Coefficients of Catalan states of lattice crossing II: Applications of ΘA-state expansions

Plucking polynomial of a plane rooted tree with a delay function $\alpha$ was introduced in 2014 by Przytycki. As shown in this paper, plucking polynomial factors when $\alpha$ satisfies additional conditions. We use this result and ${\mathrm{\Theta }}_A$-state expansion introduced in our previous work to derive new properties of coefficients $C(A)$ of Catalan states $C$ resulting from an $(m\times n)$-lattice crossing $L(m,n)$. In particular, we show that $C(A)$ factors when $C$ has arcs with some special properties. In many instances, this yields a more efficient way for computing $C(A)$. As an application, we give closed-form formulas for coefficients of Catalan states of $L(m,3)$.