Quantum topology without topology

Abstract

These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum physics helping to transfer ideas, and stimulating mutual development. They also possess deep and intriguing connections to representation theory, particularly through representations of quantum groups. These lecture notes aim to illustrate how categorical algebra provides a framework for studying both algebra and topology. Specifically, they demonstrate how quantum invariants emerge naturally from a mostly categorical perspective.