On Oct 27, 9:57 am, Philippe Saade <psa...@gmail.com> wrote:
> 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
You can find suggested fixes here: <http://trac.sagemath.org/sage_trac/
ticket/7323>
John
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