Catalan States of Lattice Crossing: Application of Plucking Polynomial

For a Catalan state \(C\) of a lattice crossing \(L\left( m,n\right) \) with no returns on one side, we find its coefficient \(C\left( A\right) \) in the Relative Kauffman Bracket Skein Module expansion of \(L\left( m,n\right) \). 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.