Contents

Introduction 1

Chapter 1. The Definition and Some Basic Properties of the Algebra H[rr] 15

Chapter 2. A Decomposition of the Algebra H[rr] when

~ E

k 21

Chapter 3. Structure of the Algebra H[rr] for Two-Generator Groups 29

Chapter 4. Absolutely Irreducible S£(2) Representations of Two-Generator

Groups 39

Chapter 5. Further Identities in the Algebra H[rr] when

~

E k 47

Chapter 6. Structure of

H+[rrn]

for Free Groups

1fn

59

Chapter 7. Quaternion Algebra Localizations of H[rr] and Absolutely Irreducible

S£(2) Representations 69

Chapter 8. Algebro-Geometric Interpretation of S£(2) Representations of

a~~ ~

Chapter 9. The Universal Matrix Representation of the Algebra H[rr] 89

Chapter 10. Some Knot Invariants Derived from the Algebra H[rr]

Appendix A. Addenda

1*. Addendum to Chapter 1

2*. Addendum to Chapter 2

3*. Addendum to Chapter 3

4*. Addendum to Chapter 4

5*. Addendum to Chapter 5

6*. Addendum to Chapter 6

7*. Addendum to Chapter 7

8*. Addendum to Chapter 8

9*. Addendum to Chapter 9

Appendix B. Afterword

Bibliography

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