3-manifolds in 6-space
This page has been accepted for publication in the Bulletin of the Manifold Atlas. |
For notation and conventions see high codimension embeddings.
Contents |
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 isotopic to the standard embedding, but is isotopic to the standard embedding.
Denote coordinates in by . The higher-dimensional trefoil knot is obtained by joining with two tubes the higher-dimensional , i.e. the three spheres given by the following three systems of equations:
See Figures 3.5 and 3.6 of [Skopenkov2006].
2 The Hopf construction of an embedding
Represent Define
It is easy to check that is an embedding. (The image of this embedding is given by the equations , .)
It would be interesting to obtain an explicit construction of an embedding which is not isotopic to the composition of the Hopf embeding with the standard inclusion . (Such an embedding is unique up to PL isotopy by the Classification Theorem of embeddings just below the stable range. The composition of with the hyperplane reflection of is PL isotopic to .)
3 Algebraic embeddings from the theory of integrable systems
Some 3-manifolds appear in the theory of integrable systems together with their embeddings into (given by a system of algebraic equations) [BolsinovFomenko2004], Chapter 14. E.g. the following system of equations corresponds to the Euler integrability case:
where and are variables while and are constants. This defines embeddings of , or into .
4 References
- [BolsinovFomenko2004] Template:BolsinovFomenko2004
- [Haefliger1962] A. Haefliger, Knotted -spheres in -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.
This page has not been refereed. The information given here might be incomplete or provisional. |