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 \(\Delta_{T(p,q)}(t) = \frac{(t^{pq}-1)(t-1)}{(t^p-1)(t^q-1)}\). This factors as \(\prod_{d|pq, d\nmid p, d\nmid q} \Phi_d(t)\), where \(\Phi_d\) are cyclotomic polynomials. Roots are roots of unity \(e^{2\pi i k/d}\) 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 \(\Phi_d\) index the orbits of the Weyl group on the weight lattice; the degree of \(\Delta\) is \((p-1)(q-1)\), which is the number of positive roots in the root system of type \(A_{p-1} \times A_{q-1}\) m Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com