On Tue, Oct 27, 2009 at 4:04 PM, John H Palmieri <jhpalmier...@gmail.com> wrote:
>> For the torus (correct me if I am wrong), the 0-th Betti number should be 1.
>> This would agree with the formula given for the Euler Characteristic
>> in that case : X = B_0 - B_1 + B_2 = 0
>
>
> The homology is reduced, so H_0 = Z, so the 0th Betti number is 0. I
> suppose it should be called the 0th reduced Betti number instead, but
> it's just the rank of H_0...
>
>
Thanks for the answer. That's what I guessed but maybe it should be
mentionned in the documentation of the method betti() because Sage is
also used by students just discovering homology and B_0 is the one
they understand most easily.
>> NB : according to the source code of SimplicialComplex, the first
>> definition should be : S = SimplicialComplex(2, [[0,1], [1,2], [0,2]])
>
> Any vertices which don't appear explicitly in simplices are ignored,
> so this is the same as
>
> SimplicialComplex(307, [[0,1], [1,2], [0,2]]) or SimplicialComplex
> (98, [[0,1], [1,2], [0,2]])
>
> or all other similar commands. As it says in the reference manual,
>
> The elements of the vertex set are not automatically contained in
> the simplicial complex: each one is only included if and only if it
> is a vertex of at least one of the specified facets.
>
> John
>
> -
I read that one too. My remark, again, was motivated by the point of
view of a newbie. My idea is that the examples in the docs should be
kept as simple as they can be.
If one reads "3", then one understands that it corresponds to the
"human" number of vertices (i.e. start counting at 1 and not 0).
In fact, here, "3" stands for "Lets take 4 vertices, but use only
3"... Misleading I think.
Lastly,
Sa = SimplicialComplex(3, [[0,1], [1,2], [0,2]]) # circle
Sa
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(1, 2), (0,
2), (0, 1)}
so vertex 3 is still present in the list (from the point of view of __repr__)
But the most important thing I have to say : thanks for the
implementation : great work and it is nice to use it.
My remarks are minor ones...
Phil
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