  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
$P\times I$
