Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

Programming an Algorithm on Calculating theNumber of Tight Contact Structures on the

Solid TorusArgonne Undergraduate Symposium

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

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

November 13, 2009

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 1 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

Outline

Introduction

Arcs and Arclists

Tightness Checking

Bypasses

Results and Conclusions

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 2 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

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:

I twistingI bendingI stretching

To illustrate this, imagine a coffee mug and a doughnut (torus).

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 3 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

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.

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 4 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

Formulating the ProblemOn 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.

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 5 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

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.

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 6 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

OverviewThe 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:

I All M vertices in a configuration must be pairedI 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!

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 7 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

Algorithm Output - Arcs and ArclistsFor 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)

(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

Computational Contact Topology - Argonne Symposium 8 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

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 the leftcutting 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.

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 9 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

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.

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 10 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

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.

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 11 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

Algorithm Output - Tightness CheckerConsider 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 = 3

1→ 1−5 mod 8 = 4

2→ 2−5 mod 8 = 5

3→ 3−5 mod 8 = 6

4→ 4−5 mod 8 = 7

5→ 5−5 mod 8 = 0

6→ 6−5 mod 8 = 1

7→ 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 , . . .

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 12 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

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 = 3

1→ 1−5 mod 8 = 4

2→ 2−5 mod 8 = 5

3→ 3−5 mod 8 = 6

4→ 4−5 mod 8 = 7

5→ 5−5 mod 8 = 0

6→ 6−5 mod 8 = 1

7→ 7−5 mod 8 = 2

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

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 , . . .

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 13 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

Brief Overview - BypassesA 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.

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 14 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

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.

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Computational Contact Topology - Argonne Symposium 15 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

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.

5. Representing arclists and bypasses as permutationmatrices and defining tightness as a certain product ofpermutation matrices (Cofer and Barkley, in preparation).

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Computational Contact Topology - Argonne Symposium 16 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

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. Searching for a formula for the case of four dividing curves.

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 17 / 18

Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

Acknowledgements

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

I Donald Barkley for helping us program the algorithms inJava.

I Argonne National Laboratory for giving me the opportunityto present my research.

Christopher L. Toni

Computational Contact Topology - Argonne Symposium 18 / 18