Paul Bird's Scientific Research

A network of belts would be equivalent to a network of directed graphs. A half twist in a belt corresponds to a directed arrow. In particular a sub structure rotating 360 degrees would reverse the arrows going into the substructure:   Can we be more clear on graph equivalences? Therefor such a patch as fermionic structure. But what are fermions? (Perhaps all nodes are fermions.)

Idea is that the state of the Universe is represented by a knot. The complex amplitude is given by a knot invariant. This is the sum of histories. A history is a series of slicings which return the knot to the unknot. Knot states related by Reidmeister moves are equivalent. The cosmological time of a knot is the 'unknotting' number. One possible flaw is that there are an infinite number of families of knots with 'unknotting number' 1. However, many of these may have very low amplitudes. The amplitudes might be given by the knot polynomials , e.g,for some constant,k,q: \[ q\psi(K_0) = e^{ik}\psi(K_+) - e^{-ik}\psi(K_-) \] and \( \psi( unknot ) = 1 \) . The question is, how would one relate a knot to a curved 3 dimensional manifold? What would the curvature be? Where would the points of space-time be? Does it satisfy: \[ \sum\limits_{K_2}\Delta(K_1,K_2)\Delta(K_2,K_3) = \Delta(K_1,K_3) \]

First is General Relativity . The action can be put in this form: \[ S_{EH'}=\frac{1}{8\kappa}\int \sqrt{-g}\left( g^{ab}g^{de}g^{cf} +2 g^{ac}g^{bf}g^{de} + 3g^{ad}g^{be}g^{cf} -6 g^{ad}g^{bf}g^{ce} \right)\partial_c g_{ab}\partial_f g_{de} dx^4 \] Where \(g^{ab} = \frac{1}{4!}\varepsilon^{apqr}\varepsilon^{auvw}g_{pu}g_{qv}g_{rw} / \det(g) \) We can get Maxwell's equations from the above by a Kaluza-Klein compactification on (D+n) dimensions. They are:  \[S_{YM} = \int\sqrt{-g}\frac{1}{2}(g^{ab}g^{cd} - g^{ac}g^{bd})(\partial_a A_b + i A_a A_b) (\partial_c A_d + i A_c A_d) dx^4 \] Where the A fields are matrices which generate the gauge group. Then Dirac's equations are: \[S_D = \int e \psi^\dagger e^a\sigma (\partial_a + i A_a.\tau + i\Omega_a)\psi + M (\psi \varepsilon\psi + \psi^\dagger \varepsilon\psi^\dagger )dx^4\] with: \[ \Omega_a = \varepsilon e_a e^b e^c \partial_b e_c \] and \(\tau\) is a generator of a gauge group, such as SU(n). M is a matrix which