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

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...

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Bibliographic Details
Published in:arXiv.org 2022-09
Main Authors: Dabkowski, Mieczyslaw K, Wu, Cheyu
Format: Article
Language:English
Subjects:
Coefficients
Plucking
Polynomials
Online Access:Get full text
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Summary: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 \(L(m,3)\).
ISSN:2331-8422