Theory Update 97

Everyone knows that it takes three dimensions to tie a knot. Modern knot theory is very important in mathematical physics, where invariants for knots are typically computed using interesting quantum field theories. A simple knot, or link, is formally a set of closed loops in a background space.

The easiest way to think of a flat Euclidean space as a collection of lines is to take an uncountable set of parallel lines. For a sphere, one could use an uncountable set of parallel loops, with two pointlike loops at the poles, but there is a nice way of creating certain spheres entirely out of loops, known as a Hopf fibration. For example, the three dimensional sphere may be thought of as flat three dimensional space with an extra point at infinity. In the diagram, the point at infinity closes the line into a loop. Another loop of the Hopf fibration is linked to this large loop, as shown.

The reason this works is that the three dimensional sphere can be embedded as a unit sphere in a four dimensional flat space, which is given the structure of a complex number plane $P = \{ z_1, z_2 \}$. The Hopf fibration is a map

$h(z_1, z_2) = (2 z_1 \overline{z}_{2} , |z_1|^2 - |z_2|^2)$

on $P$. The three dimensional sphere maps to a two dimensional sphere, such that each point on this $2$-sphere comes from a loop. We think of the $2$-sphere as the set of complex numbers plus a point at infinity. There are also Hopf fibrations for other number fields, namely the reals, quaternions and octonions. For the octonions, one maps a $15$ dimensional sphere to the $8$ dimensional one.

In the compex number case, there is a way to create loop fibrations for higher dimensional complex spaces. For the octonionic plane, however, one cannot take the $31$ dimensional sphere and obtain a fibration. But like Peano before us, we would like to build other spaces out of simple curves. We could start with other systems of numbers, like the bioctonions.