Knot Linking

In general, it is possible to link two $n$-D Hyperspheres in $(n+2)$-D space in an infinite number of inequivalent ways. In dimensions greater than $n+2$ 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 1-spheres in 3-D. The following table gives the number of nontrivial ways that two $n$-D Hyperspheres can be linked in $k$-D.

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-D Hyperspheres link up in 12, 13, 14, 15, and 16-D, then unlink in 17-D, link up again in 18, 19, 20, and 21-D. The proof of these results consists of an ``easy part'' (Zeeman 1962) and a ``hard part'' (Ravenel 1986). The hard part is related to the calculation of the (stable and unstable) Homotopy Groups of Spheres.

References

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. ``Isotopies and Knots in Manifolds.'' In Topology of 3-Manifolds and Related Topics (Ed. M. K. Fort). Englewood Cliffs, NJ: Prentice-Hall, 1962.