Catalan states of lattice crossing

For a lattice crossing L(m,n) we show which Catalan connection between 2(m+n) points on the boundary of m×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×I.