Gauge Symmetry, Fermion Generations, and Magnetic Moments from the Topology of the Trefoil Knot
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2026
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| _version_ | 1866901034402578432 |
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| author | Bhatt, Aaditya |
| author_facet | Bhatt, Aaditya |
| contents | <p>We derive the colour gauge algebra <span><span><span><span><span><span>su</span></span><span>(</span><span>3</span><span>)</span></span></span></span></span> and the leading-order magnetic moment ratio <span><span><span><span><span><span>μ</span><span><span><span><span><span><span><span>n</span></span></span></span><span></span></span></span></span></span><span>/</span><span><span>μ</span><span><span><span><span><span><span><span>p</span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span>−</span><span>2/3</span></span></span></span></span> from the topology of the trefoil knot <span><span><span><span><span>T</span><span>(</span><span>2</span><span>,</span><span>3</span><span>)</span></span></span></span></span> within the Discrete Topological Torsion Theory (DTTT) framework, and propose a topological mechanism for exactly three fermion generations and colour confinement. The derivation of <span><span><span><span><span><span>su</span></span><span>(</span><span>3</span><span>)</span></span></span></span></span> proceeds through a four-step chain of established theorems: Fox 3-colouring <span><span><span><span><span>→</span></span><span><span><span>S</span><span><span><span><span><span><span><span>3</span></span></span></span><span></span></span></span></span></span><span>→</span></span><span><span><span>A</span><span><span><span><span><span><span><span>2</span></span></span></span><span></span></span></span></span></span></span></span></span></span> root lattice (Proposition 5.1) <span><span><span><span><span>→</span></span><span><span><span>su</span></span><span>(</span><span>3</span><span>)</span></span></span></span></span> via Serre's theorem. The three-generation structure follows from the Milnor fibre genus formula <span><span><span><span><span>g</span><span>(</span><span>T</span><span>(</span><span>2</span><span>,</span><span>q</span><span>))</span><span>=</span></span><span><span>(</span><span>q</span><span>−</span></span><span><span>1</span><span>)</span><span>/2</span></span></span></span></span> combined with a topological stability bound; the Callias index computation that would complete this argument is precisely stated as an open problem (Conjecture 6.2). The sign of <span><span><span><span><span><span>m</span><span><span><span><span><span><span><span>n</span></span></span></span><span></span></span></span></span></span><span>></span></span><span><span><span>m</span><span><span><span><span><span><span><span>p</span></span></span></span><span></span></span></span></span></span></span></span></span></span> is derived from inter-lobe Coulomb sums with zero free parameters. All results operate within <span><span><span><span><span>3</span><span>+</span></span><span><span>1</span></span></span></span></span> dimensions; extra dimensions are excluded by the Poplawski theorem. Three problems - the absolute W-boson mass, the Yukawa hierarchy, and the exact path-integral derivation of <span><span><span><span><span><span>n</span><span><span><span><span><span><span><span>T</span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span>4</span></span></span></span></span> - remain irreducibly open and are stated as such.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19064152 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Gauge Symmetry, Fermion Generations, and Magnetic Moments from the Topology of the Trefoil Knot Bhatt, Aaditya su(3) Fox colouring Cosserat elasticity fermion generations Milnor fibre Poplawski exclusion magnetic moments trefoil knot topological solitons <p>We derive the colour gauge algebra <span><span><span><span><span><span>su</span></span><span>(</span><span>3</span><span>)</span></span></span></span></span> and the leading-order magnetic moment ratio <span><span><span><span><span><span>μ</span><span><span><span><span><span><span><span>n</span></span></span></span><span></span></span></span></span></span><span>/</span><span><span>μ</span><span><span><span><span><span><span><span>p</span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span>−</span><span>2/3</span></span></span></span></span> from the topology of the trefoil knot <span><span><span><span><span>T</span><span>(</span><span>2</span><span>,</span><span>3</span><span>)</span></span></span></span></span> within the Discrete Topological Torsion Theory (DTTT) framework, and propose a topological mechanism for exactly three fermion generations and colour confinement. The derivation of <span><span><span><span><span><span>su</span></span><span>(</span><span>3</span><span>)</span></span></span></span></span> proceeds through a four-step chain of established theorems: Fox 3-colouring <span><span><span><span><span>→</span></span><span><span><span>S</span><span><span><span><span><span><span><span>3</span></span></span></span><span></span></span></span></span></span><span>→</span></span><span><span><span>A</span><span><span><span><span><span><span><span>2</span></span></span></span><span></span></span></span></span></span></span></span></span></span> root lattice (Proposition 5.1) <span><span><span><span><span>→</span></span><span><span><span>su</span></span><span>(</span><span>3</span><span>)</span></span></span></span></span> via Serre's theorem. The three-generation structure follows from the Milnor fibre genus formula <span><span><span><span><span>g</span><span>(</span><span>T</span><span>(</span><span>2</span><span>,</span><span>q</span><span>))</span><span>=</span></span><span><span>(</span><span>q</span><span>−</span></span><span><span>1</span><span>)</span><span>/2</span></span></span></span></span> combined with a topological stability bound; the Callias index computation that would complete this argument is precisely stated as an open problem (Conjecture 6.2). The sign of <span><span><span><span><span><span>m</span><span><span><span><span><span><span><span>n</span></span></span></span><span></span></span></span></span></span><span>></span></span><span><span><span>m</span><span><span><span><span><span><span><span>p</span></span></span></span><span></span></span></span></span></span></span></span></span></span> is derived from inter-lobe Coulomb sums with zero free parameters. All results operate within <span><span><span><span><span>3</span><span>+</span></span><span><span>1</span></span></span></span></span> dimensions; extra dimensions are excluded by the Poplawski theorem. Three problems - the absolute W-boson mass, the Yukawa hierarchy, and the exact path-integral derivation of <span><span><span><span><span><span>n</span><span><span><span><span><span><span><span>T</span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span>4</span></span></span></span></span> - remain irreducibly open and are stated as such.</p> |
| title | Gauge Symmetry, Fermion Generations, and Magnetic Moments from the Topology of the Trefoil Knot |
| topic | su(3) Fox colouring Cosserat elasticity fermion generations Milnor fibre Poplawski exclusion magnetic moments trefoil knot topological solitons |
| url | https://doi.org/10.5281/zenodo.19064152 |