Knot Invariants, Quantum Field Theory, and Topological Phases of Matter — E8 Intelligence Research

FINDING: Knot invariants (crossing, bridge, unknotting numbers) and quantum knot invariants from zero-dimensional QFT and supersymmetric gauge theories are linked to topological phases of matter and quantum field theory. MATH: - Crossing number \( c(K) \), bridge number \( b(K) \), unknotting number \( u(K) \) — numerical invariants under Reidemeister moves. - Alexander invariants (higher-order) for homologically fibered knots factorize via Magnus matrix: \( Δ_n(t) = ∏_i P_i(t) \) where \( P_i \) are polynomials derived from the knot's homology. - Quantum knot invariants (e.g., Jones polynomial \( V_K(q) \), colored HOMFLY-PT) arise from Chern-Simons theory path integrals: \( ZCS(M) = ∫ DA \, e^{i k SCS[A]} \), with \( SCS = k/4π ∫_M Tr(A dA + 2/3 A A A) \). - Zero-dimensional QFT invariants: partition functions as sums over Feynman diagrams with weights given by knot combinatorics. CONNECTION: Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com