Calculating the first nontrivial 1-cocycle in the space of long knots

Turchin, Victor
(2006) Mathematical notes / Academy of Sciences of the U S S R — Vol. 80, n° 1-2, p. 101-108 (2006)

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  • Turchin, Victor
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Abstract
For spaces of knots in R-3, the Vassiliev theory defines the so-called cocycles of finite order. The zero-dimensional cocycles are 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 mod 2 of this cocycle on 1-cycles obtained by dragging knots one through another or by rotating a knot around a given line.
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Turchin, V. (2006). Calculating the first nontrivial 1-cocycle in the space of long knots. Mathematical notes / Academy of Sciences of the U S S R, 80(1-2), 101-108. https://doi.org/10.1007/s11006-006-0113-8 (Original work published 2006)