Catalan states of lattice crossing

For a lattice crossing L(m,n) we show which Catalan connection between 2(m+n) points on the boundary of m×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...

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Bibliographic Details
Published in:Topology and its applications 2015-03, Vol.182, p.1-15
Main Authors: Dabkowski, Mieczyslaw K., Li, Changsong, Przytycki, Jozef H.
Format: Article
Language:English
Subjects:
Catalan tangles
Kauffman bracket
Knot
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Summary:For a lattice crossing L(m,n) we show which Catalan connection between 2(m+n) points on the boundary of m×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×I.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2014.11.015