# Subfactors and Knots

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*Vaughan F. R. Jones*

A co-publication of the AMS and CBMS

This book is based on a set of lectures presented by the author at the NSF-CBMS Regional Conference, Applications of Operator Algebras to Knot Theory and Mathematical Physics, held at the U.S. Naval Academy in Annapolis in June 1988. The audience consisted of low-dimensional topologists and operator algebraists, so the speaker attempted to make the material comprehensible to both groups. He provides an extensive introduction to the theory of von Neumann algebras and to knot theory and braid groups. The presentation follows the historical development of the theory of subfactors and the ensuing applications to knot theory, including full proofs of some of the major results. The author treats in detail the Homfly and Kauffman polynomials, introduces statistical mechanical methods on knot diagrams, and attempts an analogy with conformal field theory. Written by one of the foremost mathematicians of the day, this book will give readers an appreciation of the unexpected interconnections between different parts of mathematics and physics.

#### Reviews & Endorsements

This book is a set of lectures by the author on von Neumann algebras, braid groups and invariants of knots and links. Since the author is the person who discovered (in 1984) the extraordinary connections among these subjects, it is delightful to have an exposition of these ideas from him … a unique resource.

-- Mathematical Reviews

Even a superficial perusal of the book will teach something. Belongs on every mathematician's shelf.

-- The Bulletin of Mathematics Books and Computer Software

The author includes an introduction, in his refreshing and very clear style, to the basics of each of these subjects, making these notes very readable also for non-specialists … These notes are a must for anybody who wants to familiarize himself with this fascinating circle of ideas. But also for the expert they give new insights and present many ideas more clearly than the original articles.

-- Zentralblatt MATH

Although these lectures were delivered 9 years ago and their subject matter has undergone a very rapid development since then, their informal style, freshness of presentation and unique personal insights on the surprising interconnections between apparently different topics in mathematics and mathematical physics continue to make them compulsory and highly rewarding reading. No reader wishing to gain insight into one of the most exciting mathematical developments of the past 15 years could do better than go straight to these lectures by the master (VFRJ) for inspiration.

-- Monatshefte für Mathematik

#### Table of Contents

# Table of Contents

## Subfactors and Knots

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Introduction xi12 free
- Lecture 1. von Neumann Algebras 114 free
- 1.1. Three topologies on B(H) 114
- 1.2. von Neumann's bicommutant theorem 215
- 1.3. (Concrete) von Neumann algebras 215
- 1.4. Factors 316
- 1.5. Examples of factors 417
- 1.6. Comparison of projections 619
- 1.7. Types I, II[sub(1)],III[sub(∞)], and III factors 720
- 1.8. Standard form for II[sub(1)] factors 720
- 1.9. The fundamental group of a II[sub(1)] factor 821
- 1.10. Type III factors 922
- 1.11. Hyperfiniteness 1023

- Lecture 2. Group Actions and Subfactors 1326
- 2.1. The coupling constant 1326
- 2.2. Galois theory for finite group actions 1427
- 2.3. Connes' results on automorphisms of R 1528
- 2.4. Extensions of Connes' automorphism results 1629
- 2.5. Index for subfactors 1730
- 2.6. The basic construction 1831
- 2.7. The basic construction in finite dimensions 1932
- 2.8. Two basic constructions, proof of Goldman's theorem 2033

- Lecture 3. Values of the Index, Virasoro Algebra 2336
- 3.1. Values of the index 2336
- 3.2. The Virasoro unitarity result 2437
- 3.3. The continuous series for subfactors 2538
- 3.4. Iterating the basic construction: the e[sub(i)] algebra 2639
- 3.5. Combinatorics of the e[sub(i)]'s 2639
- 3.6. The e[sub(i)] algebra is a II[sub(1)] factor 2740
- 3.7. The element e[sub(1)] V e[sub(2)] V . . . V e[sub(n)], the values 4 cos[sup(2)] π/n 2841
- 3.8. Ghosts 3043

- Lecture 4. Construction of Examples, Further Structure 3346
- 4.1. The discrete series of subfactors 3346
- 4.2. Bratteli diagrams of the e[sub(i)] algebras 3649
- 4.3. Affine Lie algebra 3851
- 4.4.Realizing the Virasoro discrete series 3952
- 4.5.The tower of relative commutants 4053
- 4.6.Examples of towers of relative commutants 4255
- 4.7. The relative commutant problem 4356

- Lecture 5. The Braid Group and Its Representations 4558
- 5.1. Definition and presentation 4558
- 5.2. Action of the braid group on the free group 4760
- 5.3. The pure braid group and the inductive structure of the braid groups 4760
- 5.4. Burau and Gassner representations 4861
- 5.5. Representations in the e[sub(i)] algebras 5063
- 5.6. Representations in the Pimsner-Popa-Temperley-Lieb algebra (PPTL) 5265
- 5.7. QISM representations of the braid group 5366
- 5.8. The Potts model and Gaussian representations 5568
- 5.9. More representations 5669

- Lecture 6. Knots and Links 5972
- 6.1. Knots and links 5972
- 6.2. The fundamental group and the Alexander module 6073
- 6.3. Seifert surfaces 6275
- 6.4. Seifert matrices, S-equivalence 6376
- 6.5. Untwisted doubles of knots have trivial Alexander module 6578
- 6.6. Skein relation for the Alexander polynomial 6679
- 6.7. Closed braids and the Burau representation 6780

- Lecture 7. The Knot Polynomial V[sub(L)] 6982
- 7.1. First definition of V[sub(L)] 6982
- 7.2. The theory of plats 7083
- 7.3. A second definition of V[sub(L)], the plat approach 7184
- 7.4. Kauffman's e[sub(i)] diagrammatics 7285
- 7.5. Skein relation, third definition of V[sub(L)] 7386
- 7.6. The skein polynomial, inductive definition 7487
- 7.7. The Kauffman polynomial 7588
- 7.8. Kauffman's "states model", fourth and best definition of V[sub(L)] 7689

- Lecture 8. Knots and Statistical Mechanics 7992
- 8.1. Statistical mechanics formalism 7992
- 8.2. Ising, Potts, Vertex, Spin, and IRF models 7992
- 8.3. Transfer matrices 8295
- 8.4. The six-vertex model, Temperley-Lieb equivalence 8396
- 8.5. Commuting transfer matrices, the Yang-Baxter equation 8598
- 8.6. Vertex models on link diagrams 87100
- 8.7. Spin models on link diagrams 89102

- Lecture 9. The Algebraic Approach 93106
- 9.1. The Hecke algebra 93106
- 9.2. The relationship between the e[sub(i)], algebra and H(q, n) 94107
- 9.3. Ocneanu's trace on H(q, n) 95108
- 9.4. Positivity considerations and subfactors from the Hecke algebra 96109
- 9.5. The Birman-Murakami-Wenzl algebra 98111
- 9.6. The Markov trace on the BMW algebra 100113
- 9.7. Structure of the BMW algebra 101114
- 9.8. Wenzl's result on Brauer's centralizer algebra 102115
- 9.9. Quantum invariant theory 103116

- Appendix 105118
- References 107120
- Back Cover Back Cover1129