Catalan States of Lattice Crossing: Application of Plucking Polynomial

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...

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Bibliographic Details
Published in:arXiv.org 2017-11
Main Authors: Dabkowski, Mieczyslaw K, Przytycki, Jozef H
Format: Article
Language:English
Subjects:
Plucking
Polynomials
Online Access:Get full text
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Summary: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.
ISSN:2331-8422