neiu permutation algebra

IntroductionPermutation Representation

ConjecturesResults and Conclusions

Using Permutations to Study a ClassificationProblem on the Solid Torus

NEIU 18th Annual Student Research Symposium

Kelly Hirschbeck Christopher Toni Dr. Tanya Cofer∗

April 16, 2010

Kelly Hirschbeck, Christopher Toni Computational Contact Topology - NEIU Symposium 1 / 30

Outline

1 Introduction

2 Permutation RepresentationArcs and ArclistsTightness CheckingBypasses

3 Conjectures

4 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.

Formulating Our Problem (cont.)

n is the number of dividing curves

p is the number of longitudinal wraps

q is the number of meridinal wraps

Arcs and ArclistsTightness CheckingBypasses

Outline

1 Introduction

3 Conjectures

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.

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.

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. �

How Do Permutations Apply Here? (Cont.)

Consider the example mentioned on the previous slide.

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).

Outline

1 Introduction

3 Conjectures

Overview - Tightness Checking

Potentially Tight Overtwisted

x→ x−nq+1 mod np

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

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.Kelly Hirschbeck, Christopher Toni Computational Contact Topology - NEIU Symposium 11 / 30

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)

Permutation Example

Given: n = 2, p = 4 ,q = 3

β−1 = (05274163)

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

Permutation Example

Given: n = 2, p = 4 ,q = 3

β−1 = (05274163)

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

Outline

1 Introduction

3 Conjectures

Abstract Bypasses

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

Two Abstract Bypasses. . No Abstract Bypasses.

Abstract Bypasses

Abstract Bypasses (Cont.)

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

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

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

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)

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)

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)

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)

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.

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)

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)

“Star Wars” Graphs

Tori for (2,4,1) and (8,1,1) Cases

.(2,4,1) torus (8,1,1) torus

Comparing the (2, p,1) and (n,1,1) Cases

(2, p,1) (n,1,1)

x→ x−nq+1 mod np x→ x−nq+1 mod np

x→ x−1 mod np x→ x+1 mod np

β = (0(np−1)(np−2) . . .321) β = (0123 . . .(np−2)(np−1))

β−1 = (01 . . .(np−2)(np−1)) β−1 = (0(np−1)(np−2) . . .1)

Example of (2, p,1) Example of (n,1,1)

.A = (01)(25)(34)(67)

(2,4,1)β = (07654321)

β−1 = (01234567)

β−1Aβ = (07)(12)(36)(45)

(8,1,1)β = (01234567)

β−1 = (07654321)

β−1Aβ = (07)(14)(23)(56)

Torus diagrams

Conjectures for the (2, p,1) and (n,1,1) cases

(2, p,1)

Every possible arc configurationis potentially tight

Every abstract bypass isan actual bypass

All arcs are actually tight

p equivalence classesand tight structures

(n,1,1)

Every possible arc configurationis potentially tight

All abstract bypasses are notactual bypasses

All arcs are actually tight

Every arc list is its ownequivalence class

Future Research

Future goals include, but not limited to:

Publication of Findings in Undergraduate Journal

Proving our Current Conjectures.

Extension of Algorithm to the two-holed torus

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

Future Research

Acknowledgements

We would like to thank:

Dr. Tanya Cofer for:

Giving us the opportunity to conduct graduate levelresearch as undergraduates.

Being an amazing person to work with.

Being supportive in every way possible.

Believing in us!!!!

NEIU for giving us the opportunity to present our research.

Acknowledgements

Believing in us!!!!

Acknowledgements

Believing in us!!!!

Acknowledgements

Believing in us!!!!

Acknowledgements

Believing in us!!!!

Acknowledgements

Believing in us!!!!

Acknowledgements

Believing in us!!!!

neiu permutation algebra

Documents