SCSE Symposium Presentation

IntroductionArcs and Arclists

Tightness CheckingBypasses

Final Results and Thoughts

Improving on an Existing Program That Checksfor Tight Contact Structures on the Solid TorusNortheastern Illinois University SCSE Research Symposium

Christopher L. Toni Kelly Hirschbeck Nathan WalterWilliam Krepelin Donald Barkley William Byrd

John Wallin Mayra Bravo-Gonzalez Banlieman KolaniDr. Tanya Cofer∗

October 2, 2009

Christopher L. Toni, Donald Barkley Computational Contact Topology 1 / 20




Outline

1 Introduction

2 Arcs and Arclists

3 Tightness Checking

4 Bypasses

5 Final Results and Thoughts





What is Topology?

Topology is a field of mathematics that does not focus on anobject’s shape, but the properties that remain consistentthrough deformations like:

1 twisting2 bending3 stretching

To illustrate this, visualize a coffee cup and a doughnut (torus).





What is Topology?


1 twisting

2 bending3 stretching






What is Topology?


1 twisting2 bending

3 stretching






What is Topology?








What is Topology?








What is Topology? (cont.)

The torus and the coffee cup are topologically equivalentobjects. We see above that through bending and stretching, thetorus can be morphed into a coffee cup and vice versa.





Formulating the Problem

On the solid torus (defined by S1×D2), dividing curves arelocated where twisting planes switch from positive to negative.

These dividing curves keep track of and allow for investigationof certain topological properties in the neighborhood of asurface.



















Formulating the Problem (cont.)

We define n to be the number of dividing curves, p to be thenumber of times the dividing curves are wrapped about thelongitudinal section of the torus, and q to be the number oftimes the dividing curves are wrapped about the meridinalsection of the torus.





Formulating the Problem (cont.)

We define n to be the number of dividing curves, p to be thenumber of times the dividing curves are wrapped about thelongitudinal section of the torus, and q to be the number oftimes the dividing curves are wrapped about the meridinalsection of the torus.





Overview

The first computational task is to generate arclists for a givennumber of vertices M, where M = np.

DefinitionAn arc is a path between vertices subject to:

1 All M vertices in a configuration must be paired2 Paths cannot cross

An arclist is a set (list) of legal pairs of arcs.

We can think of arclists for M vertices as certain permutationsof M objects.The solution is to “walk” a new element throughthe solution set for a smaller problem. There is one challenge:the algorithm is space and time intensive!





Overview










Overview



1 All M vertices in a configuration must be paired

2 Paths cannot cross







Overview










Overview










Overview





We can think of arclists for M vertices as certain permutationsof M objects.

The solution is to “walk” a new element throughthe solution set for a smaller problem. There is one challenge:the algorithm is space and time intensive!





Overview





We can think of arclists for M vertices as certain permutationsof M objects.The solution is to “walk” a new element throughthe solution set for a smaller problem.

There is one challenge:the algorithm is space and time intensive!





Overview










Algorithm - Arcs and Arclist





Algorithm - Arcs and Arclist





Algorithm Output - Arcs and Arclists

For the case of n = 2, p = 4, q = 3, we haveM = 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)

(0 5)(1 2)(3 4)(6 7)(0 7)(1 4)(2 3)(5 6)(0 1)(2 3)(4 7)(5 6)(0 5)(1 4)(2 3)(6 7)(0 7)(1 2)(3 6)(4 5)(0 7)(1 6)(2 5)(3 4)(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 Checkingmodule.






For the case of n = 2, p = 4, q = 3, we haveM = 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)

(0 5)(1 2)(3 4)(6 7)(0 7)(1 4)(2 3)(5 6)(0 1)(2 3)(4 7)(5 6)(0 5)(1 4)(2 3)(6 7)(0 7)(1 2)(3 6)(4 5)(0 7)(1 6)(2 5)(3 4)(0 7)(1 6)(2 3)(4 5)








(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)

(0 5)(1 2)(3 4)(6 7)(0 7)(1 4)(2 3)(5 6)(0 1)(2 3)(4 7)(5 6)(0 5)(1 4)(2 3)(6 7)(0 7)(1 2)(3 6)(4 5)(0 7)(1 6)(2 5)(3 4)(0 7)(1 6)(2 3)(4 5)








(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)

(0 5)(1 2)(3 4)(6 7)(0 7)(1 4)(2 3)(5 6)(0 1)(2 3)(4 7)(5 6)(0 5)(1 4)(2 3)(6 7)(0 7)(1 2)(3 6)(4 5)(0 7)(1 6)(2 5)(3 4)(0 7)(1 6)(2 3)(4 5)








(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)

(0 5)(1 2)(3 4)(6 7)(0 7)(1 4)(2 3)(5 6)(0 1)(2 3)(4 7)(5 6)(0 5)(1 4)(2 3)(6 7)(0 7)(1 2)(3 6)(4 5)(0 7)(1 6)(2 5)(3 4)(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 Checkingmodule. Christopher L. Toni, Donald Barkley Computational Contact Topology 9 / 20




Overview - Tightness Checker

Once the arclists are found, it is possible to determine how thevertices on the left and right cutting disks match up.

The formula x → x−nq+1 mod np maps the vertices on theleft cutting disk to the right cutting disk.

The formula x → x +nq−1 mod np maps the vertices on theright cutting disk to the left cutting disk.

To determine if the torus admits a tight or overtwisted contactstructure, the dividing curves and arcs need to be analyzed.
































Overview - Tightness Checker (cont.)

If a single closed curve can betraced on the torus, it isconsidered to be a potentiallytight contact structure.

If more than one closed curvecan be traced on the torus, itis considered to be anovertwisted structure.



















Algorithm - Tightness Checker

All vertices hook up to a singlecurve. Thus, the structure is

potentially tight.

Only a few vertices hook up toa curve. Thus, the structure is

overtwisted.







potentially tight.


overtwisted.







potentially tight.


overtwisted.





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 asfollows:

0→ 0−5 mod 8 = 31→ 1−5 mod 8 = 42→ 2−5 mod 8 = 53→ 3−5 mod 8 = 6

4→ 4−5 mod 8 = 75→ 5−5 mod 8 = 06→ 6−5 mod 8 = 17→ 7−5 mod 8 = 2

Using the arclist as a guide, the output be a list of numbers

0,3,6,3,6,1,0,5,4,7,2,7,2,5,4,1,0, . . .

0, 3,6 , 3,6 , 1,0 , 5,4 , 7,2 , 7,2 , 5,4 , 1,0 , . . .






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 asfollows:

0→ 0−5 mod 8 = 31→ 1−5 mod 8 = 42→ 2−5 mod 8 = 53→ 3−5 mod 8 = 6

4→ 4−5 mod 8 = 75→ 5−5 mod 8 = 06→ 6−5 mod 8 = 17→ 7−5 mod 8 = 2


0,3,6,3,6,1,0,5,4,7,2,7,2,5,4,1,0, . . .

0, 3,6 , 3,6 , 1,0 , 5,4 , 7,2 , 7,2 , 5,4 , 1,0 , . . .







0→ 0−5 mod 8 = 31→ 1−5 mod 8 = 42→ 2−5 mod 8 = 53→ 3−5 mod 8 = 6

4→ 4−5 mod 8 = 75→ 5−5 mod 8 = 06→ 6−5 mod 8 = 17→ 7−5 mod 8 = 2


0,3,6,3,6,1,0,5,4,7,2,7,2,5,4,1,0, . . .

0, 3,6 , 3,6 , 1,0 , 5,4 , 7,2 , 7,2 , 5,4 , 1,0 , . . .







0→ 0−5 mod 8 = 31→ 1−5 mod 8 = 42→ 2−5 mod 8 = 53→ 3−5 mod 8 = 6

4→ 4−5 mod 8 = 75→ 5−5 mod 8 = 06→ 6−5 mod 8 = 17→ 7−5 mod 8 = 2


0,3,6,3,6,1,0,5,4,7,2,7,2,5,4,1,0, . . .

0, 3,6 , 3,6 , 1,0 , 5,4 , 7,2 , 7,2 , 5,4 , 1,0 , . . .







0→ 0−5 mod 8 = 31→ 1−5 mod 8 = 42→ 2−5 mod 8 = 53→ 3−5 mod 8 = 6

4→ 4−5 mod 8 = 75→ 5−5 mod 8 = 06→ 6−5 mod 8 = 17→ 7−5 mod 8 = 2


0,3,6,3,6,1,0,5,4,7,2,7,2,5,4,1,0, . . .

0, 3,6 , 3,6 , 1,0 , 5,4 , 7,2 , 7,2 , 5,4 , 1,0 , . . .







0→ 0−5 mod 8 = 31→ 1−5 mod 8 = 42→ 2−5 mod 8 = 53→ 3−5 mod 8 = 6

4→ 4−5 mod 8 = 75→ 5−5 mod 8 = 06→ 6−5 mod 8 = 17→ 7−5 mod 8 = 2


0,3,6,3,6,1,0,5,4,7,2,7,2,5,4,1,0, . . .

0, 3,6 , 3,6 , 1,0 , 5,4 , 7,2 , 7,2 , 5,4 , 1,0 , . . .





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 asfollows:

0→ 0−5 mod 8 = 31→ 1−5 mod 8 = 42→ 2−5 mod 8 = 53→ 3−5 mod 8 = 6

4→ 4−5 mod 8 = 75→ 5−5 mod 8 = 06→ 6−5 mod 8 = 17→ 7−5 mod 8 = 2


0,3,2,7,0,3,2,7,0,3,2,7,0 . . .

0, 3,2 , 7,0 , 3,2 , 7,0 , 3,2 , 7,0 , . . .







0→ 0−5 mod 8 = 31→ 1−5 mod 8 = 42→ 2−5 mod 8 = 53→ 3−5 mod 8 = 6

4→ 4−5 mod 8 = 75→ 5−5 mod 8 = 06→ 6−5 mod 8 = 17→ 7−5 mod 8 = 2


0,3,2,7,0,3,2,7,0,3,2,7,0 . . .

0, 3,2 , 7,0 , 3,2 , 7,0 , 3,2 , 7,0 , . . .







0→ 0−5 mod 8 = 31→ 1−5 mod 8 = 42→ 2−5 mod 8 = 53→ 3−5 mod 8 = 6

4→ 4−5 mod 8 = 75→ 5−5 mod 8 = 06→ 6−5 mod 8 = 17→ 7−5 mod 8 = 2


0,3,2,7,0,3,2,7,0,3,2,7,0 . . .

0, 3,2 , 7,0 , 3,2 , 7,0 , 3,2 , 7,0 , . . .







0→ 0−5 mod 8 = 31→ 1−5 mod 8 = 42→ 2−5 mod 8 = 53→ 3−5 mod 8 = 6

4→ 4−5 mod 8 = 75→ 5−5 mod 8 = 06→ 6−5 mod 8 = 17→ 7−5 mod 8 = 2


0,3,2,7,0,3,2,7,0,3,2,7,0 . . .

0, 3,2 , 7,0 , 3,2 , 7,0 , 3,2 , 7,0 , . . .







0→ 0−5 mod 8 = 31→ 1−5 mod 8 = 42→ 2−5 mod 8 = 53→ 3−5 mod 8 = 6

4→ 4−5 mod 8 = 75→ 5−5 mod 8 = 06→ 6−5 mod 8 = 17→ 7−5 mod 8 = 2


0,3,2,7,0,3,2,7,0,3,2,7,0 . . .

0, 3,2 , 7,0 , 3,2 , 7,0 , 3,2 , 7,0 , . . .







0→ 0−5 mod 8 = 31→ 1−5 mod 8 = 42→ 2−5 mod 8 = 53→ 3−5 mod 8 = 6

4→ 4−5 mod 8 = 75→ 5−5 mod 8 = 06→ 6−5 mod 8 = 17→ 7−5 mod 8 = 2


0,3,2,7,0,3,2,7,0,3,2,7,0 . . .

0, 3,2 , 7,0 , 3,2 , 7,0 , 3,2 , 7,0 , . . .





Brief Overview - Bypasses

A bypass exists when a line can be drawn through three arcson a cutting disk.

There are two possiblebypasses on this cutting disk.

There are no possiblebypasses on this cutting disk.





























Brief Overview - Bypasses (cont.)

When a bypass is performed, it produces an already existingarclist!

This is crucial in determining if these arclists form a tightcontact structure on the torus.

The bypass can be viewed as an equivalence relationbetween arclists.

If one arclist is overtwisted in an equivalence class, the entireequivalence class is associated to an overtwisted structure.Thissaves time in the calculation process.



























If one arclist is overtwisted in an equivalence class, the entireequivalence class is associated to an overtwisted structure.

Thissaves time in the calculation process.














Results and Conclusions

1 Formula for computing the number of arclists for a givennumber of vertices and web implementation of this formula.

2 Software module to produce arclists For various number ofvertices.

3 Modification of succeeding software modules (bypass andtightness checking) to read these arclists as input.

4 Manually produced algorithms and results sets for variousvalues of n, p, q to be used for software testing.
































Results and Conclusions (cont.)

N[2] = 1N[4] = 2N[6] = 5N[8] = 14

N[10] = 42N[12] = 132N[14] = 429N[16] = 1430N[18] = 4862N[20] = 16796

N[22] = 58786N[24] = 208012N[26] = 742900N[28] = 2674440N[30] = 9694845N[32] = 35357670N[34] = 129644790N[36] = 477638700N[38] = 1767263190N[40] = 6564120420

Note that the number of arclists increase rapidly as the numberof vertices get larger. At M = 28, its well over a million!





Results and Conclusions (cont.)

N[2] = 1N[4] = 2N[6] = 5N[8] = 14

N[10] = 42N[12] = 132N[14] = 429N[16] = 1430N[18] = 4862N[20] = 16796

N[22] = 58786N[24] = 208012N[26] = 742900N[28] = 2674440N[30] = 9694845N[32] = 35357670N[34] = 129644790N[36] = 477638700N[38] = 1767263190N[40] = 6564120420

Note that the number of arclists increase rapidly as the numberof vertices get larger. At M = 28, its well over a million!





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.





Future Research









Future Research









Future Research









Acknowledgements

We would like to thank:

∙ The SCSE (Dept. of Education) for funding the researchover summer.

∙ Dr. Tanya Cofer for leading us through tough concepts andtedious calculations.

∙ Donald Barkley for helping us program the algorithms inJava.

Donald Barkley will now talk about the programming part of theproject.





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