Knots, i.e. embeddings of spheres
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[edit] 1 Introduction
See general introduction on embeddings, notation and conventions in [Skopenkov2016c, 1, 2].
[edit] 2 Examples
Analogously to the Haefliger trefoil knot for one constructs a smooth embedding . For even this embedding is a generator of ; 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 odd this embedding is a generator of . The last phrase of [Haefliger1962t] suggests that this is true for .
[edit] 3 Classification
For the group has been described in terms of exact sequences involving the homotopy groups of spheres and the homotopy groups of the pairs for and [Haefliger1966a], cf. [Levine1965], [Habegger1986]. Here is the space of maps of degree . Restricting an element of to identifies as a subspace of .
For some readily calculable results see [Skopenkov2006, 3.3].
(I would suggest including the classification of simple knots a la Kearton et. al. in this section.---John Klein)
[edit] 4 References
- [Habegger1986] N. Habegger, Knots and links in codimension greater than 2, Topology, 25:3 (1986) 253--260.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
- [Levine1965] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR0180981 (31 #5211) Zbl 0136.21102
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248--342. Available at the arXiv:0604045.
- [Skopenkov2016c] A. Skopenkov, High codimension embeddings: classification, submitted to Bull. Man. Atl..