Polynomial Jones Bounds on Hyperbolic Volume and Link Surfaces — E8 Intelligence Research

FINDING: Tsvietkova and Kalfagianni independently establish polynomial (Jones) bounds on hyperbolic volume and surface counts in knot complements, with Tsvietkova's later work giving explicit exponential-to-polynomial improvements for alternating links. MATH: - Jones polynomial \( V_L(t) \in \mathbb{Z}[t^{1/2}, t^{-1/2}] \), evaluated at \( t = e^{i\pi/3} \) (or \( t = -e^{i\pi/3} \)) yields \( |V_L(e^{i\pi/3})| \) as a volume bound parameter. - Kalfagianni's result: For a knot \( K \) with non-orientable genus \( \gamma(K) \), the Jones polynomial degree span bounds \( \gamma(K) \leq \frac{1}{2}(\deg_{\max} V_K - \deg_{\min} V_K) \). - Tsvietkova–Hass–Thompson (arXiv:2311.08567): For alternating link complements, the number of essential surfaces is bounded by a polynomial in the crossing number \( c \), specifically \( O(c^m) \) for fixed \( m \) (explicit \( m \) derived from Jones polynomial coefficients), replacing prior exponential \( 2^{O(c)} \) bounds. - Volume bound: Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com