Higher-genus quantum curves for exponential curves

Construct quantum curves for higher-genus exponential spectral curves, particularly those related to knot theory and whose traditional parametrizations coincide with knot A-polynomials, and determine whether the corresponding quantizations exist.

Background

The paper's quantum-curve constructions are restricted to genus-zero curves. Many exponential curves of interest in knot theory have higher genus, and the paper observes that its approximation theorem could in principle provide a route to quantization by approximating them with algebraic curves.

For traditional parametrizations of curves associated with knot A-polynomials, the relevant quantizations are explicitly stated not to be known. Thus both the implementation of the limiting construction and the existence of the desired quantizations remain unresolved.