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Keywords = Moebius strips: fiber bundles

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Article
Knots on a Torus: A Model of the Elementary Particles
by Jack S. Avrin
Symmetry 2012, 4(1), 39-115; https://doi.org/10.3390/sym4010039 - 9 Feb 2012
Cited by 9 | Viewed by 9078
Abstract
Two knots; just two rudimentary knots, the unknot and the trefoil. That’s all we need to build a model of the elementary particles of physics, one with fermions and bosons, hadrons and leptons, interactions weak and strong and the attributes of spin, isospin, [...] Read more.
Two knots; just two rudimentary knots, the unknot and the trefoil. That’s all we need to build a model of the elementary particles of physics, one with fermions and bosons, hadrons and leptons, interactions weak and strong and the attributes of spin, isospin, mass, charge, CPT invariance and more. There are no quarks to provide fractional charge, no gluons to sequester them within nucleons and no “colors” to avoid violating Pauli’s principle. Nor do we require the importation of an enigmatic Higgs boson to confer mass upon the particles of our world. All the requisite attributes emerge simply (and relativistically invariant) as a result of particle conformation and occupation in and of spacetime itself, a spacetime endowed with the imprimature of general relativity. Also emerging are some novel tools for systemizing the particle taxonomy as governed by the gauge group SU(2) and the details of particle degeneracy as well as connections to Hopf algebra, Dirac theory, string theory, topological quantum field theory and dark matter. One exception: it is found necessary to invoke the munificent geometry of the icosahedron in order to provide, as per the group “flavor” SU(3), a scaffold upon which to organize the well-known three generations—no more, no less—of the particle family tree. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
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Figure 1

Figure 1
<p>A knot encircling a torus.</p>
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<p>A Moebius strip (MS) and its boundary.</p>
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<p>An MS as a concatenation of torus knots.</p>
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<p>Braids with closure and framing.</p>
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<p>Trivial and nontrivial vector bundles.</p>
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<p>The basic set of Flattened Moebius strips (FMS).</p>
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<p>Charge <span class="html-italic">vs.</span> Twist.</p>
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<p>Synthetic approach to flattening.</p>
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<p>Up and down quirks; two views.</p>
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<p>Definition of an antiparticle.</p>
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<p>Reference “d” quirk.</p>
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<p>Rotated around “<span class="html-italic">y</span>” axis.</p>
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<p>Rotation around ‘<span class="html-italic">z</span>” axis.</p>
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<p>A “d*” quirk.</p>
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<p><a href="#symmetry-04-00039-f014" class="html-fig">Figure 14</a> seen from the back.</p>
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<p>Four Instanton characteristics.</p>
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<p>Quirk/antiquirk correspondence to instantons and reverse instantons.</p>
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<p>Fusion of <span class="html-italic">x</span> and <span class="html-italic">y</span> junctions; two views.</p>
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<p>Hopf Multiplication and comultiplication.</p>
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<p>Two mutually conjugate pions.</p>
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<p>Dirac equation output; a fermion and antifermion bound pair.</p>
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<p>Twist loci and gradient; first-order fusion.</p>
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<p>Operational diagram for convolution.</p>
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<p>Assembly of convolution products.</p>
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<p>Twist loci; second-order fusion.</p>
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<p>Orthogonal twist and charge gradients.</p>
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<p>Operational diagram for second-order convolution.</p>
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<p>Occupancy of the inclined planar twist loci of second-order fusion.</p>
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<p>Convolution output twist assemblies; second-order fusion.</p>
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<p>Twist-based quaternary number system; zeroth, first and second-order fusion.</p>
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<p>Availability of 32 binary choices in first-order fusion.</p>
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<p>Availability of 256 binary choices in second-order fusion.</p>
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<p>Composition-enhanced twist assemblies for first-order fusion</p>
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<p>Half of contingency-enhanced twist assemblies; second-order fusion.</p>
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<p>Ambiguities in the model-to-SM connection.</p>
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<p>Nucleons and excited state interactions mediated by pions.</p>
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<p>Delta particles with pion constituents.</p>
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<p>First stage of neutron decay.</p>
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<p>Second stage of neutron decay.</p>
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<p>First stage of muon decay.</p>
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<p>Second stage of muon decay.</p>
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<p>The bosonic operator matrix in SM nomenclature.</p>
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<p>Model of Deuteron stability mediated by pions.</p>
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<p>The basic fermions in order of twist.</p>
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<p>Basic fermions in “label space”.</p>
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<p>Three-label scaffold (More to come).</p>
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<p>Four-label scaffold.</p>
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<p>Six-label scaffold; that’s all!</p>
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<p>Sample “duad”.</p>
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<p>The three orthogonal duads of the first syntheme.</p>
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<p>And their normals.</p>
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<p>The Spin ½ Baryon Octet.</p>
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<p>The Spin 3/2 baryon decuplet.</p>
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<p>The Spin 0 Octet.</p>
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<p>Some bosonic twist-charge state vectors.</p>
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<p>An approach to validating the model as a Topological Quantum field Theory (TQFT).</p>
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<p>Our model of the pentaquark.</p>
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<p>Source of additional decay products.</p>
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<p>Second-order Convolution for <span class="html-italic">n</span> = −5.</p>
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<p>Free and Fused FMS Junctions.</p>
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<p>Hopf Multiplication and Comultiplication.</p>
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<p>Two Different Intersections.</p>
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<p>Intersections as Overpasses.</p>
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<p>The Difference between the Two Intersections.</p>
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<p>Before and After Antiquirks in the Beta Switch.</p>
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<p>The Difference Between the Two Antiquirks.</p>
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<p>Quaternionic matrix subdivision.</p>
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<p>Cross product and scalar components of the reduced quaternionic matrix.</p>
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<p>“Pure” quaternionic matrix for the reduced basic fermion vector.</p>
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<p>Cross product and scalar components of <a href="#symmetry-04-00039-f069" class="html-fig">Figure E3</a>.</p>
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<p>Operational model of Eqation F-2.</p>
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