M Theory Lesson 91
Recall that knots have primes. For example, all (m,n) torus knots are prime. Mike Sullivan has shown that although the universal Ghrist template contains all knots, there exist simpler templates (such as the Lorenz one) for which all the knots are prime. Zeta functions for templates have been defined.
Is there a zeta function for knots? The question is what to use in place of the factor $p$. Presumably the correct choice is the Jones (or homflypt) polynomial, or categorified versions thereof (a polynomial is really a kind of number, anyway). Nothing prevents one from simply defining a zeta function
$\zeta (z) = \prod_{K} (1 - J(K)^{-z})^{-1}$
over prime knots $K$, and the Jones polynomial is functorial with respect to knot composition. Then if the Jones polynomial were itself a zeta function for the knot diagram, this would be an iterated zeta function. This looks horrible, but at least the number of prime knots of $n$ crossings is bounded above by $e^n$ (!!) and convergence might be reasonable since, for torus knots, the invariant $J(K)$ looks like an ordinary ordinal when normalised to the simplest knot in the series. That is, for negative integers $z$ the corresponding Euler series would be a summation of counting numbers, just as for the Riemann zeta function.
Is there a zeta function for knots? The question is what to use in place of the factor $p$. Presumably the correct choice is the Jones (or homflypt) polynomial, or categorified versions thereof (a polynomial is really a kind of number, anyway). Nothing prevents one from simply defining a zeta function
$\zeta (z) = \prod_{K} (1 - J(K)^{-z})^{-1}$
over prime knots $K$, and the Jones polynomial is functorial with respect to knot composition. Then if the Jones polynomial were itself a zeta function for the knot diagram, this would be an iterated zeta function. This looks horrible, but at least the number of prime knots of $n$ crossings is bounded above by $e^n$ (!!) and convergence might be reasonable since, for torus knots, the invariant $J(K)$ looks like an ordinary ordinal when normalised to the simplest knot in the series. That is, for negative integers $z$ the corresponding Euler series would be a summation of counting numbers, just as for the Riemann zeta function.
posted by Kea | 1:03 PM
4 Comments:
Dear Kea,
thank you for an interesting posting.
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.
This in turn inspires the question whether d=1-dimensional prime knots might be in 1-1 correspondence with infinite primes: quaternionic infinite primes would be suggested by non-commutativity.
If so they could correspond to states of a super-symmetric arithmetic quantum field theory with bosonic single particle states and fermionic states labelled by quaternionic primes.
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. Also bound states are there and correspond to irreducible polynomials with roots which are algebraic numbers.
In TGD framework this hierarchy corresponds to a hierarchy defined by space-time sheets of topological condensate: space-time sheet containing a galaxy can behave like an elementary particle at the next level of hierarchy.
Could this hierarchy has 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 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.
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).
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.
08 27 07
Kea:
Here is a paper (PDF) that might be of interest to the less advanced readers (like myself);)
Someday all thes M-theory lessons would make an interesting book. Another catchy title would be "The Dark Energy Fraud."
This in turn inspires the question whether d=1-dimensional prime knots might be in 1-1 correspondence with infinite primes
Somebody has thought about this, but unfortunately I cannot remember where I saw it. Yes, it seems that knots in 3d are basic to physics. BTW this is a commutative case: I would like to see twisted ribbons for non-commutativity, like in Mulase and Waldron. These could come from templates, but with a new kind of zeta function (related to the knots contained in the template).
Thanks for the link, Mahndisa, and LOL, Louise!
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