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.

8 years ago

For the bioctonions, we can define a norm so that $N(z) = z \overline{z}$. And we don't require fibration loops to be circles, when it may be more convenient to consider other conic sections.

ReplyDeleteKea,

ReplyDeleteSuch an octonionic plane can be mapped onto a general plane.

The 15 dimensional sphere can describe the analog to a rhombidodecahedron which is the triacontrahedron, and this relationship is the reason we find reductions from 24-cell tori to hypercubes and so on...

Quantum privileging while allowed can trivially show in a same 3D system we can substitute colors.

Thus the analogy from part of the geometry (which is not simply part of the algebra when we can see these as a system of one, as is the logic and math thought possibly the same language- both notions make a dynamic space)is that the algebra of the rhombidodecahedron is the analog to the 24-cell.

So of we allow bioctonions we may need in a more general physics bi-associahedra. (so not just one per natural dimension).

I will post tables for this today. But 31 dimensions if we have the same language is presently outside the span of my graph paper.

The PeSla

The PeSla, my last paper is on bi-associahedra. This was just an introductory post.

ReplyDelete