Catalan States of Lattice Crossing

Abstract

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 $P\times I$

12 pages, 18 figures