Both Heisenberg’s honeycombs and Kapranov’s Fourier transform suggest the use of non-commutative polynomial invariants. Now discussions with Matti Pitkanen about knot invariants has led to the idea of introducing non-commutativity for twisted ribbons, as in the original Mulase and Waldron symplectic matrix ensemble.

The Jones polynomial, the two variable homflypt polynomial and the Kauffman polynomial are all commutative. The most interesting invariant in this context is the Kauffman polynomial, defined in terms of the (planar isotopy) Kauffman bracket, because the latter has always been associated to the idea of diagrams as numbers. Is there a way to associate letters to (a skein relation for) twists that would extend these invariants to a non-commutative braided ribbon invariant? There is certainly an analogy between the three basic crossing diagrams ($L_{+}$, $L_{-}$ and $L_{0}$) for the homflypt polynomial and the three possible twist elements: left handed half twist, right handed half twist and flat ribbon.

As it turns out, Bar-Natan seems to be thinking about such things! As he says on his knot wiki, “If you know what this is about, good. If not, bummer.”

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## Matti Pitkanen said,

August 31, 2007 @ 4:29 am

A couple of enthusiastic comments to the knotty speculations inspired by the usual processing of what I wrote first.

a) Connected sum for knots is of course commutative. I had in mind braids and quantum group structure might quite well imply that the unitary matrices representing knots might not commute although trace for their product is commutative and maps product of knots to a product of Jones polynomials (assuming that I have understood correctly).

b) I realized that the simplest infinite primes representing free fermion boson states are in a very natural 2-to-1 correspondence with torus knots. Both correspond to rationals but for torus knots m/n knot is isotopic with n/m knot. Therefore the hypothesis survives a very stringent test.

c) n-categories, n-groupoids, etc. emerge very naturally from the generalization of imbeddings of spheres S^d with imbeddings of a general d-manifold to d+2 manifold. Knot product is still commutative but defines d-groupoid like structure rather since the topology of product is different from the topologies of composites.

Thus infinite primes would correspond to the hierarchy of n-structures very naturally as is indeed natural since are an outcome of a repeated abstraction process: statements about statements about …..

d) The possibility that algebraic extensions of infinite primes could allow to describe the refinements related to the varying topologies of knot and imbedding space would mean a deep connection between number theory, manifold topology, and sub-manifold topology, and n-structures.

e) This would also mean that n-structures would have direct correspondence with the TGD physics assuming that repeated second quantization makes sense and corresponds to the hierarchical structure of many-sheeted space-time.