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Functorial Knot Theory cover

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.


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

Readership: Mathematicians and theoretical physicists.

Free Access
FRONT MATTER
  • Pages:i–xii

https://doi.org/10.1142/9789812810465_fmatter

No Access
Introduction I
  • Pages:7–9

https://doi.org/10.1142/9789812810465_0001

Knots and Categories


No Access
Basic Concepts
  • Pages:13–38

https://doi.org/10.1142/9789812810465_0002

No Access
Monoidal Categories, Functors and Natural Transformations
  • Pages:39–59

https://doi.org/10.1142/9789812810465_0003

No Access
A Digression on Algebras
  • Pages:61–70

https://doi.org/10.1142/9789812810465_0004

No Access
More About Monoidal Categories
  • Pages:71–82

https://doi.org/10.1142/9789812810465_0005

No Access
Knot Polynomials
  • Pages:83–85

https://doi.org/10.1142/9789812810465_0006

No Access
Categories of Tangles
  • Pages:87–95

https://doi.org/10.1142/9789812810465_0007

No Access
Smooth Tangles and PL Tangles
  • Pages:97–115

https://doi.org/10.1142/9789812810465_0008

No Access
Shum's Theorem
  • Pages:117–130

https://doi.org/10.1142/9789812810465_0009

No Access
A Little Enriched Category Theory
  • Pages:131–137

https://doi.org/10.1142/9789812810465_0010

Deformations


No Access
Introduction II
  • Pages:141–142

https://doi.org/10.1142/9789812810465_0011

No Access
Definitions
  • Pages:143–147

https://doi.org/10.1142/9789812810465_0012

No Access
Deformation Complexes of Semigroupal Categories and Functors
  • Pages:149–152

https://doi.org/10.1142/9789812810465_0013

No Access
Some Useful Cochain Maps
  • Pages:153–154

https://doi.org/10.1142/9789812810465_0014

No Access
First Order Deformations
  • Pages:155–158

https://doi.org/10.1142/9789812810465_0015

No Access
Obstructions and the Cup Product and Pre-Lie Structures on X(F)
  • Pages:159–173

https://doi.org/10.1142/9789812810465_0016

No Access
Units
  • Pages:175–179

https://doi.org/10.1142/9789812810465_0017

No Access
Extrinsic Deformations of Monoidal Categories
  • Pages:181–183

https://doi.org/10.1142/9789812810465_0018

No Access
Vassiliev Invariants, Framed and Unframed
  • Pages:185–200

https://doi.org/10.1142/9789812810465_0019

No Access
Vassiliev Theory in Characteristic 2
  • Pages:201–207

https://doi.org/10.1142/9789812810465_0020

No Access
Categorical Deformations as Proper Generalizations of Classical Notions
  • Pages:209–212

https://doi.org/10.1142/9789812810465_0021

No Access
Open Questions
  • Pages:213–217

https://doi.org/10.1142/9789812810465_0022

Free Access
BACK MATTER
  • Pages:219–230

https://doi.org/10.1142/9789812810465_bmatter

“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.”
Mathematical Reviews, 2002

“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.”
Mathematics Abstracts, 2002