A cable knot and BPS-series (Q2682663)

A two-variable series invariant \(F_{K}\) for a complement of a knot was studied in [\textit{S. Gukov} and \textit{C. Manolescu}, Quantum Topol. 12, No. 1, 1--109 (2021; Zbl 1497.57015)]. In that paper, Conjecture 1.5 proposed a formula for \(F_{K}\) in terms of the colored Jones polynomial, and Conjecture 1.6 proposed that the quantum \(A\)-polynomial annihilates a normalized \(F_{K}\). In this paper the author computes the \(F_{K}\) series for \((9,2)\)-cabling of the figure eight knot using the two conjectures. For the computations cabling formulas for colored Jones polynomial, Alexander polynomial, and quantum \(A\)-polynomial are utilized.