Braids and Knots ................................................ 1
Patrick D. Bangert
1 Physical Knots and Braids: A History and Overview ............ 2
2 Braids and the Braid Group ................................... 4
2.1 The Topological Idea .................................... 4
2.2 The Origin of Braid Theory .............................. 5
2.3 The Topological Braid ................................... 9
2.4 The Braid Group ........................................ 12
2.5 Other Presentations of the Braid Group ................. 16
2.6 The Alexander and Jones Polynomials .................... 18
2.7 Properties of the Braid Group .......................... 21
2.8 Algorithmic Problems in the Braid Groups ............... 22
3 Braids and Knots ............................................ 24
3.1 Notation for Knots ..................................... 24
3.2 Braids to Knots ........................................ 28
3.3 Example: The Torus Knots ............................... 28
3.4 Knots to Braids I: The Vogel Method .................... 29
3.5 Knots to Braids II: An Axis for the Universal
Polyhedron ............................................. 31
3.6 Peripheral Group Systems of Closed Braids .............. 39
4 Classification of Braids and Knots .......................... 45
4.1 The Word Problem I: Garside's Solution ................. 45
4.2 The Word Problem II: Rewriting Systems ................. 47
4.3 The Conjugacy Problem I: Garside's Solution ............ 52
4.4 The Conjugacy Problem II: Rewriting Systems ............ 54
4.5 Markov's Theorem ....................................... 59
4.6 The Minimal Word Problem ............................... 62
5 Open Problems ............................................... 69
References .................................................. 70
Topological Quantities: Calculating Winding, Writhing,
Linking, and Higher Order Invariants ........................... 75
Mitchell A. Berger (CIME Lecturer)
1 Introduction ................................................ 75
2 Winding Numbers ............................................. 77
2.1 Two Braided Curves between Parallel Planes ............. 77
2.2 General Curves ......................................... 78
2.3 Topological Invariance ................................. 81
3 Linking Numbers ............................................. 82
3.1 Winding Number Derivation .............................. 82
3.2 General Properties ..................................... 83
4 Twist and Writhe Numbers .................................... 84
4.1 Ribbons ................................................ 84
4.2 Twisted Tubes .......................................... 86
5 Writhe from Winding Numbers ................................. 87
5.1 The Twist as a Function of Height ...................... 88
5.2 The Local Winding Number as a Function of Height ....... 89
5.3 The Local Writhe as a Function of Height ............... 90
5.4 The Nonlocal Winding Number as a Function of Height .... 91
5.5 Example: A Trefoil Torus Knot .......................... 92
6 Writhe for Open Curves ...................................... 94
7 Higher Order Winding ........................................ 96
References .................................................. 97
Tangles, Rational Knots and DNA ................................ 99
Louis H. Kauffman (CIME Lecturer) and Sofia Lambropoulou
1 Introduction ................................................ 99
2 2-Tangles and Rational Tangles ............................. 102
3 Continued Fractions and the Classification of Rational
Tangles .................................................... 107
4 Alternate Definitions of the Tangle Fraction ............... 111
4.1 F(T) Through the Bracket Polynomial ................... 111
4.2 The Fraction through Coloring ......................... 119
4.3 The Fraction through Conductance ...................... 121
5 The Classification of Unoriented Rational Knots ............ 122
6 Rational Knots and Their Mirror Images ..................... 126
7 The Oriented Case .......................................... 127
8 Strongly Invertible Links .................................. 131
9 Applications to the Topology of DNA ........................ 132
References ................................................. 136
The Group and Hamiltonian Descriptions of Hydrodynamical
Systems ....................................................... 139
Boris Khesin (CIME Lecturer)
1 Introduction ............................................... 139
2 Euler Equations and Geodesics .............................. 140
2.1 The Euler Equation of an Ideal Incompressible Fluid ... 140
2.2 Geodesics on Lie Groups ............................... 141
2.3 Geodesic Description for Various Equations ............ 142
3 Euler Equations on Groups as Hamiltonian Systems and the
Binormal Equation .......................................... 142
3.1 Hamiltonian Reformulation of the Euler Equations ...... 142
3.2 Hamiltonian Structure of the Landau-Lifschitz
Equation .............................................. 143
3.3 Properties of the Binormal Equation ................... 145
4 The KdV-Type Equations as Euler Equations .................. 147
4.1 The Virasoro Algebra and the KdV Equation ............. 147
4.2 Similar Equations and Conservation Laws ............... 149
5 Hamiltonian Structure of the Euler Equations for an
Incompressible Fluid ....................................... 150
5.1 The Euler Hydrodynamics as a Hamiltonian Equation ..... 150
5.2 The Space of Knots and the Dual of the Lie Algebra
of Divergence-Free Vector Fields ...................... 153
References ................................................. 154
Singularities in Fluid Dynamics and their Resolution .......... 157
H.K. Moffatt (CIME Lecturer)
1 Introduction ............................................... 157
2 Boundary-Driven Singularities .............................. 158
3 Cusp Singularities at a Free Surface ....................... 160
4 A Simple Finite-Time Singularity: the Euler Disk ........... 161
5 Finite-Time Singularities at Interior Points ............... 162
References ................................................. 166
Structural Complexity and Dynamical Systems ................... 167
Renzo L. Ricca (School Director and CIME Lecturer)
1 Introduction ............................................... 167
2 Helmholtz's Work on Vortex Motion: Birth of Topological
Fluid Mechanics ............................................ 168
2.1 Multi-Valued Potentials in Multiply Connected
Regions ............................................... 168
2.2 Green's Theorem in Multiply Connected Regions ......... 172
2.3 Conservation Laws ..................................... 172
3 Measures of Structural Complexity .......................... 173
3.1 Dynamical Systems and Vector Field Analysis ........... 174
3.2 Measures of Tangle Complexity ......................... 175
4 Topological Bounds on Energy and Helicity-Crossing Number
Relations for Magnetic Knots and Links ..................... 181
4.1 Topology Bounds Energy in Ideal Fluid ................. 182
4.2 Helicity-Crossing Number Relations in Dissipative
Fluid ................................................. 184
References ................................................. 185
Random Knotting: Theorems, Simulations and Applications ....... 187
De Witt Sumners (CIME Lecturer)
1 Introduction ............................................... 187
2 The Frisch-Wasserman-Delbruck Conjecture ................... 189
3 Entanglement Complexity of Random Knots and Random Ares .... 194
4 Writhe, Signature and Chirality of Random Knots ............ 196
5 Application of Random Knotting to Viral DNA Packing ........ 201
5.1 Knot Type Probabilities for P4 DNA in Free Solution ... 204
5.2 Monte Carlo Simulation ................................ 205
5.3 Results and Discussion Knot Complexity of DNA
Molecules Extracted from Phage P4 ..................... 206
5.4 Identification of Specific Knot Types by Their
Location on the Gel ................................... 209
5.5 Monte Carlo Simulations of Random Knot Distributions
in Confined Volumes ................................... 209
References ................................................. 213
Index ......................................................... 219
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