Math-Encounter: NYC August 1, 2012


Proposed TITLE

Twisted Sandwiches, Spirally Sliced Bagles, and Knotted Klein Bottles:
-- the topology of twisted toroids.


ABSTRACT

What happens, geometrically, when we slice a bagle with a spirally twisting kinfe, or take a (several) foot-long sandwich and close it into a toroidal loop while flipping the whole composite of the various layers upside down?

This talk starts with an analysis of sculptures and puzzles resulting from splitting Moebius bands and tori with lengthwise spiralling cuts.  Topological tori and Klein bottles will then be introduced. These are surfaces that can self-intersect, but which are still smooth immersions in 3D space.  We will investigates under what circumstances such surfaces can be smoothly transformed into one another.  For instance, it is possible to turn a torus inside-out without making any punctures! Surfaces that can be transformed into one another without any cuts, or creases, or other points of infinitely high curvature, are said to be in the same regular homotopy class. There are four distinct classes of this kind for tori, and four more for Klein bottles.  Representaives of all eight classes will be introduced and compared to one another.


SIMPLER  ABSTRACT

What happens, geometrically, when we slice a bagle with a spirally twisting kinfe, or take a (several) foot-long sandwich and close it into a toroidal loop while flipping the whole composite of the various layers upside down?
This talk starts with an analysis of sculptures and puzzles resulting from splitting Moebius bands (twisted belts) and tori (donuts) with lengthwise spiralling cuts.  
The notion of a torus will then be expanded to its mathematical definition, where the surface my intersect with itself, and truly weird shapes become possible.
We then ask under what circumstances such shapes can be smoothly transformed into one another.

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MATERIAL WILL BE TAKEN FROM THE FOLLOWING SOURCES:

Bridges  2005 -- "Splitting Tori, Knots, and Moebius Bands":
http://www.cs.berkeley.edu/~sequin/TALKS/Bridges05_SplitTori.ppt

Bridges  2011 -- "Tori Story":
http://www.cs.berkeley.edu/~sequin/TALKS/2011_Bridges_Tori-Story/2011_Bridges_Tori-Story.ppt


Bridges  2012 -- "From Moebius Bands to Klein Knottles":
http://www.cs.berkeley.edu/~sequin/PAPERS/2012_Bridges_Klein.pdf

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TENTATIVE TALK OUTLINE:

Start with the sculptures by Keizo Ushio
Generalize to spirally cut toroids
Introduce Moebius bands and (twisted) "Moebius" prisms.
1. activity:  -- concerning  the above.
Introduce concept of topological tori that can self-intersect.
Smooth transformations of such tori == "regular homotopies"
How many different tori are there?
Show all 4 regular homotopy classes and their representatives.
Do the same for Klein bottles.
2. activity: -- concerning Klein bottles.