The construction of honeycombs from overlapping Y pieces may be generalised to include knot crossing pieces. This means working with knotted trivalent graphs for which Kuperberg’s spider rules are useful. For example, in the 2×2 case there is the operation and the resolution of one crossing into a >-< diagram appears on the right hand side. There is also a dual relation arising from the opposite crossing.

Thanks to Nigel for this link to a classic paper by J. C. Maxwell involving honeycomb diagrams. In a way, M theory has returned us to this aether, but induced backgrounds in measurement geometry are imposed by the observer and by no means a casual aether in the traditional sense.

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## Mahndisa S. Rigmaiden said,

April 22, 2007 @ 12:09 am

04 21 07

Yes, induced background…

## CarlBrannen said,

April 23, 2007 @ 4:14 am

Kea, could you save me some effort and tell me what the “q” is?

## Matti Pitkanen said,

April 23, 2007 @ 8:50 am

Dear Kea and Others,

a comment not directly related to this posting but to the problem of whether category theoretic picture might have direct topological counterpart at space-time level.

I have been pondering various interpretations for what darkness interpreted in terms of nonstandard value of Planck constant and the modification of imbedding space obtained by gluing together H–> H/G_axG_b, G_a and G_b discrete subgroups of SU(2) associated with Jones inclusions along their common points. G_a and G_b could be restricted to be cyclic and thus leaving the choice of quantization axis invariant. A book like structure results with different copies of H analogous to the pages of the book. Probably brane people work with analogous structures.

The most conservative form is that only field bodies of particles are dark and that particle space-time sheet to which I assign the p-adic prime p characterizing particle corresponds to its em field body. Also Compton length as determined by em interaction would characterize this field body. This option is implied by a strong hypothesis that elementary particles are maximally quantum critical meaning that they belong to subspace of H left invariant by all groups G_axG_b leaving quantization axis invariant.

The implication would be that particle possess field body associated with each interaction and extremely rich repertoire of phases emerges if these bodies are allowed to be dark and characterized by p-adic primes. Planck constant would be assigned with a particular interaction of particle rather than particle. This conforms with the formula of gravitational Planck constant hbar_gr= GMm2^{11}, whose dependence on particle masses indeed forces the assignment of this constant to the gravitational field body.

What I realized is that if elementary particles are maximally quantum critical they would be analogous to objects and field bodies mediating interactions between them would be analogous to morphisms. The basic structures of category theory would have direct implementation at the level of many-sheeted space-time.

Matti

## Kea said,

April 23, 2007 @ 9:23 pm

Hi everyone. Carl, the q is borrowed from quantum group theory, where it appears either as a (complex) deformation parameter for Hopf algebras, or as the letter of the Laurent polynomials for knot invariants, which is where it comes in here. Think of Khovanov homology as a more powerful way of looking at the skein relation rules for knots, which reduce a complicated knot diagram recursively to simple ones, so that the basic axioms generate all possible knot polynomials. In our context, I want to reformulate your idempotents in terms of this knot homology, because I believe it will shed light on the nature of the deltas etc.