# Knots, i.e. embeddings of spheres

## 1 Introduction


## 2 Examples

Analogously to the Haefliger trefoil knot for $k>1$$k>1$ one constructs a smooth embedding $t:S^{2k-1}\to\Rr^{3k}$$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$.

## 3 Readily calculable classification

For $m\ge n+3$$m\ge n+3$ the group $E^m(S^n)$$E^m(S^n)$ has been described in terms of exact sequences involving the homotopy groups of spheres and the homotopy groups of the pairs $(SG_n,SO_n)$$(SG_n,SO_n)$ for $n = n_i+1$$n = n_i+1$ and $n_i$$n_i$ [Haefliger1966a], cf. [Levine1965], [Milgram1972], [Habegger1986]. Here $SG_n$$SG_n$ is the space of maps $f \colon S^{n-1} \to S^{n-1}$$f \colon S^{n-1} \to S^{n-1}$ of degree $1$$1$. Restricting an element of $SO_n$$SO_n$ to $S^{n-1} \subset \Rr^n$$S^{n-1} \subset \Rr^n$ identifies $SO_n$$SO_n$ as a subspace of $G_n$$G_n$.

Some readily calculable corollaries of this classification are recalled in [Skopenkov2006, $\S$$\S$3.3].

## 4 Codimension 2 knots

For the best known specific case, i.e. for codimension 2 embeddings of spheres (in particular, for the classical theory of knots in $\Rr^3$$\Rr^3$), a complete readily calculable classification (in the sense of Remark 1.1 of [Skopenkov2016c]) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots, most of which does not present a readily calculable classification. See e.g. interesting papers [Farber1981], [Farber1983], [Kearton1983], [Farber1984].

On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a] and [Ranicki1998].

## 5 References

• [Farber1981] M. Sh. Farber, Classification of stable fibered knots, Mat. Sb. (N.S.), 115(157):2(6) (1981) 223–262.
• [Farber1983] M. Sh. Farber, The classification of simple knots, Russian Math. Surveys, 38:5 (1983).