Charge, Leptons, and the Genus of Torus Knots

We extend the Null Worldtube (NWT) framework by introducing a fifth discrete quantum number—the framing of a torus knot embedding—and show that the set (p, q, m, n_q, f) suffices to encode mass, electric charge, baryon number, and isospin across the particle spectrum. The framing gives the isospin projection I₃ = f/2, reproducing the Gell-Mann–Nishijima formula Q = I₃ + (B+S)/2 for all hadrons tested. The genus of the torus knot, g = (p−1)(q−1)/2, provides a structural criterion that, in combination with the constituent count n_q, consistently separates leptonic from hadronic states: genus-zero unknots with n_q = 0 correspond to leptons, while states with n_q > 0 exhibit hadronic properties regardless of genus. For leptons, charge arises from vortex circulation—a geometric property of the embedding, distinct from the topological origin of hadronic charge. The τ lepton maps onto a genus-3 configuration with n_q = 3, placing it structurally among hadrons—a model prediction subject to further physical justification. Connections are established to the Rosso–Jones formula for quantum group invariants, Legendrian contact geometry (all NWT torus knots saturate the Thurston–Bennequin bound), and Dehn surgery theory, where particle decays correspond to surgery operations and neutrinos carry the topology-change data.