Page 1
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
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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
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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
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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
Page 5
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
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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
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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
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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
Page 9
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
Page 10
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
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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
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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
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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
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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
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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
Page 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
Page 17
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
Page 18
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
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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
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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
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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
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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
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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
Page 24
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
Page 25
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
Page 26
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
Page 27
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
Page 28
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
Page 29
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
Page 30
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
Page 31
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
Page 32
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