Finding Diagrams Ken Baker Department of Mathematics University of - PowerPoint PPT Presentation

Finding Diagrams Ken Baker Department of Mathematics University of Miami Coral Gables, FL Perspectives on Dehn Surgery, ICERM July 15, 2019 • Ken Baker Finding Diagrams

Tools: Two Programs SnapPy: http://SnapPy.CompuTop.org Written by Marc Culler, Nathan Dunfield, Mattias G¨ orner, and Jeff Weeks KLO: http://KLO-Software.net Written by Frank Swenton Ken Baker Finding Diagrams

Plan: Three Examples (1) Manifold − → Knot Diagram − → (Positive) Braid (2) One Seiferter − → Many Seiferters (3) Manifold − → Tangle Quotient − → DBC Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid SnapPy M=Manifold(’t12533’) M Load manifold Quick check for S 3 filling M.dehn_fill([(1,0)]) Remove filling M.fundamental_group() M.dehn_fill([(0,0)]) Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Find link for surgery description M.identify() M.drill(0).identify() M.drill(1).idenfity() Drilling removes the i th M.drill(0).drill(0).identify() shortest geodesic loop. Its meridian has slope (1,0) . M.drill(0).drill(1).identify() Record the diagram of L . M01=M.drill(0).drill(1) L=Manifold(’L12n1968’) L.browse() Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Find surgery coefficients for surgery description M01.is_isometeric_to(L) M01.is_isometric_to(L, return_isometries=True) The meridian slopes (1,0) on the three cusps of M01 go to slopes on the three cusps of L . Calculate this by hand. Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Check surgery description L.dehn_fill([(4,1), (2,1),(0,0)]) L All’s good, so now to L.identify() KLO. L.dehn_fill([(4,1),(2,1),(1,0)]) L.fundamental_group() Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid KLO: Input surgery description New Document, Surgery Description Draw the link L . Hold Shift to crouch. Process when done. Click crossings to correct. Set surgery coefficients. Accept when done. Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid KLO: Manipulate diagram Diagram Moves Click on region for Reidemeister I, II, III simplification. Drag over/under strand to adjacent crossing. Automate with lower-left buttons. Focus for Meta-moves Focus on component with Ctrl -click, Cmd -click, or right-click. Also with Shift to add more components to focus. Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid KLO: Reduce diagram to knot... Rolfsen Twists (twisting about a circle) Remove self-crossings from unknot component Click disk to eliminate interior crossings P Parallelize strands through circle T Twist about circle, enter number Click ∞ –framed component to remove Click ∆ to add ∞ –framed unknots Band Moves Convert to Kirby diagram Do Handle-slides Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid KLO: “Clone as Knot” and check invariants Signature = -16 Alexander Polynomial 1 − t + t 4 − t 5 + t 7 − t 8 + t 9 − t 10 + t 12 − t 14 + t 15 − t 16 + t 17 − t 19 + t 20 − t 23 + t 24 Ken Baker Finding Diagrams

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Dunfield: SnapPy census manifolds that are • asymmetric and • complements of L-space knots in S 3 ’t12533’ , ’t12681’ , ’o9_38928’ , ’o9_39162’ , ’o9_40363’ , ’o9_40487’ , ’o9_40504’ , ’o9_40582’ , ’o9_42675’ Find knot diagrams of these. Which of these are complements of positive braids? Ken Baker Finding Diagrams

Ex 2: One Seiferter − → Many Seiferters Ken Baker Finding Diagrams

Ex 2: One Seiferter − → Many Seiferters Definition (Deruelle-Miyazaki-Motegi) Say m –surgery on a knot K produces a Seifert fibered space. An unknot c disjoint from K is a seiferter for the m -surgery on K if it becomes a Seifert fiber. Ken Baker Finding Diagrams

Ex 2: One Seiferter − → Many Seiferters SnapPy: Observe the asymmetric seiferter A=Manifold() Pop open PLink editor A.solution_type() to draw link polygonally “Send to SnapPy” A.symmetry_group() Check hyperbolicity Check symmetry A.is_isometric_to(A, return_isometries=True) Fill K but not c Observe SFS via π 1 : A.dehn_fill([(1,1),(0,0)]) Type D 2 ( a , b ) A.fundamental_group() Ken Baker Finding Diagrams

Ex 2: One Seiferter − → Many Seiferters Ken Baker Finding Diagrams

Ex 2: One Seiferter − → Many Seiferters Motegi: Are there other asymmetric seiferters? Explore by drilling A.dehn_fill([(0,0),(0,0)]) Drill once. Check fillings of A0=A.drill(0) K : A0.dehn_fill([(1,1),(0,0),(0,0)]) A 0( ∞ , · , · ) and A 0(1 , · , · ) A0.fundamental_group() Observe cable space π 1 s ∴ many S 1 × D 2 fillings with ∆ = 1 from cabling A0.dehn_fill([(1,0),(0,0),(0,0)]) slope A0.fundamental_group() Ken Baker Finding Diagrams

Ex 2: One Seiferter − → Many Seiferters Ken Baker Finding Diagrams

Ex 2: One Seiferter − → Many Seiferters SnapPy: Find fillings to make new seiferters Look for cabling slope of A 0( ∞ , · , · ) on last component. Should be ∆ = 1 from 1 0 . A0.dehn_fill([(1,0),(0,0),(0,1)]) A0.fundamental_group() Slope 1 1 gives π 1 = Z ∗ Z 3 A0.dehn_fill([(1,0),(0,0),(1,1)]) ∴ S 1 × D 2 ∼ = L (3 , q ) Hence want slopes n − 1 n . A0.fundamental_group() A0.dehn_fill([(1,0),(0,0),(2,3)]) A0.fundamental_group() Ken Baker Finding Diagrams

Ex 2: One Seiferter − → Many Seiferters Ken Baker Finding Diagrams

Ex 2: One Seiferter − → Many Seiferters SnapPy: Find fillings to make new seiferters Find corresponding longitude of new c . A0.dehn_fill([(1,0),(3,1),(2,3)]) A0.homology() A0.dehn_fill([(1,0),(4,1),(2,3)]) Here we choose n = 3. A0.homology() Hone in on longitude 27 8 . So 10 3 is a meridian. ... A0.dehn_fill([(1,0),(27,8),(2,3)]) A0.homology() Ken Baker Finding Diagrams

Ex 2: One Seiferter − → Many Seiferters Ken Baker Finding Diagrams

Ex 2: One Seiferter − → Many Seiferters SnapPy: Find fillings to make new seiferters Check asymmetry and compare. A0.dehn_fill([(0,0),(0,0),(2,3)]) A0.solution_type() A0F=A0.filled_triangulation() Sometimes need to ”permanently” fill to see A0F.solution_type() hyperbolic structure. May distinguish by volume. A0F.is_isometric_to(A0F, return_isometries=True) A0F.volume() A.volume() Ken Baker Finding Diagrams

Ex 2: One Seiferter − → Many Seiferters Ken Baker Finding Diagrams

Ex 2: One Seiferter − → Many Seiferters Drill again to find surgery diagram as in Example 1 Ken Baker Finding Diagrams

Ex 2: One Seiferter − → Many Seiferters Ultimately find new knot with an asymmetric seiferter Ken Baker Finding Diagrams

Ex 3: Manifold − → Tangle Quotient − → DBC Ken Baker Finding Diagrams