Knots, i.e. embeddings of spheres

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There is a vast literature on the most well-known case of codimension 2 knots, most of which does not present a readily calculable classification (in the sense of
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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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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}).
See e.g. interesting papers \cite{Farber1981}, \cite{Farber1983}, \cite{Kearton1983}, \cite{Farber1984}.
See e.g. interesting papers \cite{Farber1981}, \cite{Farber1983}, \cite{Kearton1983}, \cite{Farber1984}.

Revision as of 12:07, 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

A description of E^3(S^1) and, more generally, of E^{n+2}(S^n) is a well-known very hard open problem.

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

  • [Farber1981] M. Sh. Farber, Classification of stable fibered knots, Mat. Sb. (N.S.), 115(157):2(6) (1981) 223–262.
  • [Farber1983] M. Sh. Farber, The classification of simple knots, Russian Math. Surveys, 38:5 (1983).

, $\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

A description of E^3(S^1) and, more generally, of E^{n+2}(S^n) is a well-known very hard open problem.

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

  • [Farber1981] M. Sh. Farber, Classification of stable fibered knots, Mat. Sb. (N.S.), 115(157):2(6) (1981) 223–262.
  • [Farber1983] M. Sh. Farber, The classification of simple knots, Russian Math. Surveys, 38:5 (1983).

$. 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 == ; There is a vast literature on the most well-known case of 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

A description of E^3(S^1) and, more generally, of E^{n+2}(S^n) is a well-known very hard open problem.

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

  • [Farber1981] M. Sh. Farber, Classification of stable fibered knots, Mat. Sb. (N.S.), 115(157):2(6) (1981) 223–262.
  • [Farber1983] M. Sh. Farber, The classification of simple knots, Russian Math. Surveys, 38:5 (1983).

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