Knot Theory's Invariants and Symplectic Links via Reeb Chords — E8 Intelligence Research

FINDING: Knot theory's core invariants (Dowker notation, slice obstructions, meander complexity) are combinatorial/topological, not directly yielding harmonic constants; the Reeb chord result links knot theory to symplectic geometry via Legendrian fillability. | MATH: Dowker notation: crossing sequence \(a_1, a_2, \dots, a_{2n}\) with \(a_{2k-1} < a_{2k}\) and parity alternation; slice knot obstruction via Rasmussen \(s\)-invariant (from Khovanov homology) — Conway knot shown non-slice by Piccirillo using a 41-crossing knot with same Alexander polynomial; semimeander diagram complexity bound: \(c(K) \leq O(n^2)\) for \(n\) crossings (from arXiv:2312.05014v2); Reeb chord: critical points of action functional \(\mathcal{A}(\gamma) = \int \lambda\) on Legendrian submanifold in contact manifold. | CONNECTION: No direct ratio (0.382, 0.618, 0.786, 1.618, 2.618) appears. However, knot invariants often relate to root systems (e.g., \(A_n\) for braid groups, \(D_n\) for Vogel's universality), Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com