High codimension links
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Contents |
1 Introduction
For notation and conventions throughout this page see high codimension embeddings.
2 General position and the Hopf linking
General Position Theorem 2.1.
For each -manifold
and
, every two embeddings
are isotopic.
The restriction in Theorem 2.1 is sharp for non-connected manifolds.
Example: the Hopf linking 2.2.
For each there is an embedding
which is not isotopic to the standard embedding.
For the Hopf Linking is shown in Figure~2.1.a of [Skopenkov2006].
For arbitrary
(including
) the image of the Hopf Linking is the union of two
-spheres:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2\dots+x_{2n+1}^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2\dots+x_n^2+(x_{n+1}-1)^2=1\end{array}\right..](/images/math/1/8/b/18b1d7e000cc51571b57f92b03227c8f.png)
3 Classification/Characterization
4 Invariants
5 Further discussion
6 References
- [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.