Knots, i.e. embeddings of spheres

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A description of $E^3(S^1)$ and, more generally, of $E^{n+2}(S^n)$ is a well-known very hard open problem.
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For the best known specific case, i.e. for codimension 2 embeddings (in particular, for the classical theory of knots in $\Rr^3$), a complete readily calculable classification is neither known nor expected at the time of writing.
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On the other hand, if one studies embeddings up to the weaker relation of ''concordance'', then much is known. See e.g. \cite{Levine1969a}, \cite{Cappell&Shaneson1974} and \cite{Ranicki1998}.
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On the other hand, if one studies embeddings up to the weaker relation of ''concordance'', then much is
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known. See e.g. \cite{Levine1969a}, \cite{Cappell&Shaneson1974} and \cite{Ranicki1998}.
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There is a vast literature on codimension 2 knots, most of which does not present a readily calculable classification (in the sense of
There is a vast literature on codimension 2 knots, most of which does not present a readily calculable classification (in the sense of
[[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|Remark 1.1]] of \cite{Skopenkov2016c}).
[[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|Remark 1.1]] of \cite{Skopenkov2016c}).

Revision as of 12:37, 1 September 2017

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 Readily calculable classification

For m\ge n+3 the group E^m(S^n) has been described in terms of exact sequences involving the homotopy groups of spheres and the homotopy groups of the pairs (SG_n,SO_n) for n = n_i+1 and n_i [Haefliger1966a], cf. [Levine1965], [Milgram1972], [Habegger1986]. Here SG_n is the space of maps f \colon S^{n-1} \to S^{n-1} of degree 1. Restricting an element of SO_n to S^{n-1} \subset \Rr^n identifies SO_n as a subspace of G_n.

Some readily calculable corollaries of this classification are recalled in [Skopenkov2006, \S3.3].

4 Codimension 2 knots

For the best known specific case, i.e. for codimension 2 embeddings (in particular, for the classical theory of knots in \Rr^3), a complete readily calculable classification is neither known nor expected at the time of writing.

On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a], [Cappell&Shaneson1974] and [Ranicki1998].

There is a vast literature on codimension 2 knots, most of which does not present a readily calculable classification (in the sense of Remark 1.1 of [Skopenkov2016c]). See e.g. interesting papers [Farber1981], [Farber1983], [Kearton1983], [Farber1984].

5 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 $k=3$. == Readily calculable classification == ; For $m\ge n+3$ the group $E^m(S^n)$ has been described in terms of exact sequences involving the homotopy groups of spheres and the homotopy groups of the pairs $(SG_n,SO_n)$ for $n = n_i+1$ and $n_i$ \cite{Haefliger1966a}, cf. \cite{Levine1965}, \cite{Milgram1972}, \cite{Habegger1986}. Here $SG_n$ is the space of maps $f \colon S^{n-1} \to S^{n-1}$ of degree \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 Readily calculable classification

For m\ge n+3 the group E^m(S^n) has been described in terms of exact sequences involving the homotopy groups of spheres and the homotopy groups of the pairs (SG_n,SO_n) for n = n_i+1 and n_i [Haefliger1966a], cf. [Levine1965], [Milgram1972], [Habegger1986]. Here SG_n is the space of maps f \colon S^{n-1} \to S^{n-1} of degree 1. Restricting an element of SO_n to S^{n-1} \subset \Rr^n identifies SO_n as a subspace of G_n.

Some readily calculable corollaries of this classification are recalled in [Skopenkov2006, \S3.3].

4 Codimension 2 knots

For the best known specific case, i.e. for codimension 2 embeddings (in particular, for the classical theory of knots in \Rr^3), a complete readily calculable classification is neither known nor expected at the time of writing.

On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a], [Cappell&Shaneson1974] and [Ranicki1998].

There is a vast literature on codimension 2 knots, most of which does not present a readily calculable classification (in the sense of Remark 1.1 of [Skopenkov2016c]). See e.g. interesting papers [Farber1981], [Farber1983], [Kearton1983], [Farber1984].

5 References

$. Restricting an element of $SO_n$ to $S^{n-1} \subset \Rr^n$ identifies $SO_n$ as a subspace of $G_n$. Some readily calculable corollaries of this classification are recalled in \cite[$\S.3]{Skopenkov2006}. == Codimension 2 knots == ; A description of $E^3(S^1)$ and, more generally, of $E^{n+2}(S^n)$ is a well-known very hard open problem. On the other hand, if one studies embeddings up to the weaker relation of ''concordance'', then much is known. See e.g. \cite{Levine1969a}, \cite{Cappell&Shaneson1974} and \cite{Ranicki1998}. There is a vast literature on codimension 2 knots, most of which does not present a readily calculable classification (in the sense of [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|Remark 1.1]] of \cite{Skopenkov2016c}). See e.g. interesting papers \cite{Farber1981}, \cite{Farber1983}, \cite{Kearton1983}, \cite{Farber1984}. == 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 Readily calculable classification

For m\ge n+3 the group E^m(S^n) has been described in terms of exact sequences involving the homotopy groups of spheres and the homotopy groups of the pairs (SG_n,SO_n) for n = n_i+1 and n_i [Haefliger1966a], cf. [Levine1965], [Milgram1972], [Habegger1986]. Here SG_n is the space of maps f \colon S^{n-1} \to S^{n-1} of degree 1. Restricting an element of SO_n to S^{n-1} \subset \Rr^n identifies SO_n as a subspace of G_n.

Some readily calculable corollaries of this classification are recalled in [Skopenkov2006, \S3.3].

4 Codimension 2 knots

For the best known specific case, i.e. for codimension 2 embeddings (in particular, for the classical theory of knots in \Rr^3), a complete readily calculable classification is neither known nor expected at the time of writing.

On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a], [Cappell&Shaneson1974] and [Ranicki1998].

There is a vast literature on codimension 2 knots, most of which does not present a readily calculable classification (in the sense of Remark 1.1 of [Skopenkov2016c]). See e.g. interesting papers [Farber1981], [Farber1983], [Kearton1983], [Farber1984].

5 References

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