Catalan States of Lattice Crossing

TL;DR

This paper characterizes which Catalan connections in lattice crossings can be realized as Kauffman states and provides explicit formulas for counting and coefficients of these connections.

Contribution

It offers a complete characterization and explicit formulas for Catalan states in lattice crossings, advancing understanding of their realizability and algebraic coefficients.

Findings

Identifies realizable Catalan connections as Kauffman states.

Provides explicit counting formulas for Catalan connections.

Derives closed-form coefficients in the skein module for certain Catalan states.

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 $P\times I$