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

▶
finding diagrams
SMART_READER_LITE
LIVE PREVIEW

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

Mar 09, 2023 106 likes •677 views

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

slide-1
SLIDE 1

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

slide-2
SLIDE 2

Tools: Two Programs SnapPy: http://SnapPy.CompuTop.org

Written by Marc Culler, Nathan Dunfield, Mattias G¨

  • rner, and Jeff Weeks

KLO: http://KLO-Software.net

Written by Frank Swenton

Ken Baker Finding Diagrams

slide-3
SLIDE 3

Plan: Three Examples

(1) Manifold − → Knot Diagram − → (Positive) Braid (2) One Seiferter − → Many Seiferters (3) Manifold − → Tangle Quotient − → DBC

Ken Baker Finding Diagrams

slide-4
SLIDE 4

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-5
SLIDE 5

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

SnapPy

M=Manifold(’t12533’) M M.dehn_fill([(1,0)]) M.fundamental_group() M.dehn_fill([(0,0)]) Load manifold Quick check for S3 filling Remove filling

Ken Baker Finding Diagrams

slide-6
SLIDE 6

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-7
SLIDE 7

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Find link for surgery description

M.identify() M.drill(0).identify() M.drill(1).idenfity() M.drill(0).drill(0).identify() M.drill(0).drill(1).identify() M01=M.drill(0).drill(1) L=Manifold(’L12n1968’) L.browse() Drilling removes the ith shortest geodesic loop. Its meridian has slope (1,0). Record the diagram of L.

Ken Baker Finding Diagrams

slide-8
SLIDE 8

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-9
SLIDE 9

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

slide-10
SLIDE 10

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-11
SLIDE 11

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid Check surgery description

L.dehn_fill([(4,1), (2,1),(0,0)]) L L.identify() L.dehn_fill([(4,1),(2,1),(1,0)]) L.fundamental_group() All’s good, so now to KLO.

Ken Baker Finding Diagrams

slide-12
SLIDE 12

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-13
SLIDE 13

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

slide-14
SLIDE 14

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

slide-15
SLIDE 15

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

slide-16
SLIDE 16

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-17
SLIDE 17

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-18
SLIDE 18

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-19
SLIDE 19

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-20
SLIDE 20

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-21
SLIDE 21

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-22
SLIDE 22

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-23
SLIDE 23

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-24
SLIDE 24

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-25
SLIDE 25

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-26
SLIDE 26

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-27
SLIDE 27

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-28
SLIDE 28

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-29
SLIDE 29

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-30
SLIDE 30

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Ken Baker Finding Diagrams

slide-31
SLIDE 31

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

KLO: “Clone as Knot” and check invariants

Signature = -16 Alexander Polynomial 1−t+t4−t5+t7−t8+t9−t10+t12−t14+t15−t16+t17−t19+t20−t23+t24

Ken Baker Finding Diagrams

slide-32
SLIDE 32

Ex 1: Manifold − → Knot Diagram − → (Positive) Braid

Dunfield: SnapPy census manifolds that are

  • asymmetric and
  • complements of L-space knots

in S3

’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

slide-33
SLIDE 33

Ex 2: One Seiferter − → Many Seiferters

Ken Baker Finding Diagrams

slide-34
SLIDE 34

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

slide-35
SLIDE 35

Ex 2: One Seiferter − → Many Seiferters SnapPy: Observe the asymmetric seiferter

A=Manifold() A.solution_type() A.symmetry_group() A.is_isometric_to(A, return_isometries=True) A.dehn_fill([(1,1),(0,0)]) A.fundamental_group() Pop open PLink editor to draw link polygonally “Send to SnapPy” Check hyperbolicity Check symmetry Fill K but not c Observe SFS via π1: Type D2(a, b)

Ken Baker Finding Diagrams

slide-36
SLIDE 36

Ex 2: One Seiferter − → Many Seiferters

Ken Baker Finding Diagrams

slide-37
SLIDE 37

Ex 2: One Seiferter − → Many Seiferters Motegi: Are there other asymmetric seiferters?

Explore by drilling A.dehn_fill([(0,0),(0,0)]) A0=A.drill(0) A0.dehn_fill([(1,1),(0,0),(0,0)]) A0.fundamental_group() A0.dehn_fill([(1,0),(0,0),(0,0)]) A0.fundamental_group() Drill once. Check fillings of K: A0(∞, ·, ·) and A0(1, ·, ·) Observe cable space π1s ∴ many S1 × D2 fillings with ∆ = 1 from cabling slope

Ken Baker Finding Diagrams

slide-38
SLIDE 38

Ex 2: One Seiferter − → Many Seiferters

Ken Baker Finding Diagrams

slide-39
SLIDE 39

Ex 2: One Seiferter − → Many Seiferters

SnapPy: Find fillings to make new seiferters

Look for cabling slope of A0(∞, ·, ·) on last component. Should be ∆ = 1 from 1

0.

A0.dehn_fill([(1,0),(0,0),(0,1)]) A0.fundamental_group() A0.dehn_fill([(1,0),(0,0),(1,1)]) A0.fundamental_group() A0.dehn_fill([(1,0),(0,0),(2,3)]) A0.fundamental_group() Slope 1

1 gives π1 = Z ∗ Z3

∴ S1 × D2 ∼ = L(3, q) Hence want slopes n−1

n .

Ken Baker Finding Diagrams

slide-40
SLIDE 40

Ex 2: One Seiferter − → Many Seiferters

Ken Baker Finding Diagrams

slide-41
SLIDE 41

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)]) A0.homology() ... A0.dehn_fill([(1,0),(27,8),(2,3)]) A0.homology() Here we choose n = 3. Hone in on longitude 27

8 .

So 10

3 is a meridian.

Ken Baker Finding Diagrams

slide-42
SLIDE 42

Ex 2: One Seiferter − → Many Seiferters

Ken Baker Finding Diagrams

slide-43
SLIDE 43

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() A0F.solution_type() A0F.is_isometric_to(A0F, return_isometries=True) A0F.volume() A.volume() Sometimes need to ”permanently” fill to see hyperbolic structure. May distinguish by volume.

Ken Baker Finding Diagrams

slide-44
SLIDE 44

Ex 2: One Seiferter − → Many Seiferters

Ken Baker Finding Diagrams

slide-45
SLIDE 45

Ex 2: One Seiferter − → Many Seiferters

Drill again to find surgery diagram as in Example 1

Ken Baker Finding Diagrams

slide-46
SLIDE 46

Ex 2: One Seiferter − → Many Seiferters

Ultimately find new knot with an asymmetric seiferter

Ken Baker Finding Diagrams

slide-47
SLIDE 47

Ex 3: Manifold − → Tangle Quotient − → DBC

Ken Baker Finding Diagrams

slide-48
SLIDE 48

Ex 3: Manifold − → Tangle Quotient − → DBC

Dunfield: “A census of exceptional Dehn fillings”

The manifold ’t12036’ has four toroidal fillings on slopes ∞, 0, +1, −1. T=Manifold(’t12036’) T000=T.drill(0).drill(0).drill(0) T000.identify() U=Manifold(’L12n2084’) T000.is_isometric_to(T000, return_isometries=True) T000.is_isometric_to(U, return_isometries=True) Load manifold Drill to find a known link Check invertibility Get surgery description

Ken Baker Finding Diagrams

slide-49
SLIDE 49

Ex 3: Manifold − → Tangle Quotient − → DBC

Ken Baker Finding Diagrams

slide-50
SLIDE 50

Ex 3: Manifold − → Tangle Quotient − → DBC

Ken Baker Finding Diagrams

slide-51
SLIDE 51

Ex 3: Manifold − → Tangle Quotient − → DBC

Take quotient by hand.

Ken Baker Finding Diagrams

slide-52
SLIDE 52

Ex 3: Manifold − → Tangle Quotient − → DBC

∞ Surgery Twice-punctured disks

Ken Baker Finding Diagrams

slide-53
SLIDE 53

Ex 3: Manifold − → Tangle Quotient − → DBC

0 Surgery Annulus

Ken Baker Finding Diagrams

slide-54
SLIDE 54

Ex 3: Manifold − → Tangle Quotient − → DBC

+1 Surgery Twice-punctured disk

Ken Baker Finding Diagrams

slide-55
SLIDE 55

Ex 3: Manifold − → Tangle Quotient − → DBC

−1 Surgery Twice-punctured disk

Ken Baker Finding Diagrams