Catalan States of Lattice Crossing: Application of Plucking Polynomial

For a Catalan state $C$ of a lattice crossing $L(m,n)$ with no returns on one side, we find its coefficient $C\left(A\right)$ in the Relative Kauffman Bracket Skein Module expansion of $L(m,n)$. We show, in particular, that $C\left(A\right)$ can be found using the plucking polynomial of a rooted tree with a delay function associated to $C$. Furthermore, for $C$ with returns on one side only, we prove that $C\left( A\right) $ is a product of Gaussian polynomials, and its coefficients form a unimodal sequence.