B–type Catalan States of Lattice Crossing

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2021-08-01T05:00:00.000Z

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Abstract

M. K. Dabkowski and J. H. Przytycki defined for any realizable Catalan state C with no bottom returns, the rooted plane tree with a delay function, (TC, f), and the partially ordered set (B(C), 4) of some Kauffman states that realize C. In this dissertation, we study the properties of (B(C), 4) and establish an important relation between its rank generating function and the plucking polynomial of (TC, f). Furthermore, we show that the rank generating function of (B(C), 4) is unimodal for any realizable A–type Catalan state with no bottom returns of an A–type lattice crossing LA(m, n), where n ≤ 4. In the last part of this dissertation, we study B–type Catalan states. We show which crossingless connection between 2(m + n) outer boundary points of an annulus can be realized as Kauffman states of the B–type Lattice crossing LB (m, n). Furthermore, we give a closed-form formula for the number of realizable B–type Catalan states, and find coefficients of those obtained as Kauffman states of LB(m, 1) and LB(m, 2).

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