Bridges 2012
From Möbius Bands to Klein Knottles
Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley
What is a Möbius Band ?

A single-sided surface with a single edge:
A closed ribbon
with a 180° flip.
The “Sue-Dan-ese” M.B.,
a “bottle” with circular rim.
Deformation of a Möbius Band (ML)
-- changing its apparent twist
+180°(ccw),
540°(cw)
0°,
–180°,
–
Apparent twist, compared to a rotation-minimizing frame (RMF)
Measure the built-in twist when sweep path is a circle!
Twisted Möbius Bands in Art
Web
Max Bill
M.C. Escher
M.C. Escher
The Two Different Möbius Bands
ML
and
MR
are in two different regular homotopy classes!
What is a Klein Bottle ?
A
single-sided surface
 with
no edges or punctures
 with
Euler characteristic: V – E + F = 0
 corresponding
 always
to: genus = 2
self-intersecting in 3D
( only immersions, no embeddings )
How to Make a Klein Bottle (1)
make a “tube”
by merging the horizontal edges
of the rectangular domain
 First
How to Make a Klein Bottle (2)
 Join
tube ends with reversed order:
How to Make a Klein Bottle (3)
 Close
ends smoothly
by “inverting one sock”
Limerick
A mathematician named Klein
thought Möbius bands are divine.
Said he: "If you glue
the edges of two,
you'll get a weird bottle like mine."
2 Möbius Bands Make a Klein Bottle
KOJ
=
MR
+
ML
Classical “Inverted-Sock” Klein Bottle
Figure-8 Klein Bottle
Making a Figure-8 Klein Bottle (1)
make a “figure-8 tube”
by merging the horizontal edges
of the rectangular domain
 First
Making a Figure-8 Klein Bottle (2)
 Add
a 180° flip to the tube
before the ends are merged.
Two Different Figure-8 Klein Bottles
MR
+
MR
=
K8R
ML
+
ML
=
K8L
Yet Another Way to Match-up Numbers
The New “Double-Sock” Klein Bottle
The New “Double-Sock” Klein Bottle
Rendered with Vivid 3D (Claude Mouradian)
http://netcyborg.free.fr/
The 4th Klein Bottle ??

There are 22-χ distinct regular homotopy classes
of immersions of a surface of Euler characteristic χ
into R3.
 Thus
there must be 4 distinct Klein bottle types that
cannot be transformed smoothly into one another.
J. Hass and J. Hughes, Immersions of Surfaces in 3-Manifolds.
Topology, Vol.24, No.1, pp 97-112, 1985.
 The
first 3 Klein bottles presented clearly belong to
three different regular homotopy classes.
Lawson’s Minimum Energy Klein Bottle
Klein Bottle Analysis

A regular homotopy cannot change the twist of a MB.
Thus, left-twisting bands stay left-twisting,
and right-twisting ones stay right-twisting!

K8L and K8R have chirality.
They are mirror images of one another!

But so does the Lawson KB!
Thus, there are two different Lawson KBs.

So – if the Lawson Klein bottle were something new,
then there would be TWO new bottle types.

But this cannot be; there are only four types total;
thus the Lawson bottles transform into K8R and K8L.
“Double Sock” is NOT #4!

It turns out the “Double-Sock K.B.” also has chirality!

And thus it also comes in two forms
that transform into the respective K8R or K8L.

Thus is cannot play the role of #4.

Therefore, we need to look for a K.B.
made of ML + MR to serve as #4.

Thus #4 structurally belongs into the class KOJ.

It can only be distinguished from the classical KOJ,
if we place some markings on its surface.
Regular Homotopy Classes for Tori
Decorated Klein Bottles

The 4th type can only be distinguished through
its surface decoration (parameterization)!
Arrows come
out of hole
Added collar
on KB mouth
Arrows go
into hole
Klein Bottle: Regular Homotopy Classes
Which Type of Klein Bottle Do We Get?

It depends which of the two ends gets narrowed down.
Fancy Klein Bottles of Type KOJ
Cliff Stoll
Klein bottles by Alan Bennett
in the Science Museum in South Kensington, UK
Beyond Ordinary Klein Bottles
Glass sculptures by Alan Bennett
Science Museum in South Kensington, UK
Klein Knottles Based on KOJ
Always an odd number of “turn-back mouths”!
A Gridded Model of Trefoil Knottle
Not a Klein Bottle – But a Torus !
 An
even number of surface reversals
renders the surface double-sided and orientable.
Klein Knottles with Fig.8 Crosssections
A Gridded Model of Figure-8 Trefoil
Rendered with Vivid 3D (Claude Mouradian)
http://netcyborg.free.fr/
Rendered with Vivid 3D (Claude Mouradian)
FDM Model
http://netcyborg.free.fr/
Summary of Findings

Klein bottles are closely related to Möbius bands:
every bottle is composed of two bands.

Structurally, there are three different types of K-Bs
that can’t be smoothly transformed into one another.

When considering marked (textured) surfaces,
“inverted sock” Klein bottle splits into 2 different types:
( arrows going into, or coming out of its mouth ).
Conclusions
 Klein
bottles are fascinating surfaces.
 They
come in a wide variety of shapes,
which are not always easy to analyze.
 Many
of these shapes make
attractive constructivist sculptures . . .
=== Questions ? ===