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

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Turchin, Victor

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.

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès restreint
Publication date 2006
Language Anglais
Journal information "Mathematical notes / Academy of Sciences of the U S S R" - Vol. 80, no. 1-2, p. 101-108 (2006)
Peer reviewed yes
Publisher Consultants Bureau/springer (New York)
issn 0001-4346
Publication status Publié
Affiliation UCL
Keywords long knot ; Vassiliev invariant ; finite order cocycle ; Casson's invariant
Links
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Bibliographic reference Turchin, Victor. Calculating the first nontrivial 1-cocycle in the space of long knots. In: Mathematical notes / Academy of Sciences of the U S S R, Vol. 80, no. 1-2, p. 101-108 (2006)
Permanent URL http://hdl.handle.net/2078.1/38076