  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.
