contact topology argonne
TRANSCRIPT
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Programming an Algorithm on Calculating theNumber of Tight Contact Structures on the
Solid TorusArgonne Symposium – Argonne National Laboratory
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 1 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Outline
Introduction
Arcs and Arclists
Tightness Checking
Bypasses
Final Results and Thoughts
Christopher L. Toni
Computational Contact Topology 2 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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:
1. twisting2. bending3. stretching
To illustrate this, visualize a coffee cup and a doughnut (torus).
Christopher L. Toni
Computational Contact Topology 3 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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:
1. twisting
2. bending3. stretching
To illustrate this, visualize a coffee cup and a doughnut (torus).
Christopher L. Toni
Computational Contact Topology 3 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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:
1. twisting2. bending
3. stretching
To illustrate this, visualize a coffee cup and a doughnut (torus).
Christopher L. Toni
Computational Contact Topology 3 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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:
1. twisting2. bending3. stretching
To illustrate this, visualize a coffee cup and a doughnut (torus).
Christopher L. Toni
Computational Contact Topology 3 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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:
1. twisting2. bending3. stretching
To illustrate this, visualize a coffee cup and a doughnut (torus).
Christopher L. Toni
Computational Contact Topology 3 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 4 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 5 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 5 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 5 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 6 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 6 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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:
1. All M vertices in a configuration must be paired2. 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 7 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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:
1. All M vertices in a configuration must be paired2. 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 7 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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:
1. All M vertices in a configuration must be paired
2. 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 7 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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:
1. All M vertices in a configuration must be paired2. 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 7 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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:
1. All M vertices in a configuration must be paired2. 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 7 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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:
1. All M vertices in a configuration must be paired2. 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 7 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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:
1. All M vertices in a configuration must be paired2. 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 7 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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:
1. All M vertices in a configuration must be paired2. 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 7 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Algorithm - Arcs and Arclist
Christopher L. Toni
Computational Contact Topology 8 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Algorithm - Arcs and Arclist
Christopher L. Toni
Computational Contact Topology 8 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 9 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 9 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 9 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 9 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 9 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 10 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 10 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 10 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 10 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 11 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 11 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 11 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 12 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 12 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 12 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 13 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 13 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 13 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 13 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 13 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 13 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 14 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 14 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 14 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 14 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 14 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 14 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 15 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 15 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 15 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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 15 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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.
Christopher L. Toni
Computational Contact Topology 16 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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.
Christopher L. Toni
Computational Contact Topology 16 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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.
Christopher L. Toni
Computational Contact Topology 16 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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.
Christopher L. Toni
Computational Contact Topology 16 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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.
Christopher L. Toni
Computational Contact Topology 16 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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.
Christopher L. Toni
Computational Contact Topology 17 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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.
Christopher L. Toni
Computational Contact Topology 17 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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.
Christopher L. Toni
Computational Contact Topology 17 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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.
Christopher L. Toni
Computational Contact Topology 17 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Results and Conclusions (cont.)N[2] = 1
N[4] = 2
N[6] = 5
N[8] = 14
N[10] = 42
N[12] = 132
N[14] = 429
N[16] = 1430
N[18] = 4862
N[20] = 16796
N[22] = 58786
N[24] = 208012
N[26] = 742900
N[28] = 2674440
N[30] = 9694845
N[32] = 35357670
N[34] = 129644790
N[36] = 477638700
N[38] = 1767263190
N[40] = 6564120420
Note that the number of arclists increase rapidly as the numberof vertices get larger. At M = 28, its well over a million!
Christopher L. Toni
Computational Contact Topology 18 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Results and Conclusions (cont.)N[2] = 1
N[4] = 2
N[6] = 5
N[8] = 14
N[10] = 42
N[12] = 132
N[14] = 429
N[16] = 1430
N[18] = 4862
N[20] = 16796
N[22] = 58786
N[24] = 208012
N[26] = 742900
N[28] = 2674440
N[30] = 9694845
N[32] = 35357670
N[34] = 129644790
N[36] = 477638700
N[38] = 1767263190
N[40] = 6564120420
Note that the number of arclists increase rapidly as the numberof vertices get larger. At M = 28, its well over a million!
Christopher L. Toni
Computational Contact Topology 18 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Recent Findings
Instead of approching this problem from a combinatorialstandpoint, we now introduce a “simpler” way of generatingarclists, bypasses, and checking for tightness.
The problem can be tackled using permutation matrices!!!
Christopher L. Toni
Computational Contact Topology 19 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
Recent Findings
Instead of approching this problem from a combinatorialstandpoint, we now introduce a “simpler” way of generatingarclists, bypasses, and checking for tightness.
The problem can be tackled using permutation matrices!!!
Christopher L. Toni
Computational Contact Topology 19 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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. Enhance software (requiring a reduced memory footprint)to produce results for larger number of vertices.
Christopher L. Toni
Computational Contact Topology 20 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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. Enhance software (requiring a reduced memory footprint)to produce results for larger number of vertices.
Christopher L. Toni
Computational Contact Topology 20 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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. Enhance software (requiring a reduced memory footprint)to produce results for larger number of vertices.
Christopher L. Toni
Computational Contact Topology 20 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
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. Enhance software (requiring a reduced memory footprint)to produce results for larger number of vertices.
Christopher L. Toni
Computational Contact Topology 20 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
AcknowledgementsWe would like to thank:
∙ The SCSE (Dept. of Education) for funding the researchover summer.
∙ Dr. Tanya Cofer for leading us through tough concepts andtedious calculations.
∙ Donald Barkley for helping us program the algorithms inJava.
∙ Argonne National Laboratory for giving me the opportunityof presenting my group’s summer research.
Christopher L. Toni
Computational Contact Topology 21 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
AcknowledgementsWe would like to thank:
∙ The SCSE (Dept. of Education) for funding the researchover summer.
∙ Dr. Tanya Cofer for leading us through tough concepts andtedious calculations.
∙ Donald Barkley for helping us program the algorithms inJava.
∙ Argonne National Laboratory for giving me the opportunityof presenting my group’s summer research.
Christopher L. Toni
Computational Contact Topology 21 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
AcknowledgementsWe would like to thank:
∙ The SCSE (Dept. of Education) for funding the researchover summer.
∙ Dr. Tanya Cofer for leading us through tough concepts andtedious calculations.
∙ Donald Barkley for helping us program the algorithms inJava.
∙ Argonne National Laboratory for giving me the opportunityof presenting my group’s summer research.
Christopher L. Toni
Computational Contact Topology 21 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
AcknowledgementsWe would like to thank:
∙ The SCSE (Dept. of Education) for funding the researchover summer.
∙ Dr. Tanya Cofer for leading us through tough concepts andtedious calculations.
∙ Donald Barkley for helping us program the algorithms inJava.
∙ Argonne National Laboratory for giving me the opportunityof presenting my group’s summer research.
Christopher L. Toni
Computational Contact Topology 21 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
AcknowledgementsWe would like to thank:
∙ The SCSE (Dept. of Education) for funding the researchover summer.
∙ Dr. Tanya Cofer for leading us through tough concepts andtedious calculations.
∙ Donald Barkley for helping us program the algorithms inJava.
∙ Argonne National Laboratory for giving me the opportunityof presenting my group’s summer research.
Christopher L. Toni
Computational Contact Topology 21 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
AcknowledgementsWe would like to thank:
∙ The SCSE (Dept. of Education) for funding the researchover summer.
∙ Dr. Tanya Cofer for leading us through tough concepts andtedious calculations.
∙ Donald Barkley for helping us program the algorithms inJava.
∙ Argonne National Laboratory for giving me the opportunityof presenting my group’s summer research.
Christopher L. Toni
Computational Contact Topology 21 / 21
Introduction Arcs and Arclists Tightness Checking Bypasses Final Results and Thoughts
AcknowledgementsWe would like to thank:
∙ The SCSE (Dept. of Education) for funding the researchover summer.
∙ Dr. Tanya Cofer for leading us through tough concepts andtedious calculations.
∙ Donald Barkley for helping us program the algorithms inJava.
∙ Argonne National Laboratory for giving me the opportunityof presenting my group’s summer research.
Christopher L. Toni
Computational Contact Topology 21 / 21