B–type Catalan States of Lattice Crossing

dc.contributor.advisorDabkowski, Mieczyslaw K.
dc.contributor.advisorZheng, Jie
dc.contributor.committeeMemberTran, Ahn
dc.contributor.committeeMemberHooshyar, M. Ali
dc.contributor.committeeMemberRamakrishna, Viswanath
dc.contributor.committeeMemberDragovic, Vladimir
dc.creatorRakotomalala, Diarisoa Mihaja Andriamanisa
dc.date.accessioned2023-08-24T14:46:37Z
dc.date.available2023-08-24T14:46:37Z
dc.date.created2021-08
dc.date.issued2021-08-01T05:00:00.000Z
dc.date.submittedAugust 2021
dc.date.updated2023-08-24T14:46:39Z
dc.description.abstractM. 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).
dc.format.mimetypeapplication/pdf
dc.identifier.uri
dc.identifier.urihttps://hdl.handle.net/10735.1/9789
dc.language.isoen
dc.subjectMathematics
dc.titleB–type Catalan States of Lattice Crossing
dc.typeThesis
dc.type.materialtext
thesis.degree.collegeSchool of Natural Sciences and Mathematics
thesis.degree.departmentMathematics
thesis.degree.grantorThe University of Texas at Dallas
thesis.degree.namePHD

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