Poincaré polynomials for skein lasagna modules of CP^2

Determine the Poincaré polynomials of the Khovanov skein lasagna modules S^2_0(CP^2;0;Q) and S^2_0(CP^2;1;Q) as the explicit infinite series specified by Conjecture 6.14: S^2_0(CP^2;0;Q) has polynomial 1 + \sum_{n≥1} t^{−2n^2} q^{6n^2−4n+1} K_{2n}(t,q^2) − 1, and S^2_0(CP^2;1;Q) has polynomial \sum_{n≥1} t^{−2n^2+2n} q^{6n^2−10n+4} K_{2n−1}(t,q^2) − 1, where K_n(t,q) denotes the Poincaré polynomial of Kh(T(n,n−1);Q).

Background

A central computational goal in the paper is to understand S^2_0(CP^2), which plays a key role in constructing induced maps on Khovanov homology for cobordisms in kCP^2 and in embedding obstructions.

Conjecture 6.14 proposes exact closed-form series for the Poincaré polynomials of S^2_0(CP^2;0;Q) and S^2_0(CP^2;1;Q) expressed via the Poincaré polynomials of Khovanov homology for the torus links T(n,n−1). The authors have verified compatibility with known data for small n, and resolving this conjecture would effectively determine S^2_0(CP^2).