PARTICLE KNOTS IN TORIC MODULAR SPACE

The goal of this contribution is to relate quarks to knots or loops in a 6-space CP 3 that then ollapses into a torus in real 3-space P 3 instantaneously after the Big Bang, and massive inflation, when 3 quarks unite to form nucleons.


Introduction
Kedia et. al. in recent paper [10] investigate knotted structures in hydrodynamic fields such as current-guiding magnetic field lines in a plasma, or vortex lines of classical or quantum fields which arise naturally as excitations that carry helicity that is a measure of the knottedness of the field. In particular their Fig.2g is a trefoil which is our Fig.3 without the quadrupole (that will be seen in Section 2 to collapse into a point) and the color-coding.    The torus shown in Fig.2 is due to Marcelis [13] whose calculations in a projective space with 24 vertices appear to be unpublished, but are supported to some extent by Westy [18] (from the same school) who provides a color-coded complex map of the Riemann surface z that incorporates the phases of ω=120 degrees. Murasagi [14] Ch.7 shows that Fig.2 is a trefoil (3,2) on a torus when we choose 3 points on the ends of a cylinder that can be joined to form the torus. This brings us to the goal of this contribution which is to relate the elementary particles to knots or loops in a 6-space. Here we will be guided by the work of Coxeter [5,6] who specifically labels the vertices of the torus appearing in Fig.1 by 0,±1,ω,ω 2 where ω = exp(2iπ/3) so that a knot crosses the longitude of a torus at ω=120 degrees. Essentially this is a Galois Field GF(4) with permutations of ω raised to the powers 0,1,2,3 that will be considered in more detail in the next Section where we will show how 3 quarks in 6-space unite to become a nucleon in the projective space CP 3 which collapses to P 3 immediately after the Big Bang .Section 3 will employ the color-coding of Fig.1 as a model for Quantum Chromodynamics or QCD. Finally according to Rovelli [15] knots or loops in the 6-space described by E 6 employed by Coxeter may also describe Loop Quantum Gravity although details are beyond the scope of this contribution. Also Arvin [2] has considered knots on a torus as a model for elementary particles but excluding quarks. Again, Sundance O Billson Thompson, Smolin et.al. [17] also use knots as a model for Quantum Gravity and the Standard Model but utilize trinions instead of trefoils. Fig.1 is a torus taken from Coxeter [5] which is an alternative to the graph of su(3)c × su (3) Fig.1 is labeled by (012) indicated by 0,ω,ω 2 . Thus (023) on the same tritangent is simply a rotation through ω=120 degrees and so on. In this way we find an equilateral triangle labeled by the 3 quarks uud comprising a proton and another ddu for the neutron beginning with (120). There are 2 more tritangents (not labeled) for the anti-particles which complete the outer ring of Fig.1. But quarks also belong to a GF(4) ring and thus to a trefoil on a torus which is precisely the model adopted by Green, Schwarz and Witten [8] Section 9.5.2. The quarks at the vertices of Fig.1 are trefoils illustrated by Fig.2, but the torus in Fig.1 only becomes a trefoil after the collapse of the inner ring just after the Big Bang when quarks in the 6space CP 3 unite to build nucleons in the projective space P 3 . This is supported by Barth and Nieto [3] where only the 12 outer vertices and the center of Fig.1 are in P 3 .Specifically these authors find 15 synthemes ,where a syntheme has 6 'fix-lines' that are the edges of an invariant tetrahedron such as u u d 0 representing a proton. However because there are only 3 vertices on the face of a syntheme the outer ring of Fig.1 carries the 4 stable particles proton, neutron and their anti-particles. Also since the tritangents are invariant under rotations there are actually 3x4=12 possible synthemes on the outer ring. Specifically each syntheme consists of 3 commuting operators. Thus 3 synthemes can be chosen for spin rotations about the 3 axes of 3 space. Thus introducing triality which is a characteristic of Toric-Calabi-Yau modular spaces that carry the Hessian Polyhedra in E 6 as discussed by Lie-Yang [12] and analysed by Coxeter [5,6].

Coxeter Algebra
Specifically each rotation in 3-space is also accompanied by a corresponding rotation in a parity 4-space when we permute 1,2,3. Charge conjugation belongs to a second set of 3 synthemes with the same rotations in a 3-space but a parity 5-space in another charge space [7].
according to the relationship BSMaSS Volume 11 There is no heavy-ion decay and the same relation holds for the anti-particles. This equation is accurate if we assume that m τ = 1777 MeV and m µ = 101.4 MeV instead of the Fermi decomposition of muon decay in the weak interaction yielding 106 MeV. However in a recent publication Benjamin Brau et.al. [4] find a value of approximately 100 MeV for the mass of cosmicray muons so there is as yet some experimental uncertainty.

Quantum Chromodynamics,QCD
Returning again to Fig.1,when the inner vertices are contracted to a point at the origin the red, green and blue lines could serve as gluons on a new torus where a red upper path passes through the center before emerging at the circumference and giving way to a green gluon that in turn passes under the torus and then over to connect with a down quark and so on.The 3 color complex dimensions vanish when CP 3 → P 3 but a torus knot remains in 3-space.
However Marcelis [13] calculates the dual set of 3 paths for the anti-gluons overline (r,g,b) which appear in Fig.3 ( without the quadrupole) so the gluon,antigluon linked trefoil give us the SU(3) c color symmetry underlining QCD as described by Griffiths [9],Section 9.1.For example when another quark is added after a rotation ω a red gluon may unite with an anti-blue to build r anti-b, then a following rotation would bring r to anti-r ,and so on before blow down to P3.In this way we can find 9 gluon pairs r anti-r,r anti-b ,r anti-g;b anti-r,b anti-b,b anti-g;g anti-r,g anti-b,g anti-g that are a basis for SU(3) c symmetry. Finally Adams [1] p 273 also envisages the 3 colors r,b,g as three extra dimensions in a 6-space

Acknowledgement
This contribution is a revision of a paper already submitted to BMSA 178X that was unfortunately published without the Figures   Fig.1 The Coxeter Polytope