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# ILSAMP Contact Topology

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## ILSAMP Contact TopologyPresentation Transcript

• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and Conclusions Comparison of Methods That Check for Tight Contact Structures on the Solid Torus ILSAMP Student Research Symposium Kelly Hirschbeck Christopher L. Toni Donald Barkley Steven Jerome Dr. Tanya Cofer∗ February 13, 2009Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 1 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsOutline Introduction Overview of the Process Arcs and Arclists Tightness Checking Bypasses Method 1: Hand Calculations Tightness Checking Bypasses Method 2: Permutations Tightness Checking Bypasses Results and ConclusionsKelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 2 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsWhat is Topology? Topology is a ﬁeld of mathematics that does not focus on an object’s shape, but the properties that remain consistent through deformations like:Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 3 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsWhat is Topology? Topology is a ﬁeld of mathematics that does not focus on an object’s shape, but the properties that remain consistent through deformations like: twisting bending stretchingKelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 3 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsWhat is Topology? Topology is a ﬁeld of mathematics that does not focus on an object’s shape, but the properties that remain consistent through deformations like: twisting bending stretching To illustrate this, imagine a coffee mug and a doughnut (torus).Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 3 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsWhat is Topology? (cont.) The torus and the coffee cup are topologically equivalent objects. We see above that through bending and stretching, the torus can be morphed into a coffee cup and vice versa.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 4 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsFormulating the Problem On the solid torus (deﬁned by S1 × D2 ), dividing curves are located where twisting planes switch from positive to negative.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 5 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsFormulating the Problem On the solid torus (deﬁned by S1 × D2 ), dividing curves are located where twisting planes switch from positive to negative.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 5 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsFormulating the Problem On the solid torus (deﬁned by S1 × D2 ), dividing curves are located where twisting planes switch from positive to negative. These dividing curves keep track of and allow for investigation of certain topological properties in the neighborhood of a surface.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 5 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsArcs and ArclistsOverview The ﬁrst computational task is to generate arclists for a given number of vertices np.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 6 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsArcs and ArclistsOverview The ﬁrst computational task is to generate arclists for a given number of vertices np. Deﬁnition An arc is a path between vertices subject to: All vertices must be paired and arcs cannot intersect An arclist is a set (list) of legal pairs of arcs.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 6 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsArcs and ArclistsAlgorithm Output - Arcs and Arclists When np = 8, there are 8 vertices. The arclists that are generated are:Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 7 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsArcs and ArclistsAlgorithm Output - Arcs and Arclists When np = 8, there are 8 vertices. The arclists that are generated are: (0 1)(2 5)(3 4)(6 7) (0 5)(1 2)(3 4)(6 7) (0 1)(2 7)(3 4)(5 6) (0 7)(1 4)(2 3)(5 6) (0 3)(1 2)(4 5)(6 7) (0 1)(2 3)(4 7)(5 6) (0 1)(2 3)(4 5)(6 7) (0 5)(1 4)(2 3)(6 7) (0 1)(2 7)(3 6)(4 5) (0 7)(1 2)(3 6)(4 5) (0 3)(1 2)(4 7)(5 6) (0 7)(1 6)(2 5)(3 4) (0 7)(1 2)(3 4)(5 6) (0 7)(1 6)(2 3)(4 5)Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 7 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsTightness CheckingOverview - Tightness Checker Potentially Tight Overtwisted x → x − nq + 1 mod np This maps the dividing curves on the surface from left to right cutting disk.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 8 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsBypassesBrief Overview - Bypasses An abstract bypass exists when a line can be drawn through three arcs on a cutting disk.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 9 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsBypassesBrief Overview - Bypasses An abstract bypass exists when a line can be drawn through three arcs on a cutting disk.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 9 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsBypassesBrief Overview - Bypasses An abstract bypass exists when a line can be drawn through three arcs on a cutting disk. Two Abstract Bypasses. Zero Abstract Bypasses.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 9 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsTightness CheckingChecking for Tightness (01) (27) (36) (45) (07) (14) (23) (56) All vertices hook up to a single It takes more than one curve curve. to hook up all the vertices.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 10 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsBypassesAbstract Bypasses (05) (14) (23) (67) (01) (25) (34) (67) α α β (01) (23) (47) (56) βKelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 11 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsBypassesChecking for Actual Bypasses (05) (14) (23) (67) (01) (23) (47) (56)Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 12 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsTightness CheckingRevisiting Method One (Developed by Dr. Cofer) Recall the mapping rule: x → x − nq + 1 mod np. Therefore, the formula to check for tightness: β −1 Aβ A.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 13 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsTightness CheckingPermutation Example Given: n = 2, p = 4 ,q = 3 The mapping rule tells us x → x − 5 mod 8.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 14 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsTightness CheckingPermutation Example Given: n = 2, p = 4 ,q = 3 The mapping rule tells us x → x − 5 mod 8. Therefore, β = (03614725) β −1 = (05274163)Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 14 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsTightness CheckingPermutation 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 = (0 1)(2 7)(3 6)(4 5) A = (0 7)(1 4)(2 3)(5 6) β −1 Aβ A = (0246) β −1 Aβ A = (0)Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 14 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsBypassesExistence of Bypasses The existence of actual bypasses is checked in a similar fashion as tightness.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 15 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsBypassesExistence of Bypasses The existence of actual bypasses is checked in a similar fashion as tightness. Given: An arclist A and an abstract bypass C.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 15 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsBypassesExistence of Bypasses The existence of actual bypasses is checked in a similar fashion as tightness. Given: An arclist A and an abstract bypass C. The formula: β −1 AβCKelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 15 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsBypassesExistence of Bypasses The existence of actual bypasses is checked in a similar fashion as tightness. Given: An arclist A and an abstract bypass C. The formula: β −1 AβC A = (01)(25)(34)(67) β = (03614725) β −1 = (05274163) C1 = (05)(14)(23)(67) C2 = (01)(23)(47)(56) β −1 AβC1 = (0624) β −1 AβC2 = (0)Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 15 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsFuture Research Future goals include, but not limited to:Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 16 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsFuture Research Future goals include, but not limited to: Publication of Findings in Undergraduate JournalKelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 16 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsFuture Research Future goals include, but not limited to: Publication of Findings in Undergraduate Journal Extension of Algorithm to the two-holed torusKelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 16 / 16
• Introduction Overview of the Process Method 1: Hand Calculations Method 2: Permutations Results and ConclusionsFuture Research Future goals include, but not limited to: Publication of Findings in Undergraduate Journal Extension of Algorithm to the two-holed torus Searching for a formula for the case of four dividing curves.Kelly Hirschbeck, Christopher L. ToniComputational Contact Topology - ILSAMP Symposium 16 / 16