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Introduction         Permutation Representation            Results and ConclusionsUsing Permutations to Study a Classificat...
Introduction                 Permutation Representation                    Results and ConclusionsOutline  1   Introductio...
Introduction                Permutation Representation                   Results and ConclusionsFormulating Our Problem  O...
Introduction                Permutation Representation                   Results and ConclusionsFormulating Our Problem  O...
Introduction                Permutation Representation                   Results and ConclusionsFormulating Our Problem  O...
Introduction   Arcs and Arclists                 Permutation Representation    Tightness Checking                    Resul...
Introduction   Arcs and Arclists                 Permutation Representation    Tightness Checking                    Resul...
Introduction   Arcs and Arclists                 Permutation Representation    Tightness Checking                    Resul...
Introduction   Arcs and Arclists                Permutation Representation    Tightness Checking                   Results...
Introduction   Arcs and Arclists                 Permutation Representation    Tightness Checking                    Resul...
Introduction   Arcs and Arclists                  Permutation Representation    Tightness Checking                     Res...
Introduction   Arcs and Arclists                  Permutation Representation    Tightness Checking                     Res...
Introduction   Arcs and Arclists                Permutation Representation    Tightness Checking                   Results...
Introduction   Arcs and Arclists                   Permutation Representation    Tightness Checking                      R...
Introduction   Arcs and Arclists                   Permutation Representation    Tightness Checking                      R...
Introduction   Arcs and Arclists                 Permutation Representation    Tightness Checking                    Resul...
Introduction   Arcs and Arclists                 Permutation Representation    Tightness Checking                    Resul...
Introduction   Arcs and Arclists                 Permutation Representation    Tightness Checking                    Resul...
Introduction   Arcs and Arclists                Permutation Representation    Tightness Checking                   Results...
Introduction   Arcs and Arclists                Permutation Representation    Tightness Checking                   Results...
Introduction   Arcs and Arclists                 Permutation Representation    Tightness Checking                    Resul...
Introduction   Arcs and Arclists                 Permutation Representation    Tightness Checking                    Resul...
Introduction   Arcs and Arclists               Permutation Representation    Tightness Checking                  Results a...
Introduction   Arcs and Arclists               Permutation Representation    Tightness Checking                  Results a...
Introduction       Arcs and Arclists               Permutation Representation        Tightness Checking                  R...
Introduction       Arcs and Arclists              Permutation Representation        Tightness Checking                 Res...
Introduction   Arcs and Arclists                Permutation Representation    Tightness Checking                   Results...
Introduction   Arcs and Arclists                Permutation Representation    Tightness Checking                   Results...
Introduction   Arcs and Arclists                Permutation Representation    Tightness Checking                   Results...
Introduction   Arcs and Arclists                Permutation Representation    Tightness Checking                   Results...
Introduction   Arcs and Arclists               Permutation Representation    Tightness Checking                  Results a...
Introduction   Arcs and Arclists                 Permutation Representation    Tightness Checking                    Resul...
Introduction   Arcs and Arclists                 Permutation Representation    Tightness Checking                    Resul...
Introduction   Arcs and Arclists                Permutation Representation    Tightness Checking                   Results...
Introduction   Arcs and Arclists                 Permutation Representation    Tightness Checking                    Resul...
Introduction                 Permutation Representation                    Results and ConclusionsFuture Research  Future ...
Introduction                 Permutation Representation                    Results and ConclusionsFuture Research  Future ...
Introduction                 Permutation Representation                    Results and ConclusionsFuture Research  Future ...
Introduction                 Permutation Representation                    Results and ConclusionsFuture Research  Future ...
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ISMAA Permutation Algebra

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ISMAA Permutation Algebra

  1. 1. Introduction Permutation Representation Results and ConclusionsUsing Permutations to Study a Classification Problem on the Solid Torus Illinois Sectional MAA Meeting Christopher L. Toni Dr. Tanya Cofer∗ April 10, 2010 Christopher L. Toni Computational Contact Topology - ISMAA Meeting 1 / 19
  2. 2. Introduction Permutation Representation Results and ConclusionsOutline 1 Introduction 2 Permutation Representation Arcs and Arclists Tightness Checking Bypasses 3 Results and Conclusions Christopher L. Toni Computational Contact Topology - ISMAA Meeting 2 / 19
  3. 3. Introduction Permutation Representation Results and ConclusionsFormulating Our Problem On surfaces inside the solid torus (defined by S1 × D2 ), dividing curves are located where twisting planes switch from positive to negative. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 3 / 19
  4. 4. Introduction Permutation Representation Results and ConclusionsFormulating Our Problem On surfaces inside the solid torus (defined by S1 × D2 ), dividing curves are located where twisting planes switch from positive to negative. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 3 / 19
  5. 5. Introduction Permutation Representation Results and ConclusionsFormulating Our Problem On surfaces inside the solid torus (defined by S1 × D2 ), dividing curves are located where twisting planes switch from positive to negative. These dividing curves keep track of and allow for investigation of certain topological properties in the neighborhood of a surface. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 3 / 19
  6. 6. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesOutline 1 Introduction 2 Permutation Representation Arcs and Arclists Tightness Checking Bypasses 3 Results and Conclusions Christopher L. Toni Computational Contact Topology - ISMAA Meeting 4 / 19
  7. 7. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesOverview The first computational task is to generate arclists for a given number of vertices np. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 5 / 19
  8. 8. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesOverview The first computational task is to generate arclists for a given number of vertices np. Definition An arc is a path between vertices subject to the conditions that all vertices must be paired and arcs cannot intersect. An arclist is a set (list) of legal pairs of arcs. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 5 / 19
  9. 9. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesHow Do Permutations Apply Here? Recall that a permutation is a bijective mapping of elements from a set S to itself. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 6 / 19
  10. 10. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesHow Do Permutations Apply Here? Recall that a permutation is a bijective mapping of elements from a set S to itself. Let S = {0, 1, 2, . . . , np − 1} be the set of vertex values on a cutting disk. We can define a permutation α on S that satisfies the definition of an arc/arclist. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 6 / 19
  11. 11. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesHow Do Permutations Apply Here? Recall that a permutation is a bijective mapping of elements from a set S to itself. Let S = {0, 1, 2, . . . , np − 1} be the set of vertex values on a cutting disk. We can define a permutation α on S that satisfies the definition of an arc/arclist. Example: Consider the case np = 8. The set of vertex values would be S = {0, 1, 2, . . . , 6, 7} and α = (01) (25) (34) (67) is a permutation on the set S. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 6 / 19
  12. 12. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesHow Do Permutations Apply Here? Recall that a permutation is a bijective mapping of elements from a set S to itself. Let S = {0, 1, 2, . . . , np − 1} be the set of vertex values on a cutting disk. We can define a permutation α on S that satisfies the definition of an arc/arclist. Example: Consider the case np = 8. The set of vertex values would be S = {0, 1, 2, . . . , 6, 7} and α = (01) (25) (34) (67) is a permutation on the set S. There are 14 different permutations on this set that satisfy the definitions of an arc/arclist. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 6 / 19
  13. 13. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesHow Do Permutations Apply Here? (Cont.) Consider the example mentioned on the previous slide. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 7 / 19
  14. 14. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesHow Do Permutations Apply Here? (Cont.) Consider the example mentioned on the previous slide. 0 1 7 2 6 3 5 4 For this cutting disk, the arclist is {(0, 1), (2, 5), (3, 4), (6, 7)}. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 7 / 19
  15. 15. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesHow Do Permutations Apply Here? (Cont.) Consider the example mentioned on the previous slide. 0 1 7 2 6 3 5 4 For this cutting disk, the arclist is {(0, 1), (2, 5), (3, 4), (6, 7)}. We can easily rewrite this as the permutation α = (01) (25) (34) (67). Christopher L. Toni Computational Contact Topology - ISMAA Meeting 7 / 19
  16. 16. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesOutline 1 Introduction 2 Permutation Representation Arcs and Arclists Tightness Checking Bypasses 3 Results and Conclusions Christopher L. Toni Computational Contact Topology - ISMAA Meeting 8 / 19
  17. 17. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesOverview - Tightness Checker Potentially Tight Overtwisted x → x − nq + 1 mod np This maps the dividing curves on the surface from left to right cutting disk. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 9 / 19
  18. 18. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesUsing Permutations to Determine Tightness Let β be a permutation that represents the mapping rule x → x − nq + 1 mod np and let A be the arclist permutation. The permutation formula to check for tightness is β −1 Aβ A. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 10 / 19
  19. 19. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesPermutation Example Given: n = 2, p = 4 ,q = 3 The mapping rule tells us x → x − 5 mod 8. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 11 / 19
  20. 20. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesPermutation Example Given: n = 2, p = 4 ,q = 3 The mapping rule tells us x → x − 5 mod 8. Therefore, β = (03614725) β −1 = (05274163) Christopher L. Toni Computational Contact Topology - ISMAA Meeting 11 / 19
  21. 21. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesPermutation Example Given: n = 2, p = 4 ,q = 3 The mapping rule tells us x → x − 5 mod 8. Therefore, β = (03614725) β −1 = (05274163) A = (0 1)(2 7)(3 6)(4 5) A = (0 7)(1 4)(2 3)(5 6) β −1 Aβ A = (0246) β −1 Aβ A = (0) Christopher L. Toni Computational Contact Topology - ISMAA Meeting 11 / 19
  22. 22. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesOutline 1 Introduction 2 Permutation Representation Arcs and Arclists Tightness Checking Bypasses 3 Results and Conclusions Christopher L. Toni Computational Contact Topology - ISMAA Meeting 12 / 19
  23. 23. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesAbstract Bypasses An abstract bypass exists when a line can be drawn through three arcs on a cutting disk. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 13 / 19
  24. 24. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesAbstract Bypasses An abstract bypass exists when a line can be drawn through three arcs on a cutting disk. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 13 / 19
  25. 25. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesAbstract Bypasses An abstract bypass exists when a line can be drawn through three arcs on a cutting disk. Two Abstract Bypasses. . No Abstract Bypasses. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 13 / 19
  26. 26. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesAbstract Bypasses (Cont.) (05) (14) (23) (67) (01) (25) (34) (67) α α β (01) (23) (47) (56) β Christopher L. Toni Computational Contact Topology - ISMAA Meeting 14 / 19
  27. 27. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesExistence of Bypasses The existence of actual bypasses is checked in a similar fashion as tightness. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 15 / 19
  28. 28. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesExistence of Bypasses The existence of actual bypasses is checked in a similar fashion as tightness. Given: An arclist A and an abstract bypass C. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 15 / 19
  29. 29. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesExistence of Bypasses The existence of actual bypasses is checked in a similar fashion as tightness. Given: An arclist A and an abstract bypass C. The formula: β −1 AβC Christopher L. Toni Computational Contact Topology - ISMAA Meeting 15 / 19
  30. 30. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesExistence of Bypasses The existence of actual bypasses is checked in a similar fashion as tightness. Given: An arclist A and an abstract bypass C. The formula: β −1 AβC A = (01)(25)(34)(67) β = (03614725) β −1 = (05274163) C1 = (05)(14)(23)(67) C2 = (01)(23)(47)(56) β −1 AβC1 = (0624) β −1 AβC2 = (0) Christopher L. Toni Computational Contact Topology - ISMAA Meeting 15 / 19
  31. 31. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesAbstract Bypass Generators Question: How do we identify abstract bypasses algorithmically without the luxury of pictures? Christopher L. Toni Computational Contact Topology - ISMAA Meeting 16 / 19
  32. 32. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesAbstract Bypass Generators Question: How do we identify abstract bypasses algorithmically without the luxury of pictures? Theorem For every set of np vertices, there are special permutations that detect abstract bypasses. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 16 / 19
  33. 33. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesAbstract Bypass Generators Question: How do we identify abstract bypasses algorithmically without the luxury of pictures? Theorem For every set of np vertices, there are special permutations that detect abstract bypasses. Theorem Given np, we can generate all abstract bypass generators. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 16 / 19
  34. 34. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesAbstract Bypass Generators (cont.) In the case of np = 8, we have the following bypass generators: γ1 = (042) (153) γ5 = (062) (175) γ2 = (064) (175) γ6 = (153) (264) γ3 = (062) (173) γ7 = (064) (375) γ4 = (246) (375) γ8 = (042) (173) Christopher L. Toni Computational Contact Topology - ISMAA Meeting 17 / 19
  35. 35. Introduction Arcs and Arclists Permutation Representation Tightness Checking Results and Conclusions BypassesAbstract Bypass Generators (cont.) Consider the arclist α = (01) (25) (34) (67). Applying the abstract bypass generators on the previous slide, we get: γ1 ◦ α = (05) (14) (23) (67) γ5 ◦ α = (072165) (34) γ2 ◦ α = (0743) (1652) γ6 ◦ α = (056741) (23) γ3 ◦ α = (0725) (1634) γ7 ◦ α = (016523) (47) γ4 ◦ α = (01) (23) (47) (56) γ8 ◦ α = (076325) (14) Christopher L. Toni Computational Contact Topology - ISMAA Meeting 18 / 19
  36. 36. Introduction Permutation Representation Results and ConclusionsFuture Research Future goals include, but not limited to: Christopher L. Toni Computational Contact Topology - ISMAA Meeting 19 / 19
  37. 37. Introduction Permutation Representation Results and ConclusionsFuture Research Future goals include, but not limited to: Publication of Findings in Undergraduate Journal Christopher L. Toni Computational Contact Topology - ISMAA Meeting 19 / 19
  38. 38. Introduction Permutation Representation Results and ConclusionsFuture Research Future goals include, but not limited to: Publication of Findings in Undergraduate Journal Extension of Algorithm to the two-holed torus Christopher L. Toni Computational Contact Topology - ISMAA Meeting 19 / 19
  39. 39. Introduction Permutation Representation Results and ConclusionsFuture Research Future goals include, but not limited to: Publication of Findings in Undergraduate Journal Extension of Algorithm to the two-holed torus Searching for a formula for the case of four dividing curves. Christopher L. Toni Computational Contact Topology - ISMAA Meeting 19 / 19

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