Knots, i.e. embeddings of spheres

Revision as of 12:21, 14 October 2016

For notation and conventions throughout this page see high codimension embeddings.

1 Examples

1.1 The Haefliger trefoil knot

Analogously to the Haefliger trefoil knot for ${{Stub}} For notation and conventions throughout this page see [[High_codimension_embeddings|high codimension embeddings]]. == Examples == === The Haefliger trefoil knot === <wikitex>; Analogously to [[3-manifolds_in_6-space#The_Haefliger_trefoil_knot|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 $k=3$. </wikitex> == Classification == (I would suggest including the classification of simple knots a la Kearton et. al. in this section.---John Klein) == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]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 $k=3$.

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 $(4k-1)$-spheres in $6k$-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