Knots, i.e. embeddings of spheres
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Analogously for $k>1$ one constructs a smooth embedding $t:S^{2k-1}\to\Rr^{3k}$. | Analogously 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}. | 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 this embedding is a generator of $E_D^{3k}(S^{2k-1})\cong\Zz_2$. | + | 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 $k=3$. | The last phrase of \cite{Haefliger1962t} suggests that this is true for $k=3$. | ||
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Revision as of 10:04, 26 April 2016
This page has not been refereed. The information given here might be incomplete or provisional. |
For notation and conventions throughout this page see high codimension embeddings.
Contents |
1 Examples
1.1 The Haefliger trefoil knot
Let us construct a smooth embedding (which is a generator of ) [Haefliger1962], 4.1. A miraculous property of this embedding is that it is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to the standard embedding.
Denote coordinates in by . The higher-dimensional trefoil knot is obtained by joining with two tubes the higher-dimensional Borromean rings, i.e. the three spheres given by the following three systems of equations:
See Figures 3.5 and 3.6 of [Skopenkov2006].
Analogously 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 .
2 Classification
(I would suggest including the classification of simple knots a la Kearton et. al. in this section.---John Klein)
3 References
- [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
- [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.