The Jones-Wenzl projectors are particular elements of the Temperley-Lieb algebra essential to the construction of quantum 3-manifold invariants. As a first step toward categorifying quantum 3-manifold invariants, Cooper and Krushkal categorified these projectors. In another direction, Naisse and Putyra gave a categorification of the Temperley-Lieb algebra compatible with odd Khovanov homology, introducing new machinery called grading categories.The first goal of this thesis is to provide a generalization of Naisse and Putyra's work in the spirit of Bar-Natan's canopolies or Jones's planar algebras, replacing grading categories with grading multicategories. From this updated viewpoint, we describe an invariant of diskular tangles from odd Khovanov homology, naturally extending Naisse and Putyra's tangle theory. In this thesis, the main application of our theory for diskular tangles is a proof of the existence and uniqueness of categorified Jones-Wenzl projectors in odd Khovanov homology. These results have a nearly immediate award: the existence of a new, "odd" categorification of the colored Jones polynomial. Finally, a major motivation to develop a tangle theory for odd Khovanov homology is to ultimately determine the state of its functoriality. In forthcoming work by the author, we study this question by approximating Khovanov's argument for the original theory. In doing so, we develop a theory of Hochschild homology for modules and algebras graded by categories, indicating that the new constructions offered by grading categories are also deserving of study.
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