Knots, i.e. embeddings of spheres

Revision as of 09:46, 26 April 2016

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

1 Examples

1.1 The Haefliger trefoil knot

Let us construct a smooth embedding ${{Stub}} For notation and conventions throughout this page see [[High_codimension_embeddings|high codimension embeddings]]. == Examples == === The Haefliger trefoil knot === <wikitex>; Let us construct a smooth embedding $t:S^3\to\Rr^6$ (which is a generator of $E^6_D(S^3)\cong\Zz$) \cite{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 $\Rr^6$ by $(x,y,z)=(x_1,x_2,y_1,y_2,z_1,z_2)$. The higher-dimensional trefoil knot $t$ 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: $$\left\{\begin{array}{c} x=0\ |y|^2+2|z|^2=1\end{array}\right., \qquad \left\{\begin{array}{c} y=0\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\ |x|^2+2|y|^2=1 \end{array}\right..$$ See Figures 3.5 and 3.6 of \cite{Skopenkov2006}. Analogously for $k>1$ one constructs a smooth embedding $t:S^{2k-1}\to\Rr^{3k}$ (which is a generator of $E_D^{3k}(S^3)\cong\Zz_{(k)}$) that is not ''smoothly'' isotopic to the standard embedding, but is ''piecewise smoothly'' isotopic to it \cite{Haefliger1962}. </wikitex> === Classification === (I would suggest including the classification of simple knots a la Kearton et. al. in this section.---John Klein)t:S^3\to\Rr^6$ (which is a generator of $E^6_D(S^3)\cong\Zz$) [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 $\Rr^6$ by $(x,y,z)=(x_1,x_2,y_1,y_2,z_1,z_2)$. The higher-dimensional trefoil knot $t$ 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:

$$\displaystyle \left\{\begin{array}{c} x=0\\ |y|^2+2|z|^2=1\end{array}\right., \qquad \left\{\begin{array}{c} y=0\\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\\ |x|^2+2|y|^2=1 \end{array}\right..$$

See Figures 3.5 and 3.6 of [Skopenkov2006].

Analogously for $k>1$ one constructs a smooth embedding $t:S^{2k-1}\to\Rr^{3k}$ (which is a generator of $E_D^{3k}(S^3)\cong\Zz_{(k)}$) that is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to it [Haefliger1962].

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
• [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.