mas344 knots and surfaces what is a knot
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MAS344 Knots and surfaces What is a knot? Not a knot What is a - PowerPoint PPT Presentation

MAS344 Knots and surfaces What is a knot? Not a knot What is a knot? Not a knot A knot What is a knot? Not a knot A knot Not a knot Variants of knots a link (the Borromean rings) a tangle a braid Another picture of the Borromean rings


  1. MAS344 Knots and surfaces

  2. What is a knot? Not a knot

  3. What is a knot? Not a knot A knot

  4. What is a knot? Not a knot A knot Not a knot

  5. Variants of knots a link (the Borromean rings) a tangle a braid

  6. Another picture of the Borromean rings

  7. Knotted DNA

  8. Closing a braid to make a knot

  9. A table of knots and links

  10. The Perko pair

  11. Deforming a knot These two knots are the same:

  12. Deforming a knot

  13. Deforming a knot

  14. Deforming a knot

  15. Deforming a knot

  16. Deforming a knot

  17. Deforming a knot

  18. Deforming a knot

  19. Deforming a knot

  20. Deforming a knot

  21. Jones polynomial example The Jones polynomial of this knot is A 16 − 4 A 12 + 6 A 8 − 7 A 4 + 9 − 7 A − 4 + 6 A − 8 − 4 A − 12 + A − 16 .

  22. Some knot theorists Vaughan Jones Louis Kauffman John Conway

  23. Which of these knots are equivalent?

  24. Most basic invariant: number of components c = 1 c = 2 c = 3

  25. Not an invariant: number of crossings ≃ ≃ n = 6 n = 0 n = 4 n = 3

  26. Minimal crossing number Minimal crossing number is an invariant, but not a very useful one.

  27. Universes and diagrams link diagram link diagram link universe (trefoil knot) (unknot) oriented link diagram oriented link diagram (positive Hopf link) (negative Hopf link)

  28. All diagrams for the trefoil universe

  29. Reidemeister moves Type 1 Type 2 Type 3

  30. Reidemeister moves

  31. Reidemeister example

  32. Reidemeister example

  33. Reidemeister example

  34. Reidemeister example

  35. Reidemeister example

  36. Reidemeister example

  37. Reidemeister example

  38. Reidemeister example

  39. Reidemeister example

  40. Oriented link diagrams

  41. Oriented Reidemeister moves

  42. Sign of crossings ⊕ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ If you approach a positive crossing along the top strand, then you see the other strand pass underneath you from right to left.

  43. The skein relation D + D 0 D − A 4 f ( D + ) − A − 4 f ( D − ) = ( A − 2 − A 2 ) f ( D 0 ) .

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