Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions

Comparison of Methods That Check for TightContact Structures on the Solid Torus

ILSAMP Student Research Symposium

Kelly Hirschbeck Christopher L. Toni Donald BarkleySteven Jerome Dr. Tanya Cofer∗

February 13, 2009

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 1 / 16

Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions

OutlineIntroduction

Overview of the ProcessArcs and ArclistsTightness CheckingBypasses

Method 1: Hand CalculationsTightness CheckingBypasses

Method 2: PermutationsTightness CheckingBypasses

Results and Conclusions

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 2 / 16

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 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).

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 3 / 16

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 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).

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 3 / 16

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 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).

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 3 / 16

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

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 4 / 16

Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations 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.

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 5 / 16

Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations 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.

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 5 / 16

Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations 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.

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 5 / 16

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 givennumber of vertices np.

DefinitionAn arc is a path between vertices subject to:

I 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

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 givennumber of vertices np.

DefinitionAn arc is a path between vertices subject to:

I 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

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 aregenerated 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)

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 7 / 16

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 aregenerated 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)

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 7 / 16

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 rightcutting disk.

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 8 / 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 throughthree arcs on a cutting disk.

Two Abstract Bypasses. Zero Abstract Bypasses.

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 9 / 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 throughthree arcs on a cutting disk.

Two Abstract Bypasses. Zero Abstract Bypasses.

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 9 / 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 throughthree arcs on a cutting disk.

Two Abstract Bypasses. Zero Abstract Bypasses.

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 9 / 16

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 singlecurve.

It takes more than one curveto hook up all the vertices.

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 10 / 16

Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions

Bypasses

Abstract Bypasses

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

α

β

α

β

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

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

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 11 / 16

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

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−1AβA.Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 13 / 16

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

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 14 / 16

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

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 14 / 16

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

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 14 / 16

Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions

Bypasses

Existence of BypassesThe 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)

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 15 / 16

Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions

Bypasses

Existence of BypassesThe 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)

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 15 / 16

Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions

Bypasses

Existence of BypassesThe 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)

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 15 / 16

Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions

Bypasses

Existence of BypassesThe 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)

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 15 / 16

Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions

Future Research

Future goals include, but not limited to:

I Publication of Findings in Undergraduate Journal

I Extension of Algorithm to the two-holed torus

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

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 16 / 16

Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions

Future Research

Future goals include, but not limited to:

I Publication of Findings in Undergraduate Journal

I Extension of Algorithm to the two-holed torus

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

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 16 / 16

Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions

Future Research

Future goals include, but not limited to:

I Publication of Findings in Undergraduate Journal

I Extension of Algorithm to the two-holed torus

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

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 16 / 16

Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions

Future Research

Future goals include, but not limited to:

I Publication of Findings in Undergraduate Journal

I Extension of Algorithm to the two-holed torus

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

Kelly Hirschbeck, Christopher L. Toni

Computational Contact Topology - ILSAMP Symposium 16 / 16