Knot Polynomials from QFT and Quantum Groups — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Knot invariants from zero-dimensional QFT and quantum groups yield polynomial invariants (Jones, HOMFLY-PT, Alexander) that encode topological data of embeddings in 3-space. | MATH: Jones polynomial \( V_K(t) \) satisfies skein relation \( t^{-1}V_{L_+} - tV_{L_-} = (t^{1/2} - t^{-1/2})V_{L_0} \); Alexander polynomial \( \Delta_K(t) \) from Seifert matrix; HOMFLY-PT polynomial \( P_K(a,z) \) generalizes both. Quantum invariants from \( U_q(\mathfrak{sl}_2) \) at roots of unity give 3-manifold invariants (Witten-Reshetikhin-Turaev). | CONNECTION: Jones polynomial evaluated at \( t = e^{2\pi i/5} \) yields golden ratio \( \phi = 1.618... \) for certain knots (e.g., \( V_{T(2,5)}(e^{2\pi i/5}) = \phi \)); Alexander polynomial relates to Seifert surfaces and linking numbers, echoing base-60 sexagesimal cycles in periodicity. | DEPTH: 7 — Bridges quantum field theory, topology, and number theory; golden ratio emergence suggests deeper geometric harmony in knot spectra. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171033700
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