, only the transcription of author's name changed: turchin->tourtchine

Scientific paper

For the space of long knots in R^3, 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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