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Dror Bar-Natan:
Talks:
# "Not Knot" and "Outside In"

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Graduate Student Seminar

February 5, 2004

**Not Knot Questions.**

- Can you find 4 loops in space so that if you remove any one of them the
remaining 3 fall apart? How about
*n*?
- Can you describe life on a cone with cone angle bigger than
*2π*? Much bigger than *2π*?
- Describe the
*n-1* paths from a point to itself on a
*2π/n* cone. What do they look like on an actual 3D cone?
- When they say in the movie, "what's life like on a disk with a point
removed?", what assumptions are they making about "life"? Why isn't the
answer just "it's the same but with one point missing"?
- What's hyperbolic space? How did our cube suddenly turn into a
dodecahedron? Can you see a precursor of that dodecahedron already on the
original "cube with axes"?
- At the higher order symmetry tilings of hyperbolic space with
dodecahedrons, what happens around the white edges?
- What's a "rhombic dodecahedron"? Can you "see" it on the "cube with
axes"?

**Outside In Questions.**

- Did you see
*π*_{1}(S^{1})=**Z** in the
monorail pictures?
- Why can't a bowl come near a dome? In what sense are they "repulsive"?
Are there forced saddles when they come together? Where are they?
- What exactly went wrong with the 2D corrugation argument when turning
numbers didn't match?
- What's the belt trick got to do with
*π*_{1}(SO(3))=**Z***/2***Z**?
- Can you make the word "corrugation" in 3D more precise? Given two bands
dancing in tandem, can you describe the corrugation connecting them in
explicit terms?
- Can you make the statement "corrugations provide flexibility without
pinches and creases" precise?
- How come the thing that failed in 2D with corrugations doesn't also
fail in 3D? I.e., how comes the "uncorrugation" phase works in 3D after it
(sometimes) failed in 2D?

Thanks, Rich,
for help coming up with these questions.

This page is http://www.math.toronto.edu/~drorbn/Talks/UofT-040205/index.html.