Odd Khovanov homology, higher representation theory and higher rewriting theory

TL;DR

This thesis explores the connections between odd Khovanov homology, higher representation theory, and rewriting theory, introducing new categorical frameworks and basis theorems for graded $rak{gl}_2$-foams with broad applicability.

Contribution

It develops a higher representation theoretic approach to odd Khovanov homology and introduces a rewriting theory for higher algebras, including a basis theorem for graded $rak{gl}_2$-foams.

Findings

A graded-2-category of graded $rak{gl}_2$-foams is constructed.

A basis theorem for graded $rak{gl}_2$-foams is established.

Interconnections between quantum topology and rewriting theory are demonstrated.

Abstract

This thesis is devoted to the fields of quantum topology and rewriting theory, and their surprising interconnections. In the first part of the thesis, we develop a higher representation theoretic approach to odd Khovanov homology; this is the content of arXiv:2311.14394. One of the essential ingredients is a certain graded-2-category of graded $\mathfrak{gl}_2$-foams. In the second part of the thesis, we develop a rewriting theory suitable for higher algebras and their super or graded analogues, and use it to show a basis theorem for graded $\mathfrak{gl}_2$-foams. These techniques have the potential to be applied to a wide variety of contexts. Both parts of the thesis can be read independently. Each has its own comprehensive introduction, allowing experts from one field to get acquainted with the other field.