In general, it is possible to link two -dimensional hyperspheres in
-dimensional space in an infinite
number of inequivalent ways. In dimensions greater than in the piecewise linear category, it is true that these
spheres are themselves unknotted. However, they may still form nontrivial links.
In this way, they are something like higher dimensional analogs of two one-spheres
in three dimensions. The following table gives the number of nontrivial ways that
two -dimensional hyperspheres
can be linked in
dimensions.

D of spheres

D of space

distinct linkings

23

40

239

31

48

959

102

181

3

102

182

10438319

102

183

3

Two 10-dimensional hyperspheres link up in 12, 13, 14, 15, and 16 dimensions, unlink in 17 dimensions, then link up again in 18, 19,
20, and 21 dimensions. The proof of these results consists of an "easy part"
(Zeeman 1962) and "hard part" (Ravenel 1986). The hard part is related
to the calculation of the (stable and unstable) homotopy
groups of spheres.

Bing, R. H. The Geometric Topology of 3-Manifolds. Providence, RI: Amer. Math. Soc., 1983.Ravenel,
D. Complex
Cobordism and Stable Homotopy Groups of Spheres. New York: Academic Press,
1986.Rolfsen, D. Knots
and Links. Wilmington, DE: Publish or Perish Press, p. 7, 1976.Zeeman,
E. C. "Isotopies and Knots in Manifolds." In Topology of 3-Manifolds
and Related Topics (Ed. M. K. Fort). Englewood Cliffs, NJ: Prentice-Hall,
pp. 187-193, 1962.