Knots, i.e. embeddings of spheres

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== Introduction ==
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See [[High_codimension_embeddings#Introduction|general introduction on embeddings]], [[High_codimension_embeddings#Notation and conventions|notation and conventions]] in \cite[$\S$1, $\S$2]{Skopenkov2016c}.
See [[High_codimension_embeddings#Introduction|general introduction on embeddings]], [[High_codimension_embeddings#Notation and conventions|notation and conventions]] in \cite[$\S$1, $\S$2]{Skopenkov2016c}.
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== Examples ==
== Examples ==
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== Classification ==
== Classification ==
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For some information see \cite[$\S$3.3]{Skopenkov2006}.
For some information see \cite[$\S$3.3]{Skopenkov2006}.
(I would suggest including the classification of simple knots a la Kearton et. al. in this section.---John Klein)
(I would suggest including the classification of simple knots a la Kearton et. al. in this section.---John Klein)
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== References ==
== References ==

Revision as of 12:46, 26 October 2016

This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Introduction

See general introduction on embeddings, notation and conventions in [Skopenkov2016c, \S1, \S2].

2 Examples

Analogously to the Haefliger trefoil knot for k>1 one constructs a smooth embedding t:S^{2k-1}\to\Rr^{3k}. For k even this embedding is a generator of E_D^{3k}(S^{2k-1})\cong\Zz; it is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to it [Haefliger1962]. It would be interesting to know if for k odd this embedding is a generator of E_D^{3k}(S^{2k-1})\cong\Zz_2. The last phrase of [Haefliger1962t] suggests that this is true for k=3.

3 Classification

For some information see [Skopenkov2006, \S3.3].

(I would suggest including the classification of simple knots a la Kearton et. al. in this section.---John Klein)

4 References

, $\S]{Skopenkov2016c}. == Examples == ; Analogously to [[3-manifolds_in_6-space#Examples|the Haefliger trefoil knot]] for $k>1$ one constructs a smooth embedding $t:S^{2k-1}\to\Rr^{3k}$. For $k$ even this embedding is a generator of $E_D^{3k}(S^{2k-1})\cong\Zz$; it is not ''smoothly'' isotopic to the standard embedding, but is ''piecewise smoothly'' isotopic to it \cite{Haefliger1962}. It would be interesting to know if for $k$ odd this embedding is a generator of $E_D^{3k}(S^{2k-1})\cong\Zz_2$. The last phrase of \cite{Haefliger1962t} suggests that this is true for $k=3$. == Classification == For some information see \cite[$\S.3]{Skopenkov2006}. (I would suggest including the classification of simple knots a la Kearton et. al. in this section.---John Klein) == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S1, \S2].

2 Examples

Analogously to the Haefliger trefoil knot for k>1 one constructs a smooth embedding t:S^{2k-1}\to\Rr^{3k}. For k even this embedding is a generator of E_D^{3k}(S^{2k-1})\cong\Zz; it is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to it [Haefliger1962]. It would be interesting to know if for k odd this embedding is a generator of E_D^{3k}(S^{2k-1})\cong\Zz_2. The last phrase of [Haefliger1962t] suggests that this is true for k=3.

3 Classification

For some information see [Skopenkov2006, \S3.3].

(I would suggest including the classification of simple knots a la Kearton et. al. in this section.---John Klein)

4 References

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