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Unknotted Cycles Nathan McNew Towson University Joint with Christopher Cornwell Permutation Patterns 2018 Dartmouth College Hanover, NH July 13th, 2018 Nathan McNew Unknotted Cycles July 13th, 2018 1 Cycle Diagrams The cycle diagram of a

  1. Observation Grid diagrams are remarkably similar to cycle diagrams. Treating diagonal entries of a cycle diagram as points the only differences are: In cycle diagrams one of the two points in each row/column must lie on the diagonal, which isn’t the case for grid diagrams. In grid diagrams, it is not allowed to have a single point in a row or column as occur in cycle diagrams when there are fixed points. Restricting our attention to derangements , permutations without fixed points, every cycle diagram can be interpreted as a grid diagram, associating a link (knot) to each derangement (cycle). Nathan McNew Unknotted Cycles July 13th, 2018 6

  2. The link associated to a permutation Definition For any derangement σ , the link associated to σ is obtained by drawing the cycle diagram of σ and interpreting it as a grid diagram instead. If σ is a cycle then this is the knot associated to σ . Nathan McNew Unknotted Cycles July 13th, 2018 7

  3. The link associated to a permutation Definition For any derangement σ , the link associated to σ is obtained by drawing the cycle diagram of σ and interpreting it as a grid diagram instead. If σ is a cycle then this is the knot associated to σ . We will call any cycle associated to the unknot an unknotted cycle . Nathan McNew Unknotted Cycles July 13th, 2018 7

  4. Example: σ = 837295641 Nathan McNew Unknotted Cycles July 13th, 2018 8

  5. Example: σ = 837295641 Nathan McNew Unknotted Cycles July 13th, 2018 8

  6. Example: σ = 837295641 Nathan McNew Unknotted Cycles July 13th, 2018 8

  7. Example: σ = 837295641 Nathan McNew Unknotted Cycles July 13th, 2018 8

  8. Example: σ = 837295641 Nathan McNew Unknotted Cycles July 13th, 2018 8

  9. Example: σ = 837295641 Nathan McNew Unknotted Cycles July 13th, 2018 8

  10. Example: σ = 837295641 Nathan McNew Unknotted Cycles July 13th, 2018 8

  11. Example: σ = 837295641 Nathan McNew Unknotted Cycles July 13th, 2018 8

  12. Example: σ = 837295641 Nathan McNew Unknotted Cycles July 13th, 2018 8

  13. Example: σ = 837295641 ... the unknot! Nathan McNew Unknotted Cycles July 13th, 2018 8

  14. Example: σ = 57428163 Nathan McNew Unknotted Cycles July 13th, 2018 9

  15. Example: σ = 57428163 Nathan McNew Unknotted Cycles July 13th, 2018 9

  16. Example: σ = 57428163 Nathan McNew Unknotted Cycles July 13th, 2018 9

  17. Example: σ = 57428163 Nathan McNew Unknotted Cycles July 13th, 2018 9

  18. Example: σ = 57428163 Nathan McNew Unknotted Cycles July 13th, 2018 9

  19. Example: σ = 57428163 ... a trefoil knot. Nathan McNew Unknotted Cycles July 13th, 2018 9

  20. Counting Unknotted Cycles How many cycles of length n are unknotted? Nathan McNew Unknotted Cycles July 13th, 2018 10

  21. Counting Unknotted Cycles How many cycles of length n are unknotted? n # cycles Nathan McNew Unknotted Cycles July 13th, 2018 10

  22. Counting Unknotted Cycles How many cycles of length n are unknotted? n # cycles 2 1 Nathan McNew Unknotted Cycles July 13th, 2018 10

  23. Counting Unknotted Cycles How many cycles of length n are unknotted? n # cycles 2 1 3 2 Nathan McNew Unknotted Cycles July 13th, 2018 10

  24. Counting Unknotted Cycles How many cycles of length n are unknotted? n # cycles 2 1 3 2 4 6 Nathan McNew Unknotted Cycles July 13th, 2018 10

  25. Counting Unknotted Cycles How many cycles of length n are unknotted? n # cycles 2 1 3 2 4 6 5 22 · · · Nathan McNew Unknotted Cycles July 13th, 2018 10

  26. Counting Unknotted Cycles How many cycles of length n are unknotted? n # cycles 2 1 3 2 4 6 5 22 · · · 6 90 · · · Nathan McNew Unknotted Cycles July 13th, 2018 10

  27. Counting Unknotted Cycles Nathan McNew Unknotted Cycles July 13th, 2018 11

  28. Counting Unknotted Cycles Denote by S n the n th large Schr¨ oder number, given by the recurrence S 1 = 1 and n − 1 � S n = S n − 1 + S k S n − k . k =1 Nathan McNew Unknotted Cycles July 13th, 2018 11

  29. Counting Unknotted Cycles Denote by S n the n th large Schr¨ oder number, given by the recurrence S 1 = 1 and n − 1 � S n = S n − 1 + S k S n − k . k =1 Theorem The count of unknotted cycles of size n +1 is S n . Nathan McNew Unknotted Cycles July 13th, 2018 11

  30. Rooted-Signed-Binary-Trees Nathan McNew Unknotted Cycles July 13th, 2018 12

  31. Rooted-Signed-Binary-Trees Definition A rooted-signed-binary tree is a rooted binary tree where each non-root node is either positive or negative. Nathan McNew Unknotted Cycles July 13th, 2018 12

  32. Rooted-Signed-Binary-Trees Definition A rooted-signed-binary tree is a rooted binary tree where each non-root node is either positive or negative. Two trees are equivalent if one can be obtained from another by a series of tree rotations. Nathan McNew Unknotted Cycles July 13th, 2018 12

  33. Rooted-Signed-Binary-Trees Definition A rooted-signed-binary tree is a rooted binary tree where each non-root node is either positive or negative. Two trees are equivalent if one can be obtained from another by a series of tree rotations. The allowed tree rotations are either: Nathan McNew Unknotted Cycles July 13th, 2018 12

  34. Rooted-Signed-Binary-Trees Definition A rooted-signed-binary tree is a rooted binary tree where each non-root node is either positive or negative. Two trees are equivalent if one can be obtained from another by a series of tree rotations. The allowed tree rotations are either: A child node can be rotated into a parent with the same sign. Nathan McNew Unknotted Cycles July 13th, 2018 12

  35. Rooted-Signed-Binary-Trees Definition A rooted-signed-binary tree is a rooted binary tree where each non-root node is either positive or negative. Two trees are equivalent if one can be obtained from another by a series of tree rotations. The allowed tree rotations are either: A child node can be rotated into a parent with the same sign. A node can be rotated into the root. The new node is given the sign of the node rotated into the root. Nathan McNew Unknotted Cycles July 13th, 2018 12

  36. Example: Rooted-signed-binary-trees + − + − Nathan McNew Unknotted Cycles July 13th, 2018 13

  37. Example: Rooted-signed-binary-trees + − + − + + − − Nathan McNew Unknotted Cycles July 13th, 2018 13

  38. Example: Rooted-signed-binary-trees + − + + − + + − + − − − Nathan McNew Unknotted Cycles July 13th, 2018 13

  39. The bijection Nathan McNew Unknotted Cycles July 13th, 2018 14

  40. The bijection Key observation: Inserting a point in a cycle diagram immediately above or below the diagonal does not affect the knot type. Nathan McNew Unknotted Cycles July 13th, 2018 14

  41. The bijection Key observation: Inserting a point in a cycle diagram immediately above or below the diagonal does not affect the knot type. Use the signed tree to keep track of how points are inserted. Nathan McNew Unknotted Cycles July 13th, 2018 14

  42. The bijection Key observation: Inserting a point in a cycle diagram immediately above or below the diagonal does not affect the knot type. Use the signed tree to keep track of how points are inserted. Assign the tree with only the unsigned root to the cycle σ = 21. Nathan McNew Unknotted Cycles July 13th, 2018 14

  43. The bijection Key observation: Inserting a point in a cycle diagram immediately above or below the diagonal does not affect the knot type. Use the signed tree to keep track of how points are inserted. Assign the tree with only the unsigned root to the cycle σ = 21. Number the places a leaf could be added to the tree. To add a node to the tree into the i th position of the tree: Nathan McNew Unknotted Cycles July 13th, 2018 14

  44. The bijection Key observation: Inserting a point in a cycle diagram immediately above or below the diagonal does not affect the knot type. Use the signed tree to keep track of how points are inserted. Assign the tree with only the unsigned root to the cycle σ = 21. Number the places a leaf could be added to the tree. To add a node to the tree into the i th position of the tree: For a positive node insert i +1 prior to the element in position i . Increment each element in the cycle that had value i +1 or more. Nathan McNew Unknotted Cycles July 13th, 2018 14

  45. The bijection Key observation: Inserting a point in a cycle diagram immediately above or below the diagonal does not affect the knot type. Use the signed tree to keep track of how points are inserted. Assign the tree with only the unsigned root to the cycle σ = 21. Number the places a leaf could be added to the tree. To add a node to the tree into the i th position of the tree: For a positive node insert i +1 prior to the element in position i . Increment each element in the cycle that had value i +1 or more. For a negative node insert i after the element in position i . Increment each element in the cycle that had value i or greater. Nathan McNew Unknotted Cycles July 13th, 2018 14

  46. The bijection Before insertion After inserting ⊕ After inserting ⊖ Nathan McNew Unknotted Cycles July 13th, 2018 15

  47. Example Nathan McNew Unknotted Cycles July 13th, 2018 16

  48. Example + − + − Nathan McNew Unknotted Cycles July 13th, 2018 16

  49. Example + − + − Nathan McNew Unknotted Cycles July 13th, 2018 16

  50. Example + − + − Nathan McNew Unknotted Cycles July 13th, 2018 16

  51. Example + − + − Nathan McNew Unknotted Cycles July 13th, 2018 16

  52. Example + − + + − Nathan McNew Unknotted Cycles July 13th, 2018 16

  53. Example + − + + − Nathan McNew Unknotted Cycles July 13th, 2018 16

  54. Example + − + + + − + Nathan McNew Unknotted Cycles July 13th, 2018 16

  55. Example + − + + + − + Nathan McNew Unknotted Cycles July 13th, 2018 16

  56. Example + − + + + + − + − + Nathan McNew Unknotted Cycles July 13th, 2018 16

  57. Example + − + + + + − + − + Nathan McNew Unknotted Cycles July 13th, 2018 16

  58. Example + − + + + + − + − + Nathan McNew Unknotted Cycles July 13th, 2018 16

  59. Example + − + + + + − + − + Nathan McNew Unknotted Cycles July 13th, 2018 16

  60. Ideas of proof Nathan McNew Unknotted Cycles July 13th, 2018 17

  61. Ideas of proof Show large Scr¨ oder numbers count rooted-signed-binary-trees: Nathan McNew Unknotted Cycles July 13th, 2018 17

  62. Ideas of proof Show large Scr¨ oder numbers count rooted-signed-binary-trees: � Well-defined as to the order vertices are added to a tree: Nathan McNew Unknotted Cycles July 13th, 2018 17

  63. Ideas of proof Show large Scr¨ oder numbers count rooted-signed-binary-trees: � Well-defined as to the order vertices are added to a tree: � Nathan McNew Unknotted Cycles July 13th, 2018 17

  64. Ideas of proof Show large Scr¨ oder numbers count rooted-signed-binary-trees: � Well-defined as to the order vertices are added to a tree: � Well-defined on equivalent trees after a rotation: Nathan McNew Unknotted Cycles July 13th, 2018 17

  65. Ideas of proof Show large Scr¨ oder numbers count rooted-signed-binary-trees: � Well-defined as to the order vertices are added to a tree: � Well-defined on equivalent trees after a rotation: � Map from trees to cycles is injective: Nathan McNew Unknotted Cycles July 13th, 2018 17

  66. Ideas of proof Show large Scr¨ oder numbers count rooted-signed-binary-trees: � Well-defined as to the order vertices are added to a tree: � Well-defined on equivalent trees after a rotation: � Map from trees to cycles is injective: � Nathan McNew Unknotted Cycles July 13th, 2018 17

  67. Ideas of proof Show large Scr¨ oder numbers count rooted-signed-binary-trees: � Well-defined as to the order vertices are added to a tree: � Well-defined on equivalent trees after a rotation: � Map from trees to cycles is injective: � Map from trees to cycles is surjective: Nathan McNew Unknotted Cycles July 13th, 2018 17

  68. Ideas of proof Show large Scr¨ oder numbers count rooted-signed-binary-trees: � Well-defined as to the order vertices are added to a tree: � Well-defined on equivalent trees after a rotation: � Map from trees to cycles is injective: � Map from trees to cycles is surjective: ...not so easy. Nathan McNew Unknotted Cycles July 13th, 2018 17

  69. Ideas of proof Show large Scr¨ oder numbers count rooted-signed-binary-trees: � Well-defined as to the order vertices are added to a tree: � Well-defined on equivalent trees after a rotation: � Map from trees to cycles is injective: � Map from trees to cycles is surjective: ...not so easy. To show surjectivity, it would suffice to show that every unknotted cycle, σ , has a point on the off-diagonal, i.e | σ ( i ) − i | = 1. Nathan McNew Unknotted Cycles July 13th, 2018 17

  70. Topology Nathan McNew Unknotted Cycles July 13th, 2018 18

  71. Topology Let σ be a cycle. Nathan McNew Unknotted Cycles July 13th, 2018 18

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