Derivation of the HOMFLYPT knot polynomial via helicity and geometric quantization
Antonio Michele Miti, Mauro Spera
TL;DR
This work offers a semiclassical bridge between knot invariants and geometric quantization by treating the HOMFLYPT polynomial as a WKB wave function on Brylinski's manifold of links, guided by helicity-based hydrodynamics. It combines Maslov-Hörmander techniques with Abelian geometric quantization to encode skein relations through a phase $\\alpha = e^{2\pi i \lambda \\mathcal{H}(E_+)}$ and a component-change term $z$, yielding the invariant as a covariantly constant section $\\Psi(\\alpha,z)$ after normalization. By recovering the relation $\\alpha^{-1}\\Psi(L_+) - \\alpha\\Psi(L_-) = z\\Psi(L_0)$, the paper connects the helicity-driven phase picture to the HYPM POLYNOMIAL framework and aligns with a hydrodynamical interpretation $P \\sim t^{\\mathcal{H}}$. Overall, the authors provide a self-contained, geometrically flavored quantization perspective on knot polynomials that complements the Witten-style approach and highlights deep links between symplectic geometry, Maslov theory, and knot theory.
Abstract
In this Letter we propose a semiclassical interpretation of the HOMFLYPT polynomial building on the Liu-Ricca hydrodynamical approach to the latter and on the Besana-S. symplectic approach to framing via Brylinski's manifold of mildly singular links.