Dissertation/Thesis Abstract

Knot Theory and Algebraic Structures Motivated by and Applied to Knots
by Wang, Xiao, Ph.D., The George Washington University, 2018, 171; 10810197
Abstract (Summary)

This dissertation consists of four parts. In the first part we prove that two different types of set-theoretic Yang-Baxter homology theories lead to the same homology. In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang-Baxter operators (we will call it the “algebraic” version in this thesis). In 2012, Lebed [Leb] and Przytycki [Prz4] independently defined another homology theory for pre-Yang-Baxter operators which has a nice graphic visualization(we will call it the “graphic” version in this thesis). We show that they are equivalent when restricting to set-theoretic Yang-Baxter operators. The “graphic” definition have both one-term and two-term homology comparing to the one-term and two-term rack homology respectively. In the two-term case, we found torsion in homology of the Yang-Baxter operator that yields the Jones polynomial.

In the second part of the thesis, we focus on torsion in Khovanov Homology. Khovanov homology is a powerful link invariant that categorifies the Jones polynomial. Khovanov homology has been computed for many links, and computation results show abundance of Z2-torsion, however, torsion of orders other than two appears rare. We study Khovanov homology of twist deformations of torus links and find counterexamples to the PS braid conjecture [PS] by our method. Moreover, we provide some examples showing that the Khovanov homology of the flat 2-cabling of a given knot may have interesting torsion subgroups. For example, we found Z2r for 0 ≤ r ≤ 23We find the possibility to obtain information of Khovanov homology of more general links via Hochschild homology based on the relation discovered by Przytycki [Prz2] between Hochschild homology of the Frobenius algebra Z[x]/(x2) and Khovanov homology of (2,–n)–torus links.

In the third part, we study the Kauffman bracket skein module and algebra of the thickened sphere with four holes. Frohman and Gelca established a complete description of the multiplicative operation, leading to the famous product-to-sum formula, see [FG] for detail. We study the multiplicative structure of the Kauffman bracket skein algebra of the sphere with four holes. Namely, we present an algorithm allowing us to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves.

In the last part, we introduce our work on the plucking polynomial of rooted trees. Motivation of this work comes from Kauffman bracket skein module. When we try to understand (m × n)–lattice crossings (generalized crossings) in the Kauffman bracket skein module, the plucking polynomial in variable q is closely related to the coefficient of the Catalan state of the lattice; from this one can construct the rooted tree. We classify rooted trees which have strictly unimodal plucking polynomials. We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova [Pak-Pan] on strict unimodality of q-binomial coefficients. We discuss which polynomials can be realized as plucking polynomials and when two rooted trees have the same plucking polynomial.

Indexing (document details)
Advisor: Przytycki, Jozef H.
Commitee: Kania-Bartoszynska, Joanna, Przytycki, Jozef H., Rong, Yongwu, Shumakovitch, Alexander, Wu, Hao, Zhao, Yangxiang
School: The George Washington University
Department: Mathematics
School Location: United States -- District of Columbia
Source: DAI-B 79/12(E), Dissertation Abstracts International
Subjects: Mathematics
Keywords: Jones polynomial, Khovanov homology, Knot theory, Plucking polynomials, Skein modules and algebras, Yang-Baxter homology
Publication Number: 10810197
ISBN: 978-0-438-26707-7
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