A little crazy speculation about knots and infinite primes

Kea told about some mathematical results related to knots.

1. Knots are very algebraic objects. Product of knots is defined in terms of connected sum. Connected sum quite generally defines a commutative and associative product (or sum, as you wish), and one can decompose any knot into prime knots.

2. Knots can be mapped to Jones polynomials J(K) (for instance -there are many other polynomials and there are very general mathematical results about them which go over my head) and the product of knots is mapped to a product of corresponding polynomials. The polynomials assignable to prime knots should be prime in a well-defined sense, and one can indeed define the notion of primeness for polynomials J(K): prime polynomial does not factor to a product of polynomials of lower degree in the extension of rationals considered.

This raises the idea that one could define the notion of zeta function for knots. It would be simply the product of factors 1/(1-J(K)^-s) where K runs over prime knots. The new (to me) but very natural element in the definition would be that ordinary prime is replaced with a polynomial prime.

1. Do knots correspond to the hierarchy of infinite primes?

I have been pondering the problem how to define the counterpart of zeta for infinite primes. The idea of replacing primes with prime polynomials would resolve the problem since infinite primes can be mapped to polynomials. For some reason this idea however did not occur to me.

The correspondence of both knots and infinite primes with polynomials inspires the question whether d=1-dimensional prime knots might be in correspondence (not necessarily 1-1) with infinite primes. Rational or Gaussian rational infinite primes would be naturally selected: these are also selected by physical considerations as representatives of physical states although quaternionic and octonionic variants of infinite primes can be considered.

If so, knots could correspond to the subset of states of a super-symmetric arithmetic quantum field theory with bosonic single particle states and fermionic states labelled by quaternionic primes.

1. The free Fock states of this QFT are mapped to first order polynomials and irreducible polynomials of higher degree have interpretation as bound states so that the non-decomposability to a product in a given extension of rationals would correspond physically to the non-decomposability into many-particle state. What is fascinating that apparently free arithmetic QFT allows huge number of bound states.

2. Infinite primes form an infinite hierarchy which corresponds to an infinite hierarchy of second quantizations for infinite primes meaning that n-particle states of the previous level define single particle states of the next level. At space-time level this hierarchy corresponds to a hierarchy defined by space-time sheets of the topological condensate: space-time sheet containing a galaxy can behave like an elementary particle at the next level of hierarchy.

3. Could this hierarchy have some counterpart for knots?In one realization as polynomials, the polynomials corresponding to infinite prime hierarchy have increasing number of variables. Hence the first thing that comes into my uneducated mind is as the hierarchy defined by the increasing dimension d of knot. All knots of dimension d would in some sense serve as building bricks for prime knots of dimension d+1. A canonical construction recipe for knots of higher dimensions should exist.

4. One could also wonder whether the replacement of spherical topologies for d-dimensional knot and d+2-dimensional imbedding space with more general topologies could correspond to algebraic extensions at various levels of the hierarchy bringing into the game more general infinite primes. The units of these extensions would correspond to knots which involve in an essential manner the global topology (say knotted non-contractible circles in 3-torus). Since the knots defining the product would in general have topology different from spherical topology the product of knots should be replaced with its category theoretical generalization making higher-dimensional knots a groupoid in which spherical knots would act diagonally leaving the topology of knot invariant. The assignment of d-knots with the notion of n-category, n-groupoid, etc.. by putting d=n is a highly suggestive idea. This is indeed natural since are an outcome of a repeated abstraction process: statements about statements about .....

5. The lowest d=1, D=3 level would be the fundamental one and the rest would be somewhat boring repeated second quantization;-). This is why dimension D=3 (number theoretic braids at light-like 3-surfaces!) would be fundamental for physics.

2. Further speculations

Some further comments about the proposed structure of all structures are in order.

1. 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, sub-manifold topology, and n-category theory.

2. n-structures would have very direct correspondence with the physics of TGD Universe if one assumes that repeated second quantization makes sense and corresponds to the hierarchical structure of many-sheeted space-time where even galaxy corresponds to elementary fermion or boson at some level of hierarchy. This however requires that the unions of light-like 3-surfaces and of their sub-manifolds at different levels of topological condensate should be able to represent higher-dimensional manifolds physically albeit not in the standard geometric sense since imbedding space dimension is just 8. This might be possible.
   1. As far as physics is considered, the disjoint union of submanifolds of dimensions d₁ and d₂ behaves like a d₁+d₂-dimensional Cartesian product of the corresponding manifolds. This is of course used in standard manner in wave mechanics (the configuration space of n-particle system is identified as E³ⁿ/S_n with division coming from statistics).

   2. If the surfaces have intersection points, one has a union of Cartesian product with punctures (intersection points) and of lower-dimensional manifold corresponding to the intersection points.

   3. Note also that by posing symmetries on classical fields one can effectively obtain from a given n-manifold manifolds (and orbifolds) with quotient topologies.

   The megalomanic conjecture is that this kind of physical representation of d-knots and their imbedding spaces is possible using many-sheeted space-time. Perhaps even the entire magnificient mathematics of n-manifolds and their sub-manifolds might have a physical representation in terms of sub-manifolds of 8-D M⁴×CP₂ with dimension not higher than space-time dimension d=4. Could crazy TOE builder dream of anything more ouf of edge;-)!

3. The idea survives the most obvious killer test

All this looks nice and the question is how to give a death blow to all this reckless speculation. Torus knots are an excellent candidate for permorming this unpleasant task but the hypothesis survives!

1. Torus knots are labelled by a pair integers (m,n), which are relatively prime. These are prime knots. Torus knots for which one has m/n= r/s are isotopic so that any torus knot is isotopic with a knot for which m and n have no common prime power factors.

2. The simplest infinite primes correspond to free Fock states of the supersymmetric arithmetic QFT and are labelled by pairs (m,n) of integers such that m and n do not have any common prime factors. Thus torus knots would correspond to free Fock states! Note that the prime power p^k_p appearing in m corresponds to k_p-boson state with boson "momentum" p_k and the corresponding power in n corresponds to fermion state plus k_p-1 bosons.

3. A further property of torus knots is that (m,n) and (n,m) are isotopic: this would correspond at the level of infinite primes to the symmetry mX +n→nX+m, X product of all finite primes. Thus infinite primes are in 2→ correspondence with torus knots and the hypothesis survives also this murder attempt.

4. How to realize the representation of the braid hierarchy in many-sheeted space-time?

One can consider a concrete construction of higher-dimensional knots and braids in terms of the many-sheeted space-time concept.

1. The basic observation is that ordinary knots can be constructed as closed braids so that everything reduces to the construction of braids. In particular, any torus knot labelled by (m,n) can be made from a braid with m strands: the braid word in question is (σ₁....σ_m-1)ⁿ or by (m,n)=(n,m) equivalence from n strands. The construction of infinite primes suggests that also the notion of d-braid makes sense as a collection of d-knots in d+2-space, which move and and define d+1-braid in d+3 space (the additional dimension being defined by time coordinate).

2. The notion of topological condensate should allow a concrete construction of the pairs of d- and d+2-dimensional manifolds. The 2-D character of the fundamental objects (partons) might indeed make this possible. Also the notion of length scale cutoff fundamental for the notion of topological condensate is a crucial element of the proposed construction.

The concrete construction would proceed as follows.

1. Consider first the lowest non-trivial level in the hierarchy. One has a collection of 3-D lightlike 3-surfaces X³_i representing ordinary braids. The challenge is to assign to them a 5-D imbedding space in a natural manner. Where do the additional two dimensions come from? The obvious answer is that the new dimensions correspond to the 2-d dimensional partonic 2-surface X² assignable to the 3-D lightlike surface at which these surfaces have suffered topological condensation. The geometric picture is that X³_i grow like plants from ground defined by X² at 7-dimensional δM⁴₊×CP₂.

2. The degrees of freedom of X² should be combined with the degrees of freedom of X³_i to form a 5-dimensional space X⁵. The natural idea is that one first forms the Cartesian products X⁵_i =X³_i×X² and then the desired 5-manifold X⁵ as their union by posing suitable additional conditions. Braiding means a translational motion of X³_i inside X² defining braid as the orbit in X⁵. It can happen that X³_i and X³_j intersect in this process. At these points of the union one must obviously pose some additional conditions.

   Finite (p-adic) length scale resolution suggests that all points of the union at which an intersection between two or more light-like 3-surfaces occurs must be regarded as identical. In general the intersections would occur in a 2-d region of X² so that the gluing would take place along 5-D regions of X⁵_i and there are therefore good hopes that the resulting 5-D space is indeed a manifold. The imbedding of the surfaces X³_i to X⁵ would define the braiding.

3. At the next level one would consider the 5-d structures obtained in this manner and allow them to topologically condense at larger 2-D partonic surfaces in the similar manner. The outcome would be a hierarchy consisting of 2n+1-knots in 2n+3 spaces. A similar construction applied to partonic surfaces gives a hierarchy of 2n-knots in 2n+2-spaces.

4. The notion of length scale cutoff is an essential element of the many-sheeted space-time concept. In the recent context it suggests that d-knots represented as space-time sheets topologically condensed at the larger space-time sheet representing d+2-dimensional imbedding space could be also regarded effectively point-like objects (0-knots) and that their d-knottiness and internal topology could be characterized in terms of additional quantum numbers. If so then d-knots could be also regarded as ordinary colored braids and the construction at higher levels would indeed be very much analogous to that for infinite primes.

For details see the chapter TGD as a Generalized Number Theory III: Infinite Primes of "TGD as a Generalized Number Theory".