1. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Improving on an Existing Program That Checks
for Tight Contact Structures on the Solid Torus
Northeastern Illinois University SCSE Research Symposium
Christopher L. Toni Kelly Hirschbeck Nathan Walter
William Krepelin Donald Barkley William Byrd
John Wallin Mayra Bravo-Gonzalez Banlieman Kolani
Dr. Tanya Cofer
October 2, 2009
Christopher L. Toni, Donald Barkley Computational Contact Topology 1 / 20

2. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Outline
1 Introduction
2 Arcs and Arclists
3 Tightness Checking
4 Bypasses
5 Final Results and Thoughts
Christopher L. Toni, Donald Barkley Computational Contact Topology 2 / 20

3. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
What is Topology?
Topology is a field of mathematics that does not focus on an
object’s shape, but the properties that remain consistent
through deformations like:
Christopher L. Toni, Donald Barkley Computational Contact Topology 3 / 20

4. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
What is Topology?
Topology is a field of mathematics that does not focus on an
object’s shape, but the properties that remain consistent
through deformations like:
1 twisting
Christopher L. Toni, Donald Barkley Computational Contact Topology 3 / 20

5. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
What is Topology?
Topology is a field of mathematics that does not focus on an
object’s shape, but the properties that remain consistent
through deformations like:
1 twisting
2 bending
Christopher L. Toni, Donald Barkley Computational Contact Topology 3 / 20

6. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
What is Topology?
Topology is a field of mathematics that does not focus on an
object’s shape, but the properties that remain consistent
through deformations like:
1 twisting
2 bending
3 stretching
Christopher L. Toni, Donald Barkley Computational Contact Topology 3 / 20

7. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
What is Topology?
Topology is a field of mathematics that does not focus on an
object’s shape, but the properties that remain consistent
through deformations like:
1 twisting
2 bending
3 stretching
To illustrate this, visualize a coffee cup and a doughnut (torus).
Christopher L. Toni, Donald Barkley Computational Contact Topology 3 / 20

8. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
What is Topology? (cont.)
The torus and the coffee cup are topologically equivalent
objects. We see above that through bending and stretching, the
torus can be morphed into a coffee cup and vice versa.
Christopher L. Toni, Donald Barkley Computational Contact Topology 4 / 20

9. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Formulating the Problem
On the solid torus (defined by 1 2 ), dividing curves are
located where twisting planes switch from positive to negative.
Christopher L. Toni, Donald Barkley Computational Contact Topology 5 / 20

10. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Formulating the Problem
On the solid torus (defined by 1 2 ), dividing curves are
located where twisting planes switch from positive to negative.
Christopher L. Toni, Donald Barkley Computational Contact Topology 5 / 20

11. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Formulating the Problem
On the solid torus (defined by 1 2 ), 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, Donald Barkley Computational Contact Topology 5 / 20

12. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Formulating the Problem (cont.)
Christopher L. Toni, Donald Barkley Computational Contact Topology 6 / 20

13. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Formulating the Problem (cont.)
We define n to be the number of dividing curves, p to be the
number of times the dividing curves are wrapped about the
longitudinal section of the torus, and q to be the number of
times the dividing curves are wrapped about the meridinal
section of the torus.
Christopher L. Toni, Donald Barkley Computational Contact Topology 6 / 20

14. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview
The first computational task is to generate arclists for a given
number of vertices M, where M np.
Christopher L. Toni, Donald Barkley Computational Contact Topology 7 / 20

15. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview
The first computational task is to generate arclists for a given
number of vertices M, where M np.
Definition
An arc is a path between vertices subject to:
Christopher L. Toni, Donald Barkley Computational Contact Topology 7 / 20

16. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview
The first computational task is to generate arclists for a given
number of vertices M, where M np.
Definition
An arc is a path between vertices subject to:
1 All M vertices in a configuration must be paired
Christopher L. Toni, Donald Barkley Computational Contact Topology 7 / 20

17. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview
The first computational task is to generate arclists for a given
number of vertices M, where M np.
Definition
An arc is a path between vertices subject to:
1 All M vertices in a configuration must be paired
2 Paths cannot cross
Christopher L. Toni, Donald Barkley Computational Contact Topology 7 / 20

18. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview
The first computational task is to generate arclists for a given
number of vertices M, where M np.
Definition
An arc is a path between vertices subject to:
1 All M vertices in a configuration must be paired
2 Paths cannot cross
An arclist is a set (list) of legal pairs of arcs.
Christopher L. Toni, Donald Barkley Computational Contact Topology 7 / 20

19. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview
The first computational task is to generate arclists for a given
number of vertices M, where M np.
Definition
An arc is a path between vertices subject to:
1 All M vertices in a configuration must be paired
2 Paths cannot cross
An arclist is a set (list) of legal pairs of arcs.
We can think of arclists for M vertices as certain permutations
of M objects.
Christopher L. Toni, Donald Barkley Computational Contact Topology 7 / 20

20. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview
The first computational task is to generate arclists for a given
number of vertices M, where M np.
Definition
An arc is a path between vertices subject to:
1 All M vertices in a configuration must be paired
2 Paths cannot cross
An arclist is a set (list) of legal pairs of arcs.
We can think of arclists for M vertices as certain permutations
of M objects.The solution is to “walk” a new element through
the solution set for a smaller problem.
Christopher L. Toni, Donald Barkley Computational Contact Topology 7 / 20

21. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview
The first computational task is to generate arclists for a given
number of vertices M, where M np.
Definition
An arc is a path between vertices subject to:
1 All M vertices in a configuration must be paired
2 Paths cannot cross
An arclist is a set (list) of legal pairs of arcs.
We can think of arclists for M vertices as certain permutations
of M objects.The solution is to “walk” a new element through
the solution set for a smaller problem. There is one challenge:
the algorithm is space and time intensive!
Christopher L. Toni, Donald Barkley Computational Contact Topology 7 / 20

22. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm - Arcs and Arclist
Christopher L. Toni, Donald Barkley Computational Contact Topology 8 / 20

23. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm - Arcs and Arclist
Christopher L. Toni, Donald Barkley Computational Contact Topology 8 / 20

24. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Arcs and Arclists
For the case of n 2, p 4, q 3, we have
M np 2 4 8.
Christopher L. Toni, Donald Barkley Computational Contact Topology 9 / 20

25. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Arcs and Arclists
For the case of n 2, p 4, q 3, we have
M np 2 4 8.
The arclists for M 8 vertices are:
Christopher L. Toni, Donald Barkley Computational Contact Topology 9 / 20

26. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Arcs and Arclists
For the case of n 2, p 4, q 3, we have
M np 2 4 8.
The arclists for M 8 vertices are:
(0 1)(2 5)(3 4)(6 7)
(0 1)(2 7)(3 4)(5 6)
(0 3)(1 2)(4 5)(6 7)
(0 1)(2 3)(4 5)(6 7)
(0 1)(2 7)(3 6)(4 5)
(0 3)(1 2)(4 7)(5 6)
(0 7)(1 2)(3 4)(5 6)
Christopher L. Toni, Donald Barkley Computational Contact Topology 9 / 20

27. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Arcs and Arclists
For the case of n 2, p 4, q 3, we have
M np 2 4 8.
The arclists for M 8 vertices are:
(0 1)(2 5)(3 4)(6 7) (0 5)(1 2)(3 4)(6 7)
(0 1)(2 7)(3 4)(5 6) (0 7)(1 4)(2 3)(5 6)
(0 3)(1 2)(4 5)(6 7) (0 1)(2 3)(4 7)(5 6)
(0 1)(2 3)(4 5)(6 7) (0 5)(1 4)(2 3)(6 7)
(0 1)(2 7)(3 6)(4 5) (0 7)(1 2)(3 6)(4 5)
(0 3)(1 2)(4 7)(5 6) (0 7)(1 6)(2 5)(3 4)
(0 7)(1 2)(3 4)(5 6) (0 7)(1 6)(2 3)(4 5)
Christopher L. Toni, Donald Barkley Computational Contact Topology 9 / 20

28. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Arcs and Arclists
For the case of n 2, p 4, q 3, we have
M np 2 4 8.
The arclists for M 8 vertices are:
(0 1)(2 5)(3 4)(6 7) (0 5)(1 2)(3 4)(6 7)
(0 1)(2 7)(3 4)(5 6) (0 7)(1 4)(2 3)(5 6)
(0 3)(1 2)(4 5)(6 7) (0 1)(2 3)(4 7)(5 6)
(0 1)(2 3)(4 5)(6 7) (0 5)(1 4)(2 3)(6 7)
(0 1)(2 7)(3 6)(4 5) (0 7)(1 2)(3 6)(4 5)
(0 3)(1 2)(4 7)(5 6) (0 7)(1 6)(2 5)(3 4)
(0 7)(1 2)(3 4)(5 6) (0 7)(1 6)(2 3)(4 5)
Data files for required values of M were produced and saved.
These files were then used as input to the Tightness Checking
Christopher L. Toni, Donald Barkley Computational Contact Topology 9 / 20

29. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview - Tightness Checker
Once the arclists are found, it is possible to determine how the
vertices on the left and right cutting disks match up.
Christopher L. Toni, Donald Barkley Computational Contact Topology 10 / 20

30. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview - Tightness Checker
Once the arclists are found, it is possible to determine how the
vertices on the left and right cutting disks match up.
The formula x x nq 1 mod np maps the vertices on the
left cutting disk to the right cutting disk.
Christopher L. Toni, Donald Barkley Computational Contact Topology 10 / 20

31. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview - Tightness Checker
Once the arclists are found, it is possible to determine how the
vertices on the left and right cutting disks match up.
The formula x x nq 1 mod np maps the vertices on the
left cutting disk to the right cutting disk.
The formula x x nq 1 mod np maps the vertices on the
right cutting disk to the left cutting disk.
Christopher L. Toni, Donald Barkley Computational Contact Topology 10 / 20

32. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview - Tightness Checker
Once the arclists are found, it is possible to determine how the
vertices on the left and right cutting disks match up.
The formula x x nq 1 mod np maps the vertices on the
left cutting disk to the right cutting disk.
The formula x x nq 1 mod np maps the vertices on the
right cutting disk to the left cutting disk.
To determine if the torus admits a tight or overtwisted contact
structure, the dividing curves and arcs need to be analyzed.
Christopher L. Toni, Donald Barkley Computational Contact Topology 10 / 20

33. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview - Tightness Checker (cont.)
Christopher L. Toni, Donald Barkley Computational Contact Topology 11 / 20

34. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview - Tightness Checker (cont.)
If a single closed curve can be
traced on the torus, it is
considered to be a potentially
tight contact structure.
Christopher L. Toni, Donald Barkley Computational Contact Topology 11 / 20

35. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Overview - Tightness Checker (cont.)
If a single closed curve can be If more than one closed curve
traced on the torus, it is can be traced on the torus, it
considered to be a potentially is considered to be an
tight contact structure. overtwisted structure.
Christopher L. Toni, Donald Barkley Computational Contact Topology 11 / 20

36. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm - Tightness Checker
Christopher L. Toni, Donald Barkley Computational Contact Topology 12 / 20

37. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm - Tightness Checker
All vertices hook up to a single
curve. Thus, the structure is
potentially tight.
Christopher L. Toni, Donald Barkley Computational Contact Topology 12 / 20

38. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm - Tightness Checker
All vertices hook up to a single Only a few vertices hook up to
curve. Thus, the structure is a curve. Thus, the structure is
potentially tight. overtwisted.
Christopher L. Toni, Donald Barkley Computational Contact Topology 12 / 20

39. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Tightness Checker
Consider M np 8.
Christopher L. Toni, Donald Barkley Computational Contact Topology 13 / 20

40. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Tightness Checker
Consider M np 8. Given the arclist (0 1)(2 7)(3 6)(4 5) ,
the left cutting disk hooks up with the right cutting disk as
follows:
Christopher L. Toni, Donald Barkley Computational Contact Topology 13 / 20

41. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Tightness Checker
Consider M np 8. Given the arclist (0 1)(2 7)(3 6)(4 5) ,
the left cutting disk hooks up with the right cutting disk as
follows:
0 0 5 mod 8 3 4 4 5 mod 8 7
1 1 5 mod 8 4 5 5 5 mod 8 0
2 2 5 mod 8 5 6 6 5 mod 8 1
3 3 5 mod 8 6 7 7 5 mod 8 2
Christopher L. Toni, Donald Barkley Computational Contact Topology 13 / 20

42. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Tightness Checker
Consider M np 8. Given the arclist (0 1)(2 7)(3 6)(4 5) ,
the left cutting disk hooks up with the right cutting disk as
follows:
0 0 5 mod 8 3 4 4 5 mod 8 7
1 1 5 mod 8 4 5 5 5 mod 8 0
2 2 5 mod 8 5 6 6 5 mod 8 1
3 3 5 mod 8 6 7 7 5 mod 8 2
Using the arclist as a guide, the output be a list of numbers
Christopher L. Toni, Donald Barkley Computational Contact Topology 13 / 20

43. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Tightness Checker
Consider M np 8. Given the arclist (0 1)(2 7)(3 6)(4 5) ,
the left cutting disk hooks up with the right cutting disk as
follows:
0 0 5 mod 8 3 4 4 5 mod 8 7
1 1 5 mod 8 4 5 5 5 mod 8 0
2 2 5 mod 8 5 6 6 5 mod 8 1
3 3 5 mod 8 6 7 7 5 mod 8 2
Using the arclist as a guide, the output be a list of numbers
03636105472725410
Christopher L. Toni, Donald Barkley Computational Contact Topology 13 / 20

44. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Tightness Checker
Consider M np 8. Given the arclist (0 1)(2 7)(3 6)(4 5) ,
the left cutting disk hooks up with the right cutting disk as
follows:
0 0 5 mod 8 3 4 4 5 mod 8 7
1 1 5 mod 8 4 5 5 5 mod 8 0
2 2 5 mod 8 5 6 6 5 mod 8 1
3 3 5 mod 8 6 7 7 5 mod 8 2
Using the arclist as a guide, the output be a list of numbers
03636105472725410
0 36 36 10 54 72 72 54 10
Christopher L. Toni, Donald Barkley Computational Contact Topology 13 / 20

45. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Tightness Checker (cont.)
Consider M np 8.
Christopher L. Toni, Donald Barkley Computational Contact Topology 14 / 20

46. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Tightness Checker (cont.)
Consider M np 8. Given the arclist (0 7)(1 4)(2 3)(5 6) ,
the left cutting disk hooks up with the right cutting disk as
follows:
Christopher L. Toni, Donald Barkley Computational Contact Topology 14 / 20

47. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Tightness Checker (cont.)
Consider M np 8. Given the arclist (0 7)(1 4)(2 3)(5 6) ,
the left cutting disk hooks up with the right cutting disk as
follows:
0 0 5 mod 8 3 4 4 5 mod 8 7
1 1 5 mod 8 4 5 5 5 mod 8 0
2 2 5 mod 8 5 6 6 5 mod 8 1
3 3 5 mod 8 6 7 7 5 mod 8 2
Christopher L. Toni, Donald Barkley Computational Contact Topology 14 / 20

48. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Tightness Checker (cont.)
Consider M np 8. Given the arclist (0 7)(1 4)(2 3)(5 6) ,
the left cutting disk hooks up with the right cutting disk as
follows:
0 0 5 mod 8 3 4 4 5 mod 8 7
1 1 5 mod 8 4 5 5 5 mod 8 0
2 2 5 mod 8 5 6 6 5 mod 8 1
3 3 5 mod 8 6 7 7 5 mod 8 2
Using the arclist as a guide, the output be a list of numbers
Christopher L. Toni, Donald Barkley Computational Contact Topology 14 / 20

49. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Tightness Checker (cont.)
Consider M np 8. Given the arclist (0 7)(1 4)(2 3)(5 6) ,
the left cutting disk hooks up with the right cutting disk as
follows:
0 0 5 mod 8 3 4 4 5 mod 8 7
1 1 5 mod 8 4 5 5 5 mod 8 0
2 2 5 mod 8 5 6 6 5 mod 8 1
3 3 5 mod 8 6 7 7 5 mod 8 2
Using the arclist as a guide, the output be a list of numbers
0327032703270
Christopher L. Toni, Donald Barkley Computational Contact Topology 14 / 20

50. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Algorithm Output - Tightness Checker (cont.)
Consider M np 8. Given the arclist (0 7)(1 4)(2 3)(5 6) ,
the left cutting disk hooks up with the right cutting disk as
follows:
0 0 5 mod 8 3 4 4 5 mod 8 7
1 1 5 mod 8 4 5 5 5 mod 8 0
2 2 5 mod 8 5 6 6 5 mod 8 1
3 3 5 mod 8 6 7 7 5 mod 8 2
Using the arclist as a guide, the output be a list of numbers
0327032703270
0 32 70 32 70 32 70
Christopher L. Toni, Donald Barkley Computational Contact Topology 14 / 20

51. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Brief Overview - Bypasses
A bypass exists when a line can be drawn through three arcs
on a cutting disk.
Christopher L. Toni, Donald Barkley Computational Contact Topology 15 / 20

52. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Brief Overview - Bypasses
A bypass exists when a line can be drawn through three arcs
on a cutting disk.
Christopher L. Toni, Donald Barkley Computational Contact Topology 15 / 20

53. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Brief Overview - Bypasses
A bypass exists when a line can be drawn through three arcs
on a cutting disk.
There are two possible
bypasses on this cutting disk.
Christopher L. Toni, Donald Barkley Computational Contact Topology 15 / 20

54. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Brief Overview - Bypasses
A bypass exists when a line can be drawn through three arcs
on a cutting disk.
There are two possible There are no possible
bypasses on this cutting disk. bypasses on this cutting disk.
Christopher L. Toni, Donald Barkley Computational Contact Topology 15 / 20

55. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Brief Overview - Bypasses (cont.)
When a bypass is performed, it produces an already existing
arclist!
Christopher L. Toni, Donald Barkley Computational Contact Topology 16 / 20

56. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Brief Overview - Bypasses (cont.)
When a bypass is performed, it produces an already existing
arclist!
This is crucial in determining if these arclists form a tight
contact structure on the torus.
Christopher L. Toni, Donald Barkley Computational Contact Topology 16 / 20

57. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Brief Overview - Bypasses (cont.)
When a bypass is performed, it produces an already existing
arclist!
This is crucial in determining if these arclists form a tight
contact structure on the torus.
The bypass can be viewed as an equivalence relation
between arclists.
Christopher L. Toni, Donald Barkley Computational Contact Topology 16 / 20

58. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Brief Overview - Bypasses (cont.)
When a bypass is performed, it produces an already existing
arclist!
This is crucial in determining if these arclists form a tight
contact structure on the torus.
The bypass can be viewed as an equivalence relation
between arclists.
If one arclist is overtwisted in an equivalence class, the entire
equivalence class is associated to an overtwisted structure.
Christopher L. Toni, Donald Barkley Computational Contact Topology 16 / 20

59. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Brief Overview - Bypasses (cont.)
When a bypass is performed, it produces an already existing
arclist!
This is crucial in determining if these arclists form a tight
contact structure on the torus.
The bypass can be viewed as an equivalence relation
between arclists.
If one arclist is overtwisted in an equivalence class, the entire
equivalence class is associated to an overtwisted structure.This
saves time in the calculation process.
Christopher L. Toni, Donald Barkley Computational Contact Topology 16 / 20

60. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Results and Conclusions
1 Formula for computing the number of arclists for a given
number of vertices and web implementation of this formula.
Christopher L. Toni, Donald Barkley Computational Contact Topology 17 / 20

61. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Results and Conclusions
1 Formula for computing the number of arclists for a given
number of vertices and web implementation of this formula.
2 Software module to produce arclists For various number of
vertices.
Christopher L. Toni, Donald Barkley Computational Contact Topology 17 / 20

62. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Results and Conclusions
1 Formula for computing the number of arclists for a given
number of vertices and web implementation of this formula.
2 Software module to produce arclists For various number of
vertices.
3 Modification of succeeding software modules (bypass and
tightness checking) to read these arclists as input.
Christopher L. Toni, Donald Barkley Computational Contact Topology 17 / 20

63. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Results and Conclusions
1 Formula for computing the number of arclists for a given
number of vertices and web implementation of this formula.
2 Software module to produce arclists For various number of
vertices.
3 Modification of succeeding software modules (bypass and
tightness checking) to read these arclists as input.
4 Manually produced algorithms and results sets for various
values of n, p, q to be used for software testing.
Christopher L. Toni, Donald Barkley Computational Contact Topology 17 / 20

64. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Results and Conclusions (cont.)
N2 1 N 22 58786
N4 2 N 24 208012
N6 5 N 26 742900
N8 14 N 28 2674440
N 10 42 N 30 9694845
N 12 132 N 32 35357670
N 14 429 N 34 129644790
N 16 1430 N 36 477638700
N 18 4862 N 38 1767263190
N 20 16796 N 40 6564120420
Christopher L. Toni, Donald Barkley Computational Contact Topology 18 / 20

65. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Results and Conclusions (cont.)
N2 1 N 22 58786
N4 2 N 24 208012
N6 5 N 26 742900
N8 14 N 28 2674440
N 10 42 N 30 9694845
N 12 132 N 32 35357670
N 14 429 N 34 129644790
N 16 1430 N 36 477638700
N 18 4862 N 38 1767263190
N 20 16796 N 40 6564120420
Note that the number of arclists increase rapidly as the number
of vertices get larger. At M 28, its well over a million!
Christopher L. Toni, Donald Barkley Computational Contact Topology 18 / 20

66. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Future Research
Future goals include, but not limited to:
Christopher L. Toni, Donald Barkley Computational Contact Topology 19 / 20

67. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Future Research
Future goals include, but not limited to:
1 Publication of Findings in Undergraduate Journal
Christopher L. Toni, Donald Barkley Computational Contact Topology 19 / 20

68. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Future Research
Future goals include, but not limited to:
1 Publication of Findings in Undergraduate Journal
2 Extension of Algorithm to the two-holed torus
Christopher L. Toni, Donald Barkley Computational Contact Topology 19 / 20

69. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Future Research
Future goals include, but not limited to:
1 Publication of Findings in Undergraduate Journal
2 Extension of Algorithm to the two-holed torus
3 Enhance software (requiring a reduced memory footprint)
to produce results for larger number of vertices.
Christopher L. Toni, Donald Barkley Computational Contact Topology 19 / 20

70. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Acknowledgements
We would like to thank:
Christopher L. Toni, Donald Barkley Computational Contact Topology 20 / 20

71. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Acknowledgements
We would like to thank:
The SCSE (Dept. of Education) for funding the research
over summer.
Christopher L. Toni, Donald Barkley Computational Contact Topology 20 / 20

72. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Acknowledgements
We would like to thank:
The SCSE (Dept. of Education) for funding the research
over summer.
Christopher L. Toni, Donald Barkley Computational Contact Topology 20 / 20

73. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Acknowledgements
We would like to thank:
The SCSE (Dept. of Education) for funding the research
over summer.
Dr. Tanya Cofer for leading us through tough concepts and
tedious calculations.
Christopher L. Toni, Donald Barkley Computational Contact Topology 20 / 20

74. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Acknowledgements
We would like to thank:
The SCSE (Dept. of Education) for funding the research
over summer.
Dr. Tanya Cofer for leading us through tough concepts and
tedious calculations.
Donald Barkley for helping us program the algorithms in
Java.
Christopher L. Toni, Donald Barkley Computational Contact Topology 20 / 20

75. Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Acknowledgements
We would like to thank:
The SCSE (Dept. of Education) for funding the research
over summer.
Dr. Tanya Cofer for leading us through tough concepts and
tedious calculations.
Donald Barkley for helping us program the algorithms in
Java.
Donald Barkley will now talk about the programming part of the
project.
Christopher L. Toni, Donald Barkley Computational Contact Topology 20 / 20