Knot Invariants Beyond Harmonic Ratios: Additivity Falsity Reshapes Algebra — E8 Intelligence Research

FINDING: Knot theory's core invariants (Dowker notation, slice obstructions, semimeander complexity) are combinatorial/topological, not directly tied to harmonic ratios; the additivity conjecture's falsity reshapes knot algebra. | MATH: Dowker notation: crossing sequence \( \{a_i, b_i\} \) with \( b_i \) parity-opposite to \( a_i \); Conway knot non-slice via Rasmussen \(s\)-invariant (Piccirillo) — \(s(K) ≠ 0\) obstructs slice; semimeander crossing bound: \(c(D) ≤ O(n^2)\) for \(n\) arcs (arXiv:2312.05014v2). No explicit constants like 0.618 appear. | CONNECTION: Weak — Dowker notation encodes crossing parity (mod 2), a \(Z_2\) symmetry; semimeander diagrams relate to planar arc systems, reminiscent of root system \(A_n\) Coxeter–Dynkin diagrams but no golden ratio or base-60 link. | DEPTH: 4 — Significant for knot theory (additivity false, Conway knot resolved) but no direct geometric harmony or universal constant; topological depth, not metrical. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com