In Bar-Natan’s introduction to Khovanov homology he draws the parity cube based on the (left handed) trefoil smoothings, discussed in the last lesson. The indices of the cube correspond to a crossing of the knot. Recall that these knot crossings also correspond to the three squares in the 2-dimensional K4 Stasheff polytope of 9 sides. The interior of this polytope tiled the real moduli of the six punctured Riemann sphere.

We can therefore think of these knot crossings as being embedded in a canonical manner in the real moduli space. That is, one crossing for each generation of the particle zoo. The trefoil knot traces out a loop through these idempotents. The non-planarity of knots is thus associated with the non-planarity of complexification in the tiling of moduli.

But we need to get to quaternions and octonions before all this makes sense. Just remember that the numbers don’t get put in by hand. They always come from diagrams. Bar-Natan points out that integers can come from Khovanov morphisms for empty links.

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## nige said,

March 16, 2007 @ 10:51 am

“We can therefore think of these knot crossings as being embedded in a canonical manner in the real moduli space. That is, one crossing for each generation of the particle zoo.”

It’s good to see a mention of particle physics in a post about M-theory! Seriously, I hope you will deal with how leptons, quarks and neutrinos specifically arise from string vibrations at some point, or is this difficult to say because of the landscape? From popular accounts of M-theory, the impression given is that it predicts gravitation and the Standard Model, although there are some details to fill in, like masses. However, maybe the problem is much worse. Is there any specific evidence that M-theory accounts for chiral symmetry in the standard model?

What exactly does M-theory say about the standard model? Does it say anything at all? How far does the landscape problem go to prevent the particle physics predictions of M-theory being realized? Can anyone say anything definite about the Standard Model using M-theory, or is it completely impossible? I.e., general relativity gives some predictions which are independent of its cosmological landscape.

Gravitational redshift and time-dilation are for practical purposes independent of the small cosmological constant, etc.

Unless M-theory specifies that electrons and muons are distinguished due to so-and-so types of vibration, I don’t see how M-theory excites any interest. Does it at least say how many generations (3) the standard model has?

## Kea said,

March 16, 2007 @ 9:08 pm

…arise from string vibrations…Dear Nigel! Our version of M theory is not about strings. But we call it that because the name seems apt.

## nige said,

March 17, 2007 @ 9:42 pm

Thanks! It will be interesting to see where this M-theory ends up. Hopefully it will be more successful than the last one was.