Coefficients of Catalan States of Lattice Crossing I: $\Theta_{A}$-state Expansion

TL;DR

This paper introduces a new method using $ heta_A$-state expansion to compute coefficients of Catalan states in lattice crossings, providing an algorithm and examples, including a non-unimodal coefficient case.

Contribution

It presents a novel $ heta_A$-state expansion technique for calculating Catalan state coefficients, extending previous polynomial methods and offering an explicit algorithm.

Findings

$ heta_A$-state expansion expresses coefficients as linear combinations over $Q(A)$.

An algorithm for $ heta_A$-state expansion is developed and demonstrated.

An example of a Catalan state with a non-unimodal coefficient is provided.

Abstract

Plucking polynomial for plane rooted trees was introduced by J.H. Przytycki in 2014. As it was shown later, this polynomial can be used to find coefficients $C(A)$ of Catalan states $C$ of $m \times n$-lattice crossing $L(m,n)$ without returns on one side. In this paper, we show that $C(A)$ for any $C$ can be found by using $\Theta_{A}$-state expansion which represents $C(A)$ as a linear combination of coefficients of Catalan states with no top returns over $\mathbb{Q}(A)$. We also provide an algorithm for finding $\Theta_{A}$-state expansions and examples of its applications. Finally, an example of a Catalan state with non-unimodal coefficient is given.