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 Mat. Zametki, 2006, Volume 80, Issue 1, Pages 105–114 (Mi mz2785)

This article is cited in 3 scientific papers (total in 3 papers)

Calculating the First Nontrivial 1-Cocycle in the Space of Long Knots

V. É. Turchinab

a Independent University of Moscow
b Université catholique de Louvain

Abstract: 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, Casson's invariant

DOI: https://doi.org/10.4213/mzm2785

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English version:
Mathematical Notes, 2006, 80:1, 101–108

Bibliographic databases:

UDC: 515.164
Received: 09.09.2004

Citation: V. É. Turchin, “Calculating the First Nontrivial 1-Cocycle in the Space of Long Knots”, Mat. Zametki, 80:1 (2006), 105–114; Math. Notes, 80:1 (2006), 101–108

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Sakai K., “An integral expression of the first nontrivial one-cocycle of the space of long knots in $\mathbb R^3$”, Pacific J. Math., 250:2 (2011), 407–419
2. Mortier A., “Finite-Type 1-Cocycles of Knots Given By Polyak-Viro Formulas”, J. Knot Theory Ramifications, 24:10 (2015), 1540004
3. Mortier A., “Combinatorial cohomology of the space of long knots”, Algebr. Geom. Topol., 15:6 (2015), 3435–3465
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