IntroductionPermutation Representation

Results and Conclusions

Using Permutations to Study a ClassificationProblem on the Solid Torus

Illinois Sectional MAA Meeting

Christopher L. Toni Dr. Tanya Cofer∗

April 10, 2010

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 1 / 19

IntroductionPermutation Representation

Results and Conclusions

Outline

1 Introduction

2 Permutation RepresentationArcs and ArclistsTightness CheckingBypasses

3 Results and Conclusions

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 2 / 19

IntroductionPermutation Representation

Results and Conclusions

Formulating Our Problem

On surfaces inside the solid torus (defined by S1×D2), dividingcurves are located where twisting planes switch from positive tonegative.

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

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 3 / 19

IntroductionPermutation Representation

Results and Conclusions

Formulating Our Problem

On surfaces inside the solid torus (defined by S1×D2), dividingcurves are located where twisting planes switch from positive tonegative.

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

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 3 / 19

IntroductionPermutation Representation

Results and Conclusions

Formulating Our Problem

On surfaces inside the solid torus (defined by S1×D2), dividingcurves are located where twisting planes switch from positive tonegative.

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

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 3 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Outline

1 Introduction

2 Permutation RepresentationArcs and ArclistsTightness CheckingBypasses

3 Results and Conclusions

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 4 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Overview

The first computational task is to generate arclists for a givennumber of vertices np.

DefinitionAn arc is a path between vertices subject to the conditions thatall vertices must be paired and arcs cannot intersect. An arclistis a set (list) of legal pairs of arcs.

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 5 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Overview

The first computational task is to generate arclists for a givennumber of vertices np.

DefinitionAn arc is a path between vertices subject to the conditions thatall vertices must be paired and arcs cannot intersect. An arclistis a set (list) of legal pairs of arcs.

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 5 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

How Do Permutations Apply Here?

Recall that a permutation is a bijective mapping of elementsfrom a set S to itself.

Let S = {0,1,2, . . . ,np−1} be the set of vertex values on acutting disk. We can define a permutation α on S that satisfiesthe definition of an arc/arclist.

Example: Consider the case np = 8. The set of vertex valueswould be S = {0,1,2, . . . ,6,7} and α = (01)(25)(34)(67) is apermutation on the set S.

There are 14 different permutations on this set that satisfy thedefinitions of an arc/arclist. �

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 6 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

How Do Permutations Apply Here?

Recall that a permutation is a bijective mapping of elementsfrom a set S to itself.

Let S = {0,1,2, . . . ,np−1} be the set of vertex values on acutting disk. We can define a permutation α on S that satisfiesthe definition of an arc/arclist.

Example: Consider the case np = 8. The set of vertex valueswould be S = {0,1,2, . . . ,6,7} and α = (01)(25)(34)(67) is apermutation on the set S.

There are 14 different permutations on this set that satisfy thedefinitions of an arc/arclist. �

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 6 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

How Do Permutations Apply Here?

Recall that a permutation is a bijective mapping of elementsfrom a set S to itself.

Let S = {0,1,2, . . . ,np−1} be the set of vertex values on acutting disk. We can define a permutation α on S that satisfiesthe definition of an arc/arclist.

Example: Consider the case np = 8. The set of vertex valueswould be S = {0,1,2, . . . ,6,7} and α = (01)(25)(34)(67) is apermutation on the set S.

There are 14 different permutations on this set that satisfy thedefinitions of an arc/arclist. �

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 6 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

How Do Permutations Apply Here?

Recall that a permutation is a bijective mapping of elementsfrom a set S to itself.

Let S = {0,1,2, . . . ,np−1} be the set of vertex values on acutting disk. We can define a permutation α on S that satisfiesthe definition of an arc/arclist.

Example: Consider the case np = 8. The set of vertex valueswould be S = {0,1,2, . . . ,6,7} and α = (01)(25)(34)(67) is apermutation on the set S.

There are 14 different permutations on this set that satisfy thedefinitions of an arc/arclist. �

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 6 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

How Do Permutations Apply Here? (Cont.)

Consider the example mentioned on the previous slide.

2

10

7

6

5 4

3

For this cutting disk, the arclist is {(0,1),(2,5),(3,4),(6,7)}.

We can easily rewrite this as the permutationα = (01)(25)(34)(67).

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 7 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

How Do Permutations Apply Here? (Cont.)

Consider the example mentioned on the previous slide.

2

10

7

6

5 4

3

For this cutting disk, the arclist is {(0,1),(2,5),(3,4),(6,7)}.

We can easily rewrite this as the permutationα = (01)(25)(34)(67).

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 7 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

How Do Permutations Apply Here? (Cont.)

Consider the example mentioned on the previous slide.

2

10

7

6

5 4

3

For this cutting disk, the arclist is {(0,1),(2,5),(3,4),(6,7)}.

We can easily rewrite this as the permutationα = (01)(25)(34)(67).

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 7 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Outline

1 Introduction

2 Permutation RepresentationArcs and ArclistsTightness CheckingBypasses

3 Results and Conclusions

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 8 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Overview - Tightness Checker

Potentially Tight Overtwisted

x→ x−nq+1 mod np

This maps the dividing curves on the surface from left to rightcutting disk.

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 9 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Using Permutations to Determine Tightness

Let β be a permutation that represents the mapping rulex→ x−nq+1 mod np and let A be the arclist permutation.

The permutation formula to check for tightness is β−1AβA.Christopher L. Toni Computational Contact Topology - ISMAA Meeting 10 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

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 = (01)(27)(36)(45) A = (07)(14)(23)(56)β−1AβA = (0246) β−1AβA = (0)

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 11 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

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 = (01)(27)(36)(45) A = (07)(14)(23)(56)β−1AβA = (0246) β−1AβA = (0)

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 11 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

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 = (01)(27)(36)(45) A = (07)(14)(23)(56)β−1AβA = (0246) β−1AβA = (0)

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 11 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Outline

1 Introduction

2 Permutation RepresentationArcs and ArclistsTightness CheckingBypasses

3 Results and Conclusions

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 12 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Abstract Bypasses

An abstract bypass exists when a line can be drawn throughthree arcs on a cutting disk.

Two Abstract Bypasses. . No Abstract Bypasses.

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 13 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Abstract Bypasses

An abstract bypass exists when a line can be drawn throughthree arcs on a cutting disk.

Two Abstract Bypasses. . No Abstract Bypasses.

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 13 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Abstract Bypasses

An abstract bypass exists when a line can be drawn throughthree arcs on a cutting disk.

Two Abstract Bypasses. . No Abstract Bypasses.

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 13 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Abstract Bypasses (Cont.)

(01)(25)(34)(67)

α

β

α

β

(05)(14)(23)(67)

(01)(23)(47)(56)

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 14 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Existence of Bypasses

The existence of actual bypasses is checked in a similarfashion as tightness.

Given: An arclist A and an abstract bypass C.

The formula: β−1AβC

A = (01)(25)(34)(67)β = (03614725)

β−1 = (05274163)

C1 = (05)(14)(23)(67) C2 = (01)(23)(47)(56)β−1AβC1 = (0624) β−1AβC2 = (0)

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 15 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Existence of Bypasses

The existence of actual bypasses is checked in a similarfashion as tightness.

Given: An arclist A and an abstract bypass C.

The formula: β−1AβC

A = (01)(25)(34)(67)β = (03614725)

β−1 = (05274163)

C1 = (05)(14)(23)(67) C2 = (01)(23)(47)(56)β−1AβC1 = (0624) β−1AβC2 = (0)

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 15 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Existence of Bypasses

The existence of actual bypasses is checked in a similarfashion as tightness.

Given: An arclist A and an abstract bypass C.

The formula: β−1AβC

A = (01)(25)(34)(67)β = (03614725)

β−1 = (05274163)

C1 = (05)(14)(23)(67) C2 = (01)(23)(47)(56)β−1AβC1 = (0624) β−1AβC2 = (0)

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 15 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Existence of Bypasses

The existence of actual bypasses is checked in a similarfashion as tightness.

Given: An arclist A and an abstract bypass C.

The formula: β−1AβC

A = (01)(25)(34)(67)β = (03614725)

β−1 = (05274163)

C1 = (05)(14)(23)(67) C2 = (01)(23)(47)(56)β−1AβC1 = (0624) β−1AβC2 = (0)

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 15 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Abstract Bypass Generators

Question: How do we identify abstract bypassesalgorithmically without the luxury of pictures?

TheoremFor every set of np vertices, there are special permutations thatdetect abstract bypasses.

TheoremGiven np, we can generate all abstract bypass generators.

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 16 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Abstract Bypass Generators

Question: How do we identify abstract bypassesalgorithmically without the luxury of pictures?

TheoremFor every set of np vertices, there are special permutations thatdetect abstract bypasses.

TheoremGiven np, we can generate all abstract bypass generators.

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 16 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Abstract Bypass Generators

Question: How do we identify abstract bypassesalgorithmically without the luxury of pictures?

TheoremFor every set of np vertices, there are special permutations thatdetect abstract bypasses.

TheoremGiven np, we can generate all abstract bypass generators.

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 16 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Abstract Bypass Generators (cont.)

In the case of np = 8, we have the following bypass generators:

γ1 = (042)(153)

γ2 = (064)(175)

γ3 = (062)(173)

γ4 = (246)(375)

γ5 = (062)(175)

γ6 = (153)(264)

γ7 = (064)(375)

γ8 = (042)(173)

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 17 / 19

IntroductionPermutation Representation

Results and Conclusions

Arcs and ArclistsTightness CheckingBypasses

Abstract Bypass Generators (cont.)

Consider the arclist α = (01)(25)(34)(67). Applying theabstract bypass generators on the previous slide, we get:

γ1 ◦α = (05)(14)(23)(67)

γ2 ◦α = (0743)(1652)

γ3 ◦α = (0725)(1634)

γ4 ◦α = (01)(23)(47)(56)

γ5 ◦α = (072165)(34)

γ6 ◦α = (056741)(23)

γ7 ◦α = (016523)(47)

γ8 ◦α = (076325)(14)

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 18 / 19

IntroductionPermutation Representation

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.

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 19 / 19

IntroductionPermutation Representation

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.

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 19 / 19

IntroductionPermutation Representation

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.

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 19 / 19

IntroductionPermutation Representation

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.

Christopher L. Toni Computational Contact Topology - ISMAA Meeting 19 / 19