Functorial Knot Theory: Categories Of Tangles, Coherence, Categorical Deformations And Topological Invariants Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants

Almost since the advent of skein-theoretic invariants of knots and links (the Jones, HOMFLY, and Kauffman polynomials), the important role of categories of tangles in the connection between low-dimensional topology and quantum-group theory has been recognized. The rich categorical structures naturally arising from the considerations of cobordisms have suggested functorial views of topological field theory. 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 As part of a series exploring why knot theory serves as a nexus for math, physics, biology, and philosophy, this volume probes the quantum topology links between the pivotal objects of study in low- dimensional geometric topology--including classical knots--and such exotic algebraic objects as Hopf algebras and monoidal categories. Yetter (mathematics, Kansas State U.) covers basic concepts and categories, and provides proofs of some universally assumed "folk theorems," examples of knot diagrams, and 63 references. The section on deformations assumes some familiarity with algebraic deformation theory and homological algebra. c. Book News Inc A study of functorial knot theory. It discusses the key ideas in the discovery of monoidal categories of tangles as central objects in low-dimensional topology. The focus then turns to the deformation theory of monoidal categories and the related deformation theory of monoidal functors. An exposition of building blocks for deformation theory of braided monoidal categories, giving rise to sequences of Vasseliv invariants of framed links, clarifying the interrelations between them. David N. Yetter. Includes Bibliographical References (p. 219-224) And Index.