Graduation Year

2019

Document Type

Dissertation

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Mohamed Elhamdadi, Ph.D.

Co-Major Professor

Masahico Saito, Ph.D.

Committee Member

Dmytro Savchuk, Ph.D.

Committee Member

Boris Shekhtman, Ph.D.

Keywords

Quandle, Rack, Cohomology, Yang-Baxter Equation, Symmetric Monoidal Category, Self-distributive

Abstract

In this dissertation we investigate self-distributive algebraic structures and their cohomologies, and study their relation to topological problems in knot theory. Self-distributivity is known to be a set-theoretic version of the Yang-Baxter equation (corresponding to Reidemeister move III) and is therefore suitable for producing invariants of knots and knotted surfaces. We explore three different instances of this situation. The main results of this dissertation can be, very concisely, described as follows. We introduce a cohomology theory of topological quandles and determine a class of topological quandles for which the cohomology can be computed, at least in principle, by means of the cohomology groups of smaller and discrete quandles. We utilize a diagrammatic description of higher self-distributive structures in terms of framed links via a functorial procedure called doubling, and generalize previously known (co)homology theories to introduce a cocycle invariant of framed links. Finally, we study a class of ternary self-distributive structures called heaps, and introduce two cohomology theories that classify their extensions. We show that heap cohomology is related to both group cohomology (via a long exact sequence) and ternary self-distributive cohomology (the heap second cohomology group canonically injects into the ternary self-distributive one with modified coefficients). We also develop the theory in the context of symmetric monoidal categories.

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