Knots, i.e. embeddings of spheres

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See general introduction on embeddings, notation and conventions in [Skopenkov2016c, $\S$1, $\S$2].

1 Examples

Analogously to the Haefliger trefoil knot for $k>1$${{Stub}} See [[High_codimension_embeddings#Introduction|general introduction on embeddings]], [[High_codimension_embeddings#Notation and conventions|notation and conventions]] in \cite[$\Sk>1$ one constructs a smooth embedding $t:S^{2k-1}\to\Rr^{3k}$$t:S^{2k-1}\to\Rr^{3k}$. For $k$$k$ even this embedding is a generator of $E_D^{3k}(S^{2k-1})\cong\Zz$$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$$k$ odd this embedding is a generator of $E_D^{3k}(S^{2k-1})\cong\Zz_2$$E_D^{3k}(S^{2k-1})\cong\Zz_2$. The last phrase of [Haefliger1962t] suggests that this is true for $k=3$$k=3$.

2 Classification

For some information see [Skopenkov2006, $\S$3.3].

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

• [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, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.

, $\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 $k=3$. == Classification == For some information see \cite[$\S.3]{Skopenkov2006}. (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}$$t:S^{2k-1}\to\Rr^{3k}$. For $k$$k$ even this embedding is a generator of $E_D^{3k}(S^{2k-1})\cong\Zz$$E_D^{3k}(S^{2k-1})\cong\Zz$; it is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to it [Haefliger1962]. It would be interesting to know if for $k$$k$ odd this embedding is a generator of $E_D^{3k}(S^{2k-1})\cong\Zz_2$$E_D^{3k}(S^{2k-1})\cong\Zz_2$. The last phrase of [Haefliger1962t] suggests that this is true for $k=3$$k=3$.

2 Classification

For some information see [Skopenkov2006, $\S$3.3].

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

• [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, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.