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  • 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

    November 13, 2009

    Christopher L. Toni

    Computational Contact Topology - Argonne Symposium 1 / 18

  • Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

    Outline

    Introduction

    Arcs and Arclists

    Tightness Checking

    Bypasses

    Results and Conclusions

    Christopher L. Toni

    Computational Contact Topology - Argonne Symposium 2 / 18

  • Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

    What is Topology?

    Topology is a field of mathematics that does not focus on anobjects shape, but the properties that remain consistentthrough deformations like:

    I twistingI bendingI stretching

    To illustrate this, imagine a coffee mug and a doughnut (torus).

    Christopher L. Toni

    Computational Contact Topology - Argonne Symposium 3 / 18

  • 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.

    Christopher L. Toni

    Computational Contact Topology - Argonne Symposium 4 / 18

  • Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

    Formulating the ProblemOn the solid torus (defined by S1D2), 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 - Argonne Symposium 5 / 18

  • 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.

    Christopher L. Toni

    Computational Contact Topology - Argonne Symposium 6 / 18

  • 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!

    Christopher L. Toni

    Computational Contact Topology - Argonne Symposium 7 / 18

  • Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

    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 - Argonne Symposium 8 / 18

  • Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

    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 xnq+1 mod np maps the vertices on the leftcutting disk to the right cutting disk.

    The formula x x+nq1 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 - Argonne Symposium 9 / 18

  • Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

    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 - Argonne Symposium 10 / 18

  • Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

    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 - Argonne Symposium 11 / 18

  • 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 05 mod 8= 31 15 mod 8= 42 25 mod 8= 53 35 mod 8= 6

    4 45 mod 8= 75 55 mod 8= 06 65 mod 8= 17 75 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 - Argonne Symposium 12 / 18

  • 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 05 mod 8= 31 15 mod 8= 42 25 mod 8= 53 35 mod 8= 6

    4 45 mod 8= 75 55 mod 8= 06 65 mod 8= 17 75 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 - Argonne Symposium 13 / 18

  • Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

    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 - Argonne Symposium 14 / 18

  • 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.

    Christopher L. Toni

    Computational Contact Topology - Argonne Symposium 15 / 18

  • 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).

    Christopher L. Toni

    Computational Contact Topology - Argonne Symposium 16 / 18

  • Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

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

    Christopher L. Toni

    Computational Contact Topology - Argonne Symposium 17 / 18

  • Introduction Arcs and Arclists Tightness Checking Bypasses Results and Conclusions

    Acknowledgements

    I Dr. Tanya Cofer for leading us through tough concepts andtedious calculations.

    I Donald Barkley for helping us program the algorithms inJava.

    I Argonne National Laboratory for giving me the opportunityto present my research.

    Christopher L. Toni

    Computational Contact Topology - Argonne Symposium 18 / 18

    IntroductionArcs and ArclistsTightness CheckingBypassesResults and Conclusions