Catalan States of Lattice Crossing: Application of Plucking Polynomial

Mieczyslaw K. Dabkowski; Jozef H. Przytycki

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arXiv · 1711.05328

Catalan States of Lattice Crossing: Application of Plucking Polynomial

Mieczyslaw K. Dabkowski·Jozef H. Przytycki

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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.

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BibTeX RIS

Mieczyslaw K. Dabkowski, Jozef H. Przytycki. 2017-11-14. Catalan States of Lattice Crossing: Application of Plucking Polynomial. https://arxiv.org/abs/1711.05328

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