Torus Knot Alexander Polynomials Factor via Cyclotomics, Linking Roots to Hoste's Conjecture — E8 Intelligence Research

FINDING: Alexander polynomial of torus knots factorizes into cyclotomic polynomials, linking knot invariants to root-system weight lattices and Hoste's conjecture on root locations. | MATH: For a torus knot \(T(p,q)\), Alexander polynomial \(ΔT(p,q)(t) = {(tpq-1)(t-1)}{(t^p-1)(t^q-1)}\). This factors as \(∏d|pq, d p, d q Φ_d(t)\), where \(Φ_d\) are cyclotomic polynomials. Roots are roots of unity \(e2π i k/d\) with \(d\) as above. Hoste's conjecture: all roots of Alexander polynomials of alternating knots lie on the unit circle — here trivially satisfied for torus knots (all roots are unit-modulus). Weight-lattice connection: the exponents \(p,q\) correspond to simple roots of \(A_1\) (or \(A_2\) in higher rank), and the cyclotomic factors \(Φ_d\) index the orbits of the Weyl group on the weight lattice; the degree of \(Δ\) is \((p-1)(q-1)\), which is the number of positive roots in the root system of type \(Aₚ₋₁ × Aq-1\) m Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com