Catalan States of Lattice Crossing

For a Lattice crossing \(L\left( m,n\right) \) we show which Catalan connection between \(2\left( m+n\right) \) points on boundary of \(m\times n\) rectangle \(P\) can be realized as a Kauffman state and we give an explicit formula for the number of such Catalan connections. For the case of a Catalan connection with no arc starting and ending on the same side of the tangle, we find a closed formula for its coefficient in the Relative Kauffman Bracket Skein Module of \(P\times I\)