Twisted Torus Knots: Closed-Form Alexander Polynomials and Fibonacci Determinants — E8 Intelligence Research

FINDING: Alexander polynomials of twisted torus knots admit explicit closed-form formulas via knot group presentations, with Fibonacci numbers emerging as determinant sequences in special torus-knot cases. MATH: - Alexander polynomial Δ(t) for torus knot T(p,q): Δ(t) = ((tpq−1)(t−1)) / ((t^p−1)(t^q−1)). - For T(2,n): Δ(t) = (t^n+1)/(t+1) = 1 − t + t² − … + tⁿ⁻¹ (alternating sum). - Determinant |Δ(−1)| for T(2,n) = n (odd) — but for Fibonacci-type braid closures, determinants follow Fₖ₊₁ (Fibonacci numbers) when Seifert matrix has characteristic polynomial t²−t−1. - Twisted torus knot formula (arXiv:2411.13003): Δ(t) expressed as product of cyclotomic-like factors with exponents determined by twist parameters (full twists m on adjacent strands). - Fibonacci matrix Q = [[1,1],[1,0]] has det(Q) = −1, trace = 1, eigenvalues φ = (1+√5)/2 and ψ = (1−√5)/2. Q^n yields Fₙ₊₁, F_n. CONNECTION: - Golden ratio φ = 1.618 appears as spectral radius of the Fibonacci matrix Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com