(Virtual) Bridges Conference, 2020 - PowerPoint Presentation

Slide1

(Virtual) Bridges Conference, 2020

Everted Embeddings

Carlo H. SéquinEECS, Computer Science Div., U.C. Berkeley

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“Impossible

” Sphere EversionA. Shapiro and T. Phillips: Scientific American, May 1966

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Simpler Sphere Eversions

Bernard Morin came up with a simpler solution . . .

Pugh: chicken-wire models; N. Max: Movie “… Inside Out”;

Sullivan: Movie “Optiverse”; Levy: Movie “… Outside In”.

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My Sphere Eversion Models

“Turning a Snowball Inside Out”

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A. Chéritat: Sphere Eversion Models

Bridges 2019: Stackable, layered, model 20 x 12 x 12 cmSLS 3d-printed nylon powder and acrylic paint.

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Torus Eversion: Shapiro & Phillips

(1966)Everting a 2-manifold of genus 1

Step C

e f

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Torus Eversion: Arnaud

Chéritat (2008)Everting a 2-manifold of genus 1

Lower half of a torus, green on the outside, red on the inside:

https://www.youtube.com/watch?v=jA86M6fdm_Q

Triple-Fold

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Greg McShane : Torus Skin-Eversion

https://www.youtube.com/watch?v=S4ddRPvwcZI

Reverses roles of meridians and laterals

Start:

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Handle-Body Shapes of Genus 3

“3-hole Donut” “3-ear Disk” “3-handle Stem”

“4-pillar Pagoda” “4-hole Sphere” “4-way Junction”

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Skin-Eversion of a Genus-2 Surface

Bag model shaped as a 2-hole Donut, or as a 3-pillar Pagoda

Genus-2 Lawson surface

Everted

into

a 3-way

Junction

puncture

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Cross-Sectional View of Skin-Eversion

Cross-sections of a skin-everted 2-hole Donut

Start with a

small puncture

Pull puncture around

the whole handle-body

Resulting in a

3-way Junction

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Eversion of a Tetrus

Handle-Body ?I could not do this directly with a plastic-bag model !

Introduce puncture here:

Result:

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Two Specific Eversion Problems

Borromean rings, and their embedding in a 3-hole donut.

Knot 818, and its embedding

in a 4-hole torus.

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Problem Statement on the Web:

I've been working with the intersection between the topology of knots and surfaces and I have some specific questions about the involution of torus with multiple holes and the effect on the knots written on then. Involution of a simple torus, by turning it inside-out through a hole So, if I turn inside-out a torus that inscribes a

4,3 torus knot, it becomes a torus that presents a 3,4 torus knot. Right? (It seems logical to me but I don't have the demonstration for this and would appreciate it, if anyone could help me with it)But the real deal is: Can anyone tell - and if possible prove it by showing the procedure through drawing – what results from the involution made through a hole for the case of:a triple torus which inscribes a borromean linka 4-holed torus which inscribes the 818 knotThanks to: Pedro Henrique Affonso (Brazil) !

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Surface

Eversions Through a Pin-hole

Two Case Studies (as requested on this web page):https://math.stackexchange.com/questions/2612627/involution-of-the-3-and-4-holed-torus-and-its-effects-on-some-knots-and-linksSkin-evert a genus-4 surface with embedded Knot 818Skin-evert a genus-3 surface carrying Borromean rings

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Genus-4 Plastic-Bag Model

the tube parts one 5-way Junction 4-handle Stem

hole for

eversion

partially

everted

completely

everted

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Put Knot 818 on a 4-handle Stem Surface

Knot 818 embedded on a Disk with 4 Ears

Rotate ears CW through 90°…

Various 4-handle Stem shapes



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Evert Stem-shape Carrying Knot 818

Show embeddings on different handle-bodies:

Redrawn on a

corresponding

spherical shell

with 5 holes.

Circle-inverted into

a 4-hole donut:

Central hole

becomes perimeter.

On the

everted

5-way Junction,

copied from

plastic-bag model.

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Knot 818 Everted on 4-hole Donut

Comparison of initial and everted embeddings

4 perimeter crossings.

3 strands pass through each hole.

8 perimeter crossings.

2 strands pass

through each hole.

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Twisting the Ears the Other Way

New starting geometry;  new eversion results:

4-handle Stem model,

after CCW ear-rotation

2 strands still pass

through each tunnel.

16 perimeter crossings!

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Another Model …

Circle-inverted 4-hole embedding to be mapped on 5-way Junction.This is still the same initial embedding!

5-way Junction model

Back-

everted

Stem model

Simplified model



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Knot 818

Eversions  Eversion

:

Shell with

5 holes

4-hole

Donut



5-way

Junction

Initial

Embedding

Turn ears

CW or CCW:

Circle-

Invert:

Do the equivalent steps:

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Borromean Rings on Genus-3 Surface

This is the other one of the stated problems

Embedding on a Tetrus

Mapped on a 3-hole Donut

(maintain 3-fold symmetry)

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Borr. Rings on Genus-3 Handle-Body

Convert to 3-handle Stem structure:

3-ear disk (CW)  3-handle Stem: 3D-print and paper.

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Eversion of the Borromean Rings

Stem to Junction Eversion:

3-handle Stem bag model 4-way Junction  4-hole Shell  3-hole Donut  . . .

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Borromean Rings: Eversion

Results Initial Embedding

CW – Eversion CCW – Eversion

3

perim. crossings 6

perim

. crossings 12

perim

. crossings

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Borr

. Rings on Genus-3 Handle-Body

For a different symmetry, convert the 3-hole Donut into a 4-pillar Pagoda.

3-hole Donut 4-hole “cylinder” 4-pillar Pagoda

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Borr. Rings on Genus-3 Handle-Body

Convert this to a 4-hole Shell structure

3-hole Donut 4-pillar Pagoda Shell with 4 holes

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Borr. Rings on 4-Pillar Pagoda

I also made a soft, pliable model out of plastic bags,onto which I transferred the colored traces.

4-pillar Pagoda: Bottom & Top of bag model.

opening

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Everted Borr

. Rings on 4-Way JunctionThis model can be everted

through an opening at its top:4 folded tubes.

Initial

4-pillar Pagoda



everted

4-way tubular Junction



Bottom & Top of the

everted 4-way Junction model.

opening

opening

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Physical Surface Skin-Eversion

Using plastic-bag model

Folded top view. Stretched side view. Open top puncture. Start pull-through.

More pull-through. Move flaps outward. Pull handles thru. Flipped final view.

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Everted Borromean

RingsAfter the plastic-bag model has been everted

, I read off the changed embedding of the 3 rings.Plastic-bag model Paper modelof 4-way tube Junction

4-pillar Pagoda

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Everted Borromean

Rings: ResultsOriginal embedding after eversion

:Changed features: Three (rather than 2)different strands

now share each pillar;Two (rather than 3) different strands enter each tunnel;Red and Blue strandscross

themselves on top and bottom plates.

Shell with 4

holes (seen from top)

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Everted Borromean

Rings: ResultsOriginal embedding after eversion

:

4-pillar Pagoda Shell with 4 holes

Tetrus shape

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“Back-Eversion” for

Borromean RingsTetrus structure is closely related to a 4-way Junction

Everted

into

a 3-handle

Stem shape

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Eversion Tracking with Paper Models

Paper Cut-out

Reversal Fold

Fold in half

5-way Junction

4-handle Stem

Double-wall structure

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Embedded Borromean Rings

Virtual Art

Ribbons on semi-transparent Tetrus body

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Embedded Borromean Rings

Physical Model3D-printed

Chenile-stem pipe-cleaners wound around a plastic body.

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Art Exhibit

“4 Different Embeddings of the Borromean Rings”

2D-print

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I would like to thank . . .

Pedro Henrique Affonso

Brazilfor steering me to this website:https://math.stackexchange.com/questions/2612627/involution-of-the-3-and-4-holed-torus-and-its-effects-on-some-knots-and-linksand for several stimulating subsequent e-mail exchanges.

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E X T R A S

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Turning a Surface Inside-Out (1)

Computer Graphics Mirroring: Just negate the values of one coordinate axis.

This surface passes trough itself when all x-values are zero.

This process creates geometrical singularities!

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Turning a Surface Inside-Out (2)

Physical reversal: Pull the whole surface through one small puncture.

Easy for a genus-0 surface!

For a torus this a bit more tricky!

The roles of the external hole and

of the internal tunnel get reversed.

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Turning a Surface Inside-Out (3)

Torus eversion: schematic view in the equatorial plane.Yields a left-right reversal in the cross sections

that pass through a Klein-bottle mouth

Homotopic

Eversion

:

No cuts and no creases!!

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Turning Embedded Knots Inside-Out

There are different ways of everting a surface;May have different effects on embedded knots!

(1) Computer graphics mirroring: Just negate the values of one coordinate axis.  Knot is mirrored.(2) Physical reversal: Pull the whole surface through one small puncture.  Knot is left unchanged !(3) Homotopic eversion: Use a regular homotopy as in Cheritat’s torus eversion.  Knot is mirrored.