Catalan States of Lattice Crossing: Application of Plucking Polynomial

TL;DR

This paper establishes a method to compute coefficients of Catalan states in lattice crossings using plucking polynomials and Gaussian polynomials, revealing unimodal coefficient sequences and advancing understanding of skein module expansions.

Contribution

It introduces a novel application of plucking polynomials to determine coefficients in skein module expansions for Catalan states with specific boundary conditions.

Findings

Coefficients can be computed via plucking polynomials of rooted trees.

For states with returns on one side, coefficients are products of Gaussian polynomials.

Coefficient sequences are proven to be unimodal.

Abstract

For a Catalan state $C$ of a lattice crossing $L\left( m,n\right) $ with no returns on one side, we find its coefficient $C\left( A\right) $ in the Relative Kauffman Bracket Skein Module expansion of $L\left( m,n\right) $. We show, in particular, that $C\left( A\right) $ can be found using the plucking polynomial of a rooted tree with a delay function associated to $C$. Furthermore, for $C$ with returns on one side only, we prove that $C\left( A\right) $ is a product of Gaussian polynomials, and its coefficients form a unimodal sequence.