Geodesic
Calculating the First Nontrivial 1-Cocycle
Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 105-114

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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
Mots-clés : Casson's invariant. 
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     author = {V. \'E. Turchin},
     title = {Calculating the {First} {Nontrivial} {1-Cocycle}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {105--114},
     publisher = {mathdoc},
     volume = {80},
     number = {1},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a12/}
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V. É. Turchin. Calculating the First Nontrivial 1-Cocycle. Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 105-114. http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a12/