Coefficients of Catalan States of Lattice Crossing II: Applications of $\Theta_{A}$-state Expansions

TL;DR

This paper explores the properties and factorization of coefficients of Catalan states derived from lattice crossings, utilizing $ heta_A$-state expansions and plucking polynomials to improve computation methods and derive closed-form formulas.

Contribution

It introduces new factorization properties of coefficients of Catalan states using $ heta_A$-state expansions and plucking polynomials, leading to more efficient computation methods.

Findings

Coefficients $C(A)$ factor under certain conditions.

Derived closed-form formulas for coefficients of Catalan states in $L(m,3)$.

Provided a more efficient computational approach for these coefficients.

Abstract

Plucking polynomial of a plane rooted tree with a delay function $\alpha$ was introduced in 2014 by J.H.~Przytycki. As shown in this paper, plucking polynomial factors when $\alpha$ satisfies additional conditions. We use this result and $\Theta_{A}$-state expansion introduced in our previous work to derive new properties of coefficients $C(A)$ of Catalan states $C$ resulting from an $m \times n$-lattice crossing $L(m,n)$. In particular, we show that $C(A)$ factors when $C$ has arcs with some special properties. In many instances, this yields a more efficient way for computing $C(A)$. As an application, we give closed-form formulas for coefficients of Catalan states of $L(m,3)$.