Quantum Topology Via Modified Traces From Semisimple to Nonsemisimple

This book presents a unified framework for quantum topology, demonstrating how the classical semisimple theory extends naturally to the non-semisimple setting through the use of modified traces. It offers a rigorous, systematic treatment of both the foundational theory and the major advances that have shaped the field. The exposition introduces each classical semisimple construction before developing its non-semisimple counterpart. In most cases, the proofs in the non-semisimple setting recover the semisimple theory as a special case, revealing the conceptual unity of the two frameworks and underscoring the naturality of the non-semisimple generalizations. The book provides complete definitions and constructions of most of the principal quantum invariants developed over the past three decades, including the Reshetikhin-Turaev, Turaev-Viro, Hennings, and involutive Kuperberg invariants. By placing these theories within a single coherent framework, it delivers a comprehensive and unified perspective on modern quantum topology. Written for graduate students and researchers with backgrounds in basic category theory and low-dimensional topology, this volume serves both as an accessible introduction to semisimple and non-semisimple techniques and as an authoritative reference for current developments in the field.