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  • Teaching – Knots
    • Diagrams and Notes
      • Chapter 1: Introduction
        • Chapter 1: Introduction – Notes
          • Chapter 1: Introduction – Trefoil Construction
          • Chapter 1: Introduction – Trefoil
          • Chapter 1: Introduction – Unknot-Unlink-Hopf Link
          • Chapter 1: Introduction – Trefoil Figure Eight
          • Chapter 1: Introduction – One Crossing
          • Chapter 1: Introduction – One Crossing Construction
          • Chapter 1: Introduction – Figure Eight
          • Chapter 1: Introduction – Whitehead
          • Chapter 1: Introduction – Satellite
          • Chapter 1: Introduction – Two Components
      • Chapter 2: Projections
        • Chapter 2: Projections – Notes
        • Chapter 2: Projections – Introduction
        • Chapter 2: Projections – Projections
        • Chapter 2: Projections – Trefoil – Hopf Link – Borromean Rings
        • Chapter 2: Projections – No Crossings
        • Chapter 2: Projections – Jordan Curve Theorem
        • Chapter 2: Projections – Two Crossings
      • Chapter 3: Reidemeister’s Theorem
        • Chapter 3: Reidemeister’s Theorem – Notes
        • Chapter 3: Reidemeister’s Theorem – Reidemeister’s Theorem
        • Chapter 3: Reidemeister’s Theorem – Type I Move
        • Chapter 3: Reidemeister’s Theorem – Type II Move
        • Chapter 3: Reidemeister’s Theorem – Type III Move
      • Chapter 4: Ribbons
        • Chapter 4: Ribbons – Notes
        • Chapter 4: Ribbons – Ribbons
        • Chapter 4: Ribbons – Kink Removal 1
        • Chapter 4: Ribbons – Kink Removal 2
      • Chapter 5: Orientation
        • Chapter 5: Orientation – Notes
          • Chapter 5: Orientation – Archiral
        • Chapter 5: Orientation – Colouring Notions
        • Chapter 5: Orientation – Notes on Chirality and Invertibility
        • Chapter 5: Orientation – Diagrams for Chirality and Invertibility
        • Chapter 5: Orientation – Crossings in Link
        • Chapter 5: Orientation – Isolated Crossings
        • Chapter 5: Orientation – Crossing Orientation
        • Chapter 5: Orientation – Positive Negative Crossings
        • Chapter 5: Orientation – Isolated Positive Negative Labels
        • Chapter 5: Orientation – Positive Negative Labels in Link
        • Chapter 5: Orientation – Hopf Link Orientations
        • Chapter 5: Orientation – Hopf Link Writhe
        • Chapter 5: Orientation – Definitions of Writhe and Linking Number
        • Chapter 5: Orientation – Reidemeister Move II Writhe
        • Chapter 5: Orientation – Reidemeister Move III
        • Chapter 5: Orientation – Reidemeister Move III Writhe
        • Chapter 5: Orientation – Reidemeister Move I Writhe Compensation
        • Chapter 5: Orientation – Hopf Link
        • Chapter 5: Orientation – Whitehead
        • Chapter 5: Orientation – Ribbon Computation
      • Chapter 6: Colouring
        • Chapter 6: Colouring – Notes
        • Chapter 6: Colouring – Tricolour
        • Chapter 6: Colouring – Reidemeister Move I Colour
        • Chapter 6: Colouring – Reidemeister Move II Colour
        • Chapter 6: Colouring – Reidemeister Move II Colour Again
        • Chapter 6: Colouring – n=2 Example
        • Chapter 6: Colouring – n=3 Example
        • Chapter 6: Colouring – Trefoil
        • Chapter 6: Colouring – n-Colourable Trefoil
        • Chapter 6: Colouring – n-Colouring Group
        • Chapter 6: Colouring – Theorem
        • Chapter 6: Colouring – Colouring Invariance
        • Chapter 6: Colouring – Colouring Invariance Again
        • Chapter 6: Colouring – Colouring Invariants
        • Chapter 6: Colouring – Cramer
        • Chapter 6: Colouring – Colouring Group
        • Chapter 6: Colouring – Colouring Group Again
        • Chapter 6: Colouring – Rules for Diagonalising
        • Chapter 6: Colouring – Solomon
        • Chapter 6: Colouring – Solomon Computation
        • Chapter 6: Colouring – Trefoil Colouring
        • Chapter 6: Colouring – Trefoil Colouring Group
      • Chapter 7: Brackets
        • Chapter 7: Brackets – Notes
        • Chapter 7: Brackets – States
        • Chapter 7: Brackets – Trefoil States
        • Chapter 7: Brackets – Trefoil States 2
        • Chapter 7: Brackets – Labelling
        • Chapter 7: Brackets – Labelling Again
        • Chapter 7: Brackets – Trefoil Labelling
        • Chapter 7: Brackets – Not Global
        • Chapter 7: Brackets – States and Bracket Polynomial
        • Chapter 7: Brackets – Unknot
        • Chapter 7: Brackets – One Crossing
        • Chapter 7: Brackets – Trefoil Split Recursion
        • Chapter 7: Brackets – Trefoil 1
        • Chapter 7: Brackets – Trefoil 2
        • Chapter 7: Brackets – Trefoil 3
        • Chapter 7: Brackets – Trefoil 4
        • Chapter 7: Brackets – Trefoil Bracket
        • Chapter 7: Brackets – Examples
        • Chapter 7: Brackets – Hopf Link
        • Chapter 7: Brackets – Two-Crossing Twist
        • Chapter 7: Brackets – Trefoil Recursion
        • Chapter 7: Brackets – Bracket Constraints
        • Chapter 7: Brackets – Brackets and Reidemeister Moves
        • Chapter 7: Brackets – Move II Invariance Proof
        • Chapter 7: Brackets – Move II Invariance Implies Move III Invariance
      • Chapter 8: Jones Polynomial
        • Chapter 8: Jones Polynomial – Notes
        • Chapter 8: Jones Polynomial – Definition
        • Chapter 8: Jones Polynomial – Move I Invariance
        • Chapter 8: Jones Polynomial – Jones Mirror 1
        • Chapter 8: Jones Polynomial – Jones Mirror 2
        • Chapter 8: Jones Polynomial – Jones Mirror 3
        • Chapter 8: Jones Polynomial – Recursion
        • Chapter 8: Jones Polynomial – Properties
        • Chapter 8: Jones Polynomial – Skein Proof
        • Chapter 8: Jones Polynomial – Kink Compensation
        • Chapter 8: Jones Polynomial – Hopf Recursion
        • Chapter 8: Jones Polynomial – Trefoil Recursion
        • Chapter 8: Jones Polynomial – Trefoil Bracket Writhe
        • Chapter 8: Jones Polynomial – Trefoil from Definition
        • Chapter 8: Jones Polynomial – Hopf and Trefoil
        • Chapter 8: Jones Polynomial – Trivial Case
      • Chapter 9: Composites
        • Chapter 9: Composites – Notes
        • Chapter 9: Composites – Sum of Knots
        • Chapter 9: Composites – Composite Knot
        • Chapter 9: Composites – 817 Composed
        • Chapter 9: Composites – Reef Knot, Granny Knot
        • Chapter 9: Composites – Unknot Inverse?
      • Chapter 10: HOMFLY Polynomial
        • Chapter 10: HOMFLY Polynomial – Notes
        • Chapter 10: HOMFLY Polynomial – Definition
        • Chapter 10: HOMFLY Polynomial – Hopf Links
        • Chapter 10: HOMFLY Polynomial – Unknot Union
        • Chapter 10: HOMFLY Polynomial – Multiplicative Properties
        • Chapter 10: HOMFLY Polynomial – HOMFLY-Jones
        • Chapter 10: HOMFLY Polynomial – HOMFLY-Conway
        • Chapter 10: HOMFLY Polynomial – Conway
        • Chapter 10: HOMFLY Polynomial – Conway-Alexander
        • Chapter 10: HOMFLY Polynomial – Hopf Link, Trefoil
        • Chapter 10: HOMFLY Polynomial – Solomon
        • Chapter 10: HOMFLY Polynomial – Whitehead
        • Chapter 10: HOMFLY Polynomial – Alexander
        • Chapter 10: HOMFLY Polynomial – Trefoil
        • Chapter 10: HOMFLY Polynomial – Trefoil Details
        • Chapter 10: HOMFLY Polynomial – Solomon Exercise
        • Chapter 10: HOMFLY Polynomial – Hopf Link
        • Chapter 10: HOMFLY Polynomial – Crossing Regions
        • Chapter 10: HOMFLY Polynomial – Alexander Alternative
        • Chapter 10: HOMFLY Polynomial – Figure8
      • Chapter 11: Matrices
        • Chapter 11: Matrices – Notes
        • Chapter 11: Matrices – Graphical Representation
        • Chapter 11: Matrices – Matrix Operations 1
        • Chapter 11: Matrices – Matrix Operations 2
        • Chapter 11: Matrices – Matrix Operations 3
        • Chapter 11: Matrices – Matrix Operations 4
        • Chapter 11: Matrices – Tensor Product 1
        • Chapter 11: Matrices – Tensor Product 2
        • Chapter 11: Matrices – Matrix Summary
        • Chapter 11: Matrices – Row Column Notation
        • Chapter 11: Matrices – Row Column Operations 1
        • Chapter 11: Matrices – Row Column Operations 2
        • Chapter 11: Matrices – Trace
        • Chapter 11: Matrices – Examples: Identity
        • Chapter 11: Matrices – Examples: Row Column
        • Chapter 11: Matrices – Matrix Units
        • Chapter 11: Matrices – Exercise
      • Chapter 12: R Model
        • Chapter 12: Statistical Mechanical Model for the bracket polynomial: R Model – Notes
        • Chapter 12: R Model – Matrix Labels
        • Chapter 12: R Model – Colour Example
        • Chapter 12: R Model – R R
        • Chapter 12: R Model – Colour Arcs
        • Chapter 12: R Model – R Partition Function
        • Chapter 12: R Model – Example 1
        • Chapter 12: R Model – Example 2
        • Chapter 12: R Model – Parallel 1
        • Chapter 12: R Model – Parallel 2
        • Chapter 12: R Model – Antiparallel 1
        • Chapter 12: R Model – Antiparallel 2
        • Chapter 12: R Model – YBE 1
        • Chapter 12: R Model – YBE 2
        • Chapter 12: R Model – Theorem
        • Chapter 12: R Model – Noncyclic
        • Chapter 12: R Model – Bracket
        • Chapter 12: R Model – R Formulation of Bracket
        • Chapter 12: R Model – 180 Rotation
        • Chapter 12: R Model – 90 Rotation
        • Chapter 12: R Model – RR=1
        • Chapter 12: R Model – YBE
        • Chapter 12: R Model – U Algebra
        • Chapter 12: R Model – Type II Algebraic
        • Chapter 12: R Model – TL Relations 1
        • Chapter 12: R Model – TL Relations 2
        • Chapter 12: R Model – YBE Verification
      • Chapter 13: R-M Model
        • Chapter 13: R-M Model – Notes
        • Chapter 13: R-M Model – Layering 1
        • Chapter 13: R-M Model – Layering 2
        • Chapter 13: R-M Model – Layering 3
        • Chapter 13: R-M Model – Layering 4
        • Chapter 13: R-M Model – Type 2
        • Chapter 13: R-M Model – Topological Twist
        • Chapter 13: R-M Model – Theorem
        • Chapter 13: R-M Model – Formulation of Bracket
        • Chapter 13: R-M Model – Bracket 2
        • Chapter 13: R-M Model – YBE Bracket
        • Chapter 13: R-M Model – Bracket Twist
        • Chapter 13: R-M Model – Bracket Solution
      • Chapter 14: Braids
        • Chapter 14: Braids – Notes
          • Chapter 14: Braids – A Braid
            • Chapter 14: Braids – Fig 1
            • Chapter 14: Braids – Fig 2
            • Chapter 14: Braids – Fig 3
            • Chapter 14: Braids – Fig 4
            • Chapter 14: Braids – Fig 5
            • Chapter 14: Braids – Step 1
            • Chapter 14: Braids – Step 2
            • Chapter 14: Braids – Step 3
            • Chapter 14: Braids – Step 4
            • Chapter 14: Braids – Step 5
            • Chapter 14: Braids – Markov Type 1
            • Chapter 14: Braids – Markov Type 2
            • Chapter 14: Braids – Markov Type 2.
            • Chapter 14: Braids – Invariants 1
            • Chapter 14: Braids – Invariants 2
            • Chapter 14: Braids – Invariants 3
            • Chapter 14: Braids – Invariants 4
            • Chapter 14: Braids – Invariants 5
            • Chapter 14: Braids – Invariants 6
            • Chapter 14: Braids – Invariants 7
            • Chapter 14: Braids – Invariants 8
      • Chapter 15: Tangles
        • Chapter 15: Tangles – Notes
        • Chapter 15: Tangles – Definition
        • Chapter 15: Tangles – Addition Multiplication 1
        • Chapter 15: Tangles – Addition Multiplication 2
        • Chapter 15: Tangles – Numerator
        • Chapter 15: Tangles – Additional Integers
        • Chapter 15: Tangles – Sample 1
        • Chapter 15: Tangles – Sample 2
        • Chapter 15: Tangles – Sample 3
        • Chapter 15: Tangles – Rational Identification
        • Chapter 15: Tangles – Determinant
        • Chapter 15: Tangles – 3.2.1
        • Chapter 15: Tangles – 8/5
        • Chapter 15: Tangles – 8/5 Again
        • Chapter 15: Tangles – Whitehead
      • Exercises
        • Sheet 1
          • Sheet 1 – Solution 1
          • Sheet 1 – Solution 2
          • Sheet 1 – Solution 3
        • Sheet 2
          • Sheet 2 – Solution 1
          • Sheet 2 – Solution 2
          • Sheet 2 – Solution 3
          • Sheet 2 – Solution 4
          • Sheet 2 – Solution 5
        • Sheet 3
          • Sheet 3 – Solution 1
          • Sheet 3 – Solution 2
          • Sheet 3 – Solution 3
        • Sheet 4
          • Sheet 4 – Solution 1
          • Sheet 4 – Solution 2
          • Sheet 4 – Solution 3
          • Sheet 4 – Solution 4
          • Sheet 4 – Solution 5
          • Sheet 4 – Solution 6
          • Sheet 4 – Solution 7
          • Sheet 4 – Solution 8
        • Samples
      • Acknowledgements and References
      • Frequently Asked Questions
        • Examination Paper : Use of colour
      • How to draw Knots and Links

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Professor David E Evans

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Professor David E. Evans
School of Mathematics
Senghennydd Road
Cardiff University
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