Catalan states of lattice crossing: An application of plucking polynomial

For a Catalan state C of a lattice crossing L(m,n) with no returns on one side, we find its coefficient C(A) in the Relative Kauffman Bracket Skein Module expansion of L(m,n). We show, in particular, that C(A) can be found using the plucking polynomial of a rooted tree with a delay function associat...

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Bibliographic Details
Published in:Topology and its applications 2019-03, Vol.254, p.12-28
Main Authors: Dabkowski, Mieczyslaw K., Przytycki, Jozef H.
Format: Article
Language:English
Subjects:
Catalan states
Gaussian polynomial
Kauffman bracket
Knot
Lattice crossing
Link
Plucking polynomial
Rooted tree
Skein module
Unimodal polynomial
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Summary:For a Catalan state C of a lattice crossing L(m,n) with no returns on one side, we find its coefficient C(A) in the Relative Kauffman Bracket Skein Module expansion of L(m,n). We show, in particular, that C(A) 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(A) is a product of Gaussian polynomials, and its coefficients form a unimodal sequence.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2018.12.003