Algebraic knots and generalized Feynman diagrams

Witten has been working with knots- also 2-knots - recently (for the latest popular lecture see this) and I think that his interest on these objects is not purely mathematical. Braids are central also for quantum TGD and quite recently I learned about what are called algebraic knots and braids and realized that they might provide a new mathematical building block allowing to sharpen the vision about generalized Feynman diagrams resulting as via the fusion of ordinary Feynman diagrams and braid diagrams.

This strenghens the hopes (or raises the fears?) that a collective concentrated effort could soon transform TGD from art to a mechanical calculation recipes. In the following I tell how braid theory a la TGD differs from standard braid theory, discuss briefly the notion of algebraic knots, and summarize the interpretation of generalized Feynman diagrams as a generalized braid diagrams.

How braid theory in TGD Universe differs from the standard theory?

In TGD framework knots and braids associated with preferred 3-surfaces and possibly also 2-knots and 2-braids defined by string world sheets having braids at their ends appear naturally. This because space-time dimension is D=4. Standard braid theory must be modified to sub-manifold braid theory. Braids reside at 3-surfaces with varying topologies and knot projection must be performed to a preferred 2-surface of M⁴× CP₂. To preferred Minkowski plane M² or sphere S² at light-cone boundary.

What this means from the point of view of braid theory? A typical new situation is the one in which 3-surface is locally a product of higher genus 2-surface and R so that knot strand can wind around the 2-surface: M-theorist would talk about wrapping of branes. This gives rise to what is called non-planar braid diagrams for which projection to plane produces non-standard crossings. How to cope with this kind of situation?

Algebraic knots

The answer to the question emerged as Ulla sent me a link to an article telling about algebraic knots. The introduction of the self intersection of knot - virtual crossing - besides the usual crossing below or above can be applied to non-planar braids appearing in sub-manifold braid theory. Virtual crossing combined with the algebraization of the basic moves for braids leads to completely new and very general mathematical concepts such as kei, quandle, rack, and biquandle applying also to other mathematical structures. The basic algebraic operation is a representation of the "result for x going under y". One example is group endowed with the product a*b= bⁿab^-n for n some integer. The interesting thing is that rack is aset for which the multiplication acts as automorphism. In quantum TGD completely analogous situation occurs: zero energy states define infinitesimal symmetries of S-matrix appearing as building block of M-matrices defining zero energy states.

I see bi-quandle applying to oriented knots as the most interesting structure. It is defined as operation mapping a pair of strand portions before crossing to a pair of strand portions after crossing and satisfying also set theoretic version of Yang-Baxter equation with tensor product replaced with Cartesian product. Note that Cartesian is the classical physics correlate for tensor product used to construct many particle states in quantum physics.

Why I find algebraic knots so interesting?

Why I regard algebraic knots and braids and related mathematical structures as highly interesting notions?

1. All generalized Feynman diagrams are reduced to sub-manifold braid diagrams at microscopic level by bosonic emergence (bosons as pairs of fermionic wormhole throats). Three-vertices appear only for entire braids and are purely topological whereas braid strands carrying quantum numbers are just re-distributed in vertices. No 3-vertices at the really microscopic level! This is an additional nail to the coffin of divergences in TGD Universe.

2. By projecting the braid strands of generalized Feynman diagrams to preferred plane M², one obtains a unified description of non-planar Feynman diagrams and braid diagrams (note that for Feynman diagrams the intersections have purely combinatorial origin coming from representations as 2-D diagrams: for braid diagrams intersections have different origin and non-planarity has different meaning). The necessity to choose preferred plane M² looks strange from QFT point of view but is in TGD forced by the number theoretic vision in which M² represents hyper-complex plane of sub-space of hyper-octonions which is subspace of complexified octonions. It is also forced by the condition that the choice of quantization axes has a geometric correlate both at the level of imbedding space geometry and the geometry of the "world of classical worlds".

3. The obvious conjecture is that Feynman amplitudes are a analogous to knot invariants constructible by gradually reducing non-planar Feynman diagrams to planar ones after which the already existing twistor theoretical machinery of N=4 SYMs would apply (for this conjecture see the earlier posting).

These ideas are not ready to press. I have just started to play with them and I cannot guarantee that the views about what is essential might not change. I am however becoming more and more convinced that quantum TGD as almost topological QFT reduces to the analog of braid theory with topological invariance replaced by symplectic invariance bringing in those aspects of physics for which the notion of length is essential.

For twistorial ideas see the chapter Yangian symmetry, twistors, and TGD of "Towards M-Matrix" and the chapter Knots and TGD of "Physics and Geometry of Infinite-Dimensional Geometry".