The document compares two methods for checking tight contact structures on the solid torus: hand calculations and permutations. It outlines generating arcs and arclists, checking for tightness and bypasses, and applying each method. Future work includes publishing the findings, extending the algorithm to more complex surfaces, and searching for formulas to handle additional dividing curves.

1. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Comparison of Methods That Check for Tight
Contact Structures on the Solid Torus
ILSAMP Student Research Symposium
Kelly Hirschbeck Christopher L. Toni Donald Barkley
Steven Jerome Dr. Tanya Cofer∗
February 13, 2009
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 1 / 16

2. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Outline
Introduction
Overview of the Process
Arcs and Arclists
Tightness Checking
Bypasses
Method 1: Hand Calculations
Tightness Checking
Bypasses
Method 2: Permutations
Tightness Checking
Bypasses
Results and Conclusions
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 2 / 16

3. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
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:
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 3 / 16

4. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
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:
twisting
bending
stretching
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 3 / 16

5. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
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:
twisting
bending
stretching
To illustrate this, imagine a coffee mug and a doughnut (torus).
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 3 / 16

6. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
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.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 4 / 16

7. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Formulating the Problem
On the solid torus (defined by S1 × D2 ), dividing curves are
located where twisting planes switch from positive to negative.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 5 / 16

8. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Formulating the Problem
On the solid torus (defined by S1 × D2 ), dividing curves are
located where twisting planes switch from positive to negative.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 5 / 16

9. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Formulating the Problem
On 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.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 5 / 16

10. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Arcs and Arclists
Overview
The first computational task is to generate arclists for a given
number of vertices np.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 6 / 16

11. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Arcs and Arclists
Overview
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:
All vertices must be paired and arcs cannot intersect
An arclist is a set (list) of legal pairs of arcs.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 6 / 16

12. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Arcs and Arclists
Algorithm Output - Arcs and Arclists
When np = 8, there are 8 vertices. The arclists that are
generated are:
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 7 / 16

13. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Arcs and Arclists
Algorithm Output - Arcs and Arclists
When np = 8, there are 8 vertices. The arclists that are
generated 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)
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 7 / 16

14. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Tightness Checking
Overview - 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.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 8 / 16

15. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Bypasses
Brief Overview - Bypasses
An abstract bypass exists when a line can be drawn through
three arcs on a cutting disk.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 9 / 16

16. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Bypasses
Brief Overview - Bypasses
An abstract bypass exists when a line can be drawn through
three arcs on a cutting disk.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 9 / 16

17. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Bypasses
Brief Overview - Bypasses
An abstract bypass exists when a line can be drawn through
three arcs on a cutting disk.
Two Abstract Bypasses. Zero Abstract Bypasses.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 9 / 16

18. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Tightness Checking
Checking for Tightness
(01) (27) (36) (45) (07) (14) (23) (56)
All vertices hook up to a single It takes more than one curve
curve. to hook up all the vertices.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 10 / 16

19. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Bypasses
Abstract Bypasses (05) (14) (23) (67)
(01) (25) (34) (67)
α
α
β
(01) (23) (47) (56)
β
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 11 / 16

20. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Bypasses
Checking for Actual Bypasses
(05) (14) (23) (67) (01) (23) (47) (56)
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 12 / 16

21. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Tightness Checking
Revisiting Method One (Developed by Dr. Cofer)
Recall the mapping rule: x → x − nq + 1 mod np.
Therefore, the formula to check for tightness: β −1 Aβ A.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 13 / 16

22. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Tightness Checking
Permutation Example
Given: n = 2, p = 4 ,q = 3
The mapping rule tells us x → x − 5 mod 8.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 14 / 16

23. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Tightness Checking
Permutation Example
Given: n = 2, p = 4 ,q = 3
The mapping rule tells us x → x − 5 mod 8.
Therefore, β = (03614725)
β −1 = (05274163)
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 14 / 16

24. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Tightness Checking
Permutation 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)
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 14 / 16

25. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Bypasses
Existence of Bypasses
The existence of actual bypasses is checked in a similar
fashion as tightness.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 15 / 16

26. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Bypasses
Existence of Bypasses
The existence of actual bypasses is checked in a similar
fashion as tightness.
Given: An arclist A and an abstract bypass C.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 15 / 16

27. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Bypasses
Existence 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
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 15 / 16

28. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Bypasses
Existence 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)
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 15 / 16

29. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Future Research
Future goals include, but not limited to:
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 16 / 16

30. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Future Research
Future goals include, but not limited to:
Publication of Findings in Undergraduate Journal
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 16 / 16

31. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Future Research
Future goals include, but not limited to:
Publication of Findings in Undergraduate Journal
Extension of Algorithm to the two-holed torus
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 16 / 16

32. Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions
Future 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.
Kelly Hirschbeck, Christopher L. Toni
Computational Contact Topology - ILSAMP Symposium 16 / 16