M Theory Lesson 29
Morrison and Nieh's paper on su(3) knot homology uses some relations of Kuperberg's, namely 
which involves trivalent vertices that we wish to add to the complexes of the homology. That is, instead of a smooth cobordism between the pieces on the right hand side of the third relation, we allow a 4 vertex diagram >-< at the top or bottom of the bordism. In the skein relation, whereas the smooth pair of lines has a coefficient of $q^2$, the 4 vertex diagram has a coefficient of $q^3$. The objects of their bordism category are webs generated by trivalent vertices with directed edges, either all ingoing or all outgoing.
Oh, that reminds me. I fixed up the Terence Tao hexagons:
These trivalent vertices satisfy a triangle condition $a + b + c = 0$ on the numbers assigned to each edge of the honeycomb. The semi-infinite edges correspond to eigenvalues of the triangular set of 3x3 Hermitean matrices.
which involves trivalent vertices that we wish to add to the complexes of the homology. That is, instead of a smooth cobordism between the pieces on the right hand side of the third relation, we allow a 4 vertex diagram >-< at the top or bottom of the bordism. In the skein relation, whereas the smooth pair of lines has a coefficient of $q^2$, the 4 vertex diagram has a coefficient of $q^3$. The objects of their bordism category are webs generated by trivalent vertices with directed edges, either all ingoing or all outgoing. Oh, that reminds me. I fixed up the Terence Tao hexagons:
These trivalent vertices satisfy a triangle condition $a + b + c = 0$ on the numbers assigned to each edge of the honeycomb. The semi-infinite edges correspond to eigenvalues of the triangular set of 3x3 Hermitean matrices.
posted by Kea | 2:20 PM
4 Comments:
Dear Kea,
I hope that I understood correctly. Do these pictures mean that one allows also splitting of braid strands and that the basic rules say that loops can be eliminated by the generalized moves?
I discussed for few years ago this kind of generalization of braid diagrams in the chapter
Equivalence of Loop Diagrams with Tree Diagrams and Cancellation of Infinities in Quantum TGD of "TGD as a Generalized Number teory" as a braidy counterpart for the vanishing of loop corrections.
The objection against mere tree diagrams was that photon scattering involves necessarily a loop so that one should generalize this by allowing besides planar trees also minimal diagrams imbeddable on 2-surfaces of various genera and containing homologically non-trivial loops.
My recent view is that maxima of Kaehler function give only the minimal generalized Feynman diagrams allowing the reaction. Also the sum over loops in ordinary sense vanishes for functional (not path-) integral over the small deformations of lightlike 3-surfaces by huge symmetries implying generalized symmetric space property of the "world of classical worlds".
Braids would replicate at vertices which correspond to 2-D partonic surfaces and this would be enough to get a non-trivial S-matrix.
Braid strands could disappear in a pairwise manner if the zeros of the polynomial corresponding to braid strands as its zeros become complex. 3-vertex in this sense would correspond to the emergence of a degenerate zero of polynomial and does not look plausible. This assuming that braid time evolution is given by a homotopy preserving polynomial as a polynomial all the time. If coefficients are required to be rational, the situation trivializes in this sense and only replication remains.
Matti
03 18 07
Hello Kea:
Thanks for a wonderful lesson. I found the generating paper by Kuperberg, in which he really explains this stuff in depth. You may have cited it already but here it is for others who are lazier than me;). He said a lot of interesting things for sure!
Pages 13-15 of the paper
One thing that I noticed is the symmetry of the spider web relations and how they are constructed. Apparently, the Reidermeister moves can be used to construct such representations of numbers...I am still reading...thanks for the resources!
And to sorta piggyback on what Matti alludes to; what is equivalence between Feynman diagrams and other types of diagrammatic representations of quantities?
Hi Matti and Mahndisa. Matti, I'm still not clear on exactly what you mean, but I think we might be getting closer. From my point of view, whatever the correct diagrammatics is, it must be about higher topos theory. So it must obey clear logical (in the higher categorical sense) rules with a physical interpretation separate from, but equivalent to, the number theory. The philosophy of TGD sounds similar to me in many ways, except for the heavy emphasis that I put on categorical thinking.
Thanks for the link, Mahndisa. As you know, in M theory we replace Feynman diagrams with twistor analogues, which are then categorified until we get octonionic number theory.
03 18 07
"As you know, in M theory we replace Feynman diagrams with twistor analogues, which are then categorified until we get octonionic number theory. "
Well Kea:
I wasn't really sure, hehehehehe You see everytime you place links to a paper of interest to me, I obsess over it then I forget the context in which it fits into M theory. I appreciate the latest posts you have done with the diagrams and brief explanations because it puts the information more into context for me. Otherwise, I find that I skip around from various concepts but do not marry the ideas as they should be in the categorified framework.
I learn when I visit here, thanks for the clarification:)
One thing I really seem to love is the number theoretic aspects that pop up in all forms of physics. The diagrammatic reasoning that results has direct tie in to M theory. So thanks again, and back to the learning board for me:)
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