CS 39R: Symmetry & Topology
Lecture #9 -- Mon. 4/8, 2013.

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Preparation:

Watch some of the eversion movies and think about what shapes can or cannot be everted.

Try to accomplish the following curve-shape-changes in the plane with regular (smooth) homotopies:
(Draw a sequence of smooth key-frame shapes that would make up a continuous movie)

Change this right-arm "Klein-bottle profile" into a left-arm "Klein-bottle profile".	Simplify this "double-8" curve as much as possible.	Try to turn a circle inside-out, (reversing arrow direction).

Solution Solution NO Solution: incompatible turning numbers

Try to give a definition for TOPOLOGY !

Regular Homotopies (a more specific classification of smooth manifolds):

Which surfaces are transformable into one another through a "Regular Homotopy",
i.e., a deformation that allows surface regions to pass through one another,
but does not allow any cuts, or tears, or formation of creases or other singular points with infinite curvature?
(With this definition, it is possible to turn a sphere or a torus inside out -- but it is not easy!).

Regular Homotopies of curves in the Plane (see warm-up problems above).
The turning number of a closed planar curve -- and its role in the above problems.

Examples of 2-Manifold Eversions:

Torus eversion by Cheritat (cut open, to see inside);

Turning a sphere inside out by Max;

Turning a sphere outside in by Thurston (more details Levy, Maxwell, Munzner);

Energetically optimal sphere eversion by Sullivan, Francis, Levy.

Knots

A wonderful, "must-have" resource: Colin C. Adams: "The Knot Book", W. H. Freeman and Co., New York, 1994.

What is a knot ?

When are two knots the same ?

Knot Tables

Simplifying knots: Reidemeister moves

Beautifying knots. . . -- Symmetry?

What is this particular knot ?

Applications in Nano Technology

New Homework Assignments:-- due next class, April 15, 2013, noon:

Figure out which knot it is that is depicted above.
Send your Knot solutions to me by e-mail before noon, 4/15/2013.

Work on your projects! ( ... a crucial part of getting a passing grade in this course!)
Be ready for an 8-minute formal presentation with visuals (and possibly models)
for Monday, 4/22/2013.