Quantum topology without topology

TL;DR

This paper explores how categorical algebra provides a framework for understanding quantum invariants, revealing deep connections between topology, algebra, number theory, and quantum physics without relying on traditional topological methods.

Contribution

It introduces a categorical algebra approach to quantum invariants, illustrating their emergence and connections across multiple mathematical and physical disciplines.

Findings

Quantum invariants connect topology, algebra, and physics.

Categorical algebra offers a natural framework for quantum invariants.

Deep links to representation theory and quantum groups are demonstrated.

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.