Knots, i.e. embeddings of spheres
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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 (which is a generator of ) [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 by . The higher-dimensional trefoil knot 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:
See Figures 3.5 and 3.6 of [Skopenkov2006].
Analogously for one constructs a smooth embedding (which is a generator of ) that is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to it [Haefliger1962].
1.2 Classification
(I would suggest including the classification of simple knots a la Kearton et. al. in this section.---John Klein)