Catalan States of Lattice Crossing

For a Lattice crossing \(L\left( m,n\right) \) we show which Catalan connection between \(2\left( m+n\right) \) points on boundary of \(m\times n\) rectangle \(P\) can be realized as a Kauffman state and we give an explicit formula for the number of such Catalan connections. For the case of a Catala...

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Bibliographic Details
Published in:arXiv.org 2014-09
Main Authors: Dabkowski, Mieczyslaw K, Li, Changsong, Przytycki, Jozef H
Format: Article
Language:English
Online Access:Get full text
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Summary:For a Lattice crossing \(L\left( m,n\right) \) we show which Catalan connection between \(2\left( m+n\right) \) points on boundary of \(m\times n\) rectangle \(P\) can be realized as a Kauffman state and we give an explicit formula for the number of such Catalan connections. For the case of a Catalan connection with no arc starting and ending on the same side of the tangle, we find a closed formula for its coefficient in the Relative Kauffman Bracket Skein Module of \(P\times I\)
ISSN:2331-8422