This book begins with a detailed exposition of the key ideas in the discovery of monoidal categories of tangles as central objects of study in low-dimensional topology. The focus then turns to the deformation theory of monoidal categories and the related deformation theory of monoidal functors, which is a proper generalization of Gerstenhaber's deformation theory of associative algebras. These serve as the building blocks for a deformation theory of braided monoidal categories which gives rise to sequences of Vassiliev invariants of framed links, and clarify their interrelations.

Contents:

• Knots and Categories:
• Monoidal Categories, Functors and Natural Transformations
• A Digression on Algebras
• Knot Polynomials
• Smooth Tangles and PL Tangles
• A Little Enriched Category Theory
• Deformations:
• Deformation Complexes of Semigroupal Categories and Functors
• First Order Deformations
• Units
• Extrinsic Deformations of Monoidal Categories
• Categorical Deformations as Proper Generalizations of Classical Notions
• and other papers

"The book explains clearly and in one place the key ideas concerning categories of tangles and their relations with classical knot theory on one hand, and with knot polynomials and Vassiliev invariants on the other."

"This is a very nicely written book, mostly self-contained. All the definitions from both knot theory and category theory are included, as well as proofs of many basic results which are rarely to be found written down elsewhere. The beautifully simple way in which the author combines topology with category theory makes the book recommended reading for anyone interested in quantum topology."