Contents Preface ix Acknowledgments xiii Chapter 1. Temperley-Lieb-Jones Theories 1 1.1. Generic Temperley-Lieb-Jones algebroids 1 1.2. Jones algebroids 13 1.3. Yang-Lee theory 16 1.4. Unitarity 17 1.5. Ising and Fibonacci theory 19 1.6. Yamada and chromatic polynomials 22 1.7. Yang-Baxter equation 22 Chapter 2. Quantum Circuit Model 25 2.1. Quantum framework 26 2.2. Qubits 27 2.3. n-qubits and computing problems 29 2.4. Universal gate set 29 2.5. Quantum circuit model 32 2.6. Simulating quantum physics 32 Chapter 3. Approximation of the Jones Polynomial 35 3.1. Jones evaluation as a computing problem 35 3.2. FP#P-completeness of Jones evaluation 36 3.3. Quantum approximation 37 3.4. Distribution of Jones evaluations 39 Chapter 4. Ribbon Fusion Categories 41 4.1. Fusion rules and fusion categories 41 4.2. Graphical calculus of RFCs 44 4.3. Unitary fusion categories 49 4.4. Link and 3-manifold invariants 49 4.5. Frobenius-Schur indicators 51 4.6. Modular tensor categories 53 4.7. Classification of MTCs 55 Chapter 5. (2+1)-TQFTs 57 5.1. Quantum field theory 58 5.2. Witten-Chern-Simons theories 60 5.3. Framing anomaly 61 5.4. Axioms for TQFTs 61 vii
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