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 Izv. RAN. Ser. Mat., 2005, Volume 69, Issue 5, Pages 3–52 (Mi izv654)

First-order invariants and cohomology of spaces of embeddings of self-intersecting curves in $\mathbb R^n$

V. A. Vassilievab

a Steklov Mathematical Institute, Russian Academy of Sciences
b Independent University of Moscow

Abstract: We study the cohomology of the space of generic immersions $\mathbb R^1\to\mathbb R^n$, $n\geqslant3$, with a fixed set of transversal self-intersections. In particular, we study isotopy invariants of such immersions when $n=3$, calculate the lower cohomology groups of this space for $n>3$, and define and calculate the groups of first-order invariants of such immersions for $n=3$. We investigate the representability of these invariants by rational combinatorial formulae that generalize the classical formula for the linking number of two curves in $\mathbb R^3$. We prove the existence of such combinatorial formulae with half-integer coefficients and construct the topological obstruction to their integrality. As a corollary, it is proved that one of the basic 4th order knot invariants cannot be represented by an integral Polyak–Viro formula. The structure of the cohomology groups under investigation depends on the existence of a planar curve with a given self-intersection type. On the other hand, one can use the self-intersection type to construct automatically a chain complex calculating these cohomology groups. This gives a simple homological criterion for the existence of such a planar curve.

DOI: https://doi.org/10.4213/im654

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English version:
Izvestiya: Mathematics, 2005, 69:5, 865–912

Bibliographic databases:

UDC: 515.16
MSC: 55R80, 57M25

Citation: V. A. Vassiliev, “First-order invariants and cohomology of spaces of embeddings of self-intersecting curves in $\mathbb R^n$”, Izv. RAN. Ser. Mat., 69:5 (2005), 3–52; Izv. Math., 69:5 (2005), 865–912

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv654
• https://doi.org/10.4213/im654
• http://mi.mathnet.ru/eng/izv/v69/i5/p3

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This publication is cited in the following articles:
1. Proc. Steklov Inst. Math., 266 (2009), 142–176
2. D. P. Ilyutko, V. O. Manturov, I. M. Nikonov, “Parity in knot theory and graph-links”, Journal of Mathematical Sciences, 193:6 (2013), 809–965
3. V.O.legovich Manturov, “Framed 4-valent graph minor theory I Introduction: A planarity criterion and linkless embeddability”, J. Knot Theory Ramifications, 2014, 1460002
4. Igor Nikonov, “A new proof of Vassiliev's conjecture”, J. Knot Theory Ramifications, 2014, 1460005
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