Odd Khovanov homology : higher representation theory and higher rewriting theory
(2024)
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- Authors
- Supervisors
- Vaz, Pedro
- Abstract
- Quantum topology studies the interplay between low-dimensional topology and representation theory; rewriting theory studies algebraic structures via algorithmic means. This thesis is devoted to both fields, and their surprising interconnections. The common thread is the realization that certain commutative theories admit odd (or super) analogues, where the commutative rule only holds up to signs. In topology, the last decades have seen the emergence of powerful homology theories of knots, starting with the pioneer work of Khovanov and his celebrated Khovanov homology. Their study, together with their connections to the representation theory of quantum groups, form the field of modern quantum topology. Surprisingly, Khovanov homology admits an odd analogue, known as odd Khovanov homology. While the former builds upon polynomial algebras, the latter has anti-commutative features. This led to the discovery of similar odd analogues in related fields. Despite these successes, a grasp on odd Khovanov homology has remained elusive. In the first part of this thesis, we relate odd Khovanov homology to higher representation theory, leading to a natural framework for its understanding. The underlying structure of our construction is the categorical analogue of a super algebra. Unfortunately, the increased complexity of working with such structures exhausts the standard toolkit of the higher algebraist. To solve this problem, we develop the rewriting theory of diagrammatic algebras, and use it to prove a basis theorem for our construction. The techniques developed are general and have the potential to be applied to a wide range of situations. The two parts of the thesis can be read independently.
- Affiliations
Citations
Schelstraete, L. (2024). Odd Khovanov homology : higher representation theory and higher rewriting theory. https://hdl.handle.net/2078.5/235428