This document outlines research on calculating the number of tight contact structures on a solid torus using computational topology. It discusses generating arc and arclist data structures to represent dividing curves on the torus for given values of n, p, and q. It then describes using a tightness checking algorithm to determine how the dividing curves match up between the left and right cutting disks of the torus based on a mapping formula. The results are used to identify tight contact structures for different parameter values.

1. Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Programming an Algorithm on Calculating the
Number of Tight Contact Structures on the
Solid Torus
Argonne Symposium – Argonne National Laboratory
Christopher L. Toni Kelly Hirschbeck Nathan Walter
William Krepelin Donald Barkley William Byrd
John Wallin Mayra Bravo-Gonzalez Banlieman Kolani
Dr. Tanya Cofer
November 13, 2009
Christopher L. Toni
Computational Contact Topology 1 / 21

2. Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Outline
Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Christopher L. Toni
Computational Contact Topology 2 / 21

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
Computational Contact Topology 3 / 21

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
Computational Contact Topology 3 / 21

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
Computational Contact Topology 3 / 21

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
Computational Contact Topology 3 / 21

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
Computational Contact Topology 3 / 21

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
Computational Contact Topology 4 / 21

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
Computational Contact Topology 5 / 21

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
Computational Contact Topology 5 / 21

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
Computational Contact Topology 5 / 21

12. Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Formulating the Problem (cont.)
Christopher L. Toni
Computational Contact Topology 6 / 21

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
Computational Contact Topology 6 / 21

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
Computational Contact Topology 7 / 21

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
Computational Contact Topology 7 / 21

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
Computational Contact Topology 7 / 21

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
Computational Contact Topology 7 / 21

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
Computational Contact Topology 7 / 21

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
Computational Contact Topology 7 / 21

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
Computational Contact Topology 7 / 21

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
Computational Contact Topology 7 / 21

22. Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Algorithm - Arcs and Arclist
Christopher L. Toni
Computational Contact Topology 8 / 21

23. Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Algorithm - Arcs and Arclist
Christopher L. Toni
Computational Contact Topology 8 / 21

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
Computational Contact Topology 9 / 21

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
Computational Contact Topology 9 / 21

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
Computational Contact Topology 9 / 21

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
Computational Contact Topology 9 / 21

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
module.
Christopher L. Toni
Computational Contact Topology 9 / 21

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
Computational Contact Topology 10 / 21

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
Computational Contact Topology 10 / 21

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
Computational Contact Topology 10 / 21

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
Computational Contact Topology 10 / 21

33. Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Overview - Tightness Checker (cont.)
Christopher L. Toni
Computational Contact Topology 11 / 21

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
Computational Contact Topology 11 / 21

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
Computational Contact Topology 11 / 21

36. Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Algorithm - Tightness Checker
Christopher L. Toni
Computational Contact Topology 12 / 21

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
Computational Contact Topology 12 / 21

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
Computational Contact Topology 12 / 21

39. Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Algorithm Output - Tightness Checker
Consider M np 8.
Christopher L. Toni
Computational Contact Topology 13 / 21

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
Computational Contact Topology 13 / 21

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
Computational Contact Topology 13 / 21

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
Computational Contact Topology 13 / 21

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
Computational Contact Topology 13 / 21

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
Computational Contact Topology 13 / 21

45. Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Algorithm Output - Tightness Checker (cont.)
Consider M np 8.
Christopher L. Toni
Computational Contact Topology 14 / 21

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
Computational Contact Topology 14 / 21

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
Computational Contact Topology 14 / 21

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
Computational Contact Topology 14 / 21

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
Computational Contact Topology 14 / 21

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
Computational Contact Topology 14 / 21

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
Computational Contact Topology 15 / 21

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
Computational Contact Topology 15 / 21

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
Computational Contact Topology 15 / 21

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
Computational Contact Topology 15 / 21

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
Computational Contact Topology 16 / 21

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
Computational Contact Topology 16 / 21

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
Computational Contact Topology 16 / 21

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
Computational Contact Topology 16 / 21

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
Computational Contact Topology 16 / 21

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
Computational Contact Topology 17 / 21

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
Computational Contact Topology 17 / 21

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
Computational Contact Topology 17 / 21

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
Computational Contact Topology 17 / 21

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
Computational Contact Topology 18 / 21

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
Computational Contact Topology 18 / 21

66. Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Recent Findings
Instead of approching this problem from a combinatorial
standpoint, we now introduce a “simpler” way of generating
arclists, bypasses, and checking for tightness.
Christopher L. Toni
Computational Contact Topology 19 / 21

67. Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Recent Findings
Instead of approching this problem from a combinatorial
standpoint, we now introduce a “simpler” way of generating
arclists, bypasses, and checking for tightness.
The problem can be tackled using permutation matrices!!!
Christopher L. Toni
Computational Contact Topology 19 / 21

68. Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Future Research
Future goals include, but not limited to:
Christopher L. Toni
Computational Contact Topology 20 / 21

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
Christopher L. Toni
Computational Contact Topology 20 / 21

70. 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
Computational Contact Topology 20 / 21

71. 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
Computational Contact Topology 20 / 21

72. Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Acknowledgements
We would like to thank:
Christopher L. Toni
Computational Contact Topology 21 / 21

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.
Christopher L. Toni
Computational Contact Topology 21 / 21

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.
Christopher L. Toni
Computational Contact Topology 21 / 21

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.
Christopher L. Toni
Computational Contact Topology 21 / 21

76. 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
Computational Contact Topology 21 / 21

77. 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.
Argonne National Laboratory for giving me the opportunity
of presenting my group’s summer research.
Christopher L. Toni
Computational Contact Topology 21 / 21

78. 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.
Argonne National Laboratory for giving me the opportunity
of presenting my group’s summer research.
Christopher L. Toni
Computational Contact Topology 21 / 21