Linking Spheres

The links pictured on this page consist of unknotted components. Looking at the examples below it's not hard to believe that two unknotted 1-spheres (circles) can be knotted in 3-dimensions in an infinite number of inequivalent ways (click on the picture above to find out what it represents).

Indeed it can be proven that each of the above links of two components is a true link, that is there is no smooth deformation in three dimensions that can convert one of these links into the trivial link of two components:

or into any other of the links pictured. In higher dimensions it is also possible to create non-trivial links of spheres. Generally it is possible to link two N-dimensional spheres in (N+2)-dimensional space in an infinite number of inequivalent ways.

In dimensions greater than (N+2) it is true (at least in the PL category) that these spheres are themselves unknotted. However, they may still form non-trivial links. In this way they are something like higher dimensional analogues of the above links of two 1-spheres in 3-dimensions. One of the truly amazing achievements of modern mathematics is that is it possible to demonstrate that:

• There are 239 non-trivial ways to link two 23-dimensional spheres in 40-space.
• There are 959 non-trivial ways to link two 31-dimensional spheres in 48-space.
• There are 3 non-trivial ways to link two 102-dimensional spheres in 181-space.
• There are 1048319 non-trivial ways to link two 102-dimensional spheres in 182-space.
• There are 3 non-trivial ways to link two 102-dimensional spheres in 183-space.
• Two 10-dimensional spheres link up in 12, 13, 14, 15, and 16 dimensions, then unlink in 17 dimensions, link up again in 18, 19, 20, and 21 dimensions.

Whew! These results can be rigorously proven and software exists to compute these facts. The proof consists of an "easy part" and a "hard part". For the easy part see the article by Zeeman Isotopies and Knots in Manifolds in the book Topology of 3-Manifolds and related topics, edited by M. K. Fort (Prentice-Hall, 1962).

The hard part is very hard indeed, and is related to the calculation of the (stable and unstable) homotopy groups of spheres. To get an idea of the difficulties see the book by Douglas Ravenel Complex cobordism and stable homotopy groups of spheres, Academic Press, 1986.

I was first alerted to this odd topic by an aside made on page 7 of Dale Rolfsen's book Knots and Links, Publish or Perish, 1976. The Linking Spheres got onto the "Uselessness of Wackos" page, however, everything you see here is really true.