Knots, i.e. embeddings of spheres

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

Contents

[edit] 1 Introduction

See general introduction on embeddings, notation and conventions in [Skopenkov2016c, \S1, \S3].

[edit] 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.

[edit] 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].

[edit] 4 Codimension 2 knots

For the best known specific case, i.e. for codimension 2 embeddings of spheres (in particular, for the classical theory of knots in \Rr^3), a complete readily calculable classification (in the sense of Remark 1.1 of [Skopenkov2016c]) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots, most of which does not present a readily calculable classification. See e.g. interesting papers [Farber1981], [Farber1983], [Kearton1983], [Farber1984].

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

[edit] 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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