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.

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