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Plucking polynomial of a plane rooted tree with a delay function $\alpha$ wasintroduced in 2014 by J.H.~Przytycki. As shown in this paper, pluckingpolynomial factors when $\alpha$ satisfies additional conditions. We use thisresult and $\Theta_{A}$-state expansion introduced in our previous work toderive new properties of coefficients $C(A)$ of Catalan states $C$ resultingfrom 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 manyinstances, this yields a more efficient way for computing $C(A)$. As anapplication, we give closed-form formulas for coefficients of Catalan states of$L(m,3)$.

Coefficients of Catalan States of Lattice Crossing II: Applications of $\Theta_{A}$-state Expansions