Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions Programming an Algorithm on Calculating the Number of Tight Contact Structures on the Solid Torus Argonne Undergraduate Symposium Christopher L. Toni Kelly Hirschbeck Nathan Walter William Krepelin Donald Barkley William Byrd John Wallin Mayra Bravo-Gonzalez Banlieman Kolani Dr. Tanya Cofer * November 13, 2009 Christopher L. Toni Computational Contact Topology - Argonne Symposium 1 / 18
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions
Programming an Algorithm on Calculating theNumber of Tight Contact Structures on the
Solid TorusArgonne Undergraduate Symposium
Christopher L. Toni Kelly Hirschbeck Nathan WalterWilliam Krepelin Donald Barkley William Byrd
John Wallin Mayra Bravo-Gonzalez Banlieman KolaniDr. Tanya Cofer∗
Introduction Arcs and Arclists Tightness Checking Bypasses 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.
Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions
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.
Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions
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:
I All M vertices in a configuration must be pairedI 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!
Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions
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
Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions
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
Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions
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.
Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions
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.
5. Representing arclists and bypasses as permutationmatrices and defining tightness as a certain product ofpermutation matrices (Cofer and Barkley, in preparation).