Min-frame rotatopes

Discussion of shapes with curves and holes in various dimensions.

Min-frame rotatopes

Postby PWrong » Sat Nov 28, 2009 6:45 am

It turns out my conjecture about these was wrong. I expected that you could get the n'th homology group of a rotatope just by counting the n-spheres, except with an extra Z at the beginning and end. I've just found a few exceptions.

H222 = 1,3,0,1 as expected.
H322 = 1,2,3,0,1 where we expected 1,2,1,0,1
H332 = 1,1,2,4,0,1 where we expect 1,1,2,0,0,1

I can't see any connection yet. I'll have to do a few more.
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Re: Min-frame rotatopes

Postby PWrong » Sun Nov 29, 2009 5:13 am

Ok I've invented a new notation for homology groups. Usually we would say
Z, 5Z, 0, 0, 0, Z.
This is annoying because you have to count all the zeroes. Instead we can write
h0 + 5h1 + h5.

Now I've also found formulae for some more rotatopes.
I write Tn for the Cartesian product of n circles e.g. T^2 = 22.

H T^n = h_0 + n h_1 + h_n
H 3T^n = h_0 + n h_1 + h_2 + 2h_n + h_n+2

I've also got 33T^n and I'm working on 4T^n.
All of these only work for n>1.
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Re: Min-frame rotatopes

Postby PWrong » Wed Dec 02, 2009 4:04 am

Everything in this thread is wrong now :P.
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