Homology

[ramanath]

I am working with a simplicial complex, and would like to print out just a particular homology group. Doing ->HOMOLOGY prints all the groups, and for a large dimension complex one has lots of 0's.

I see that one can do $sc->F2_VECTOR->(2,2) etc. to access particular values, is there something similar like for Homology? (I tried a couple obvious parallels, but didn't work). Any tips/tricks?

Basically I want to print out the Betti number sequence.

Thanks

-Ram

[joswig]

The data type of the HOMOLOGY property is a bit special; actually it is Array<HomologyGroup<Integer>>. This is how you can address an individual group:

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topaz > $rp2=projective_plane(); topaz > print $rp2->HOMOLOGY->[1]; {(2 1)} 0

Each homology group more or less works like an array (of a pair and a number); the rational Betti number is the latter.

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topaz > print $rp2->HOMOLOGY->[1]->[0]; {(2 1)} topaz > print $rp2->HOMOLOGY->[1]->[1]; 0

You can also play with the torsion and its degree.

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topaz > print $rp2->HOMOLOGY->[1]->[0]->[0]; 2 1 topaz > print $rp2->HOMOLOGY->[1]->[0]->[0]->[0]; 2

[ramanath]

Exactly what I need, thanks very much!

Ram

[gawrilow]

For convenience, there are also named accessors:

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print $sc->HOMOLOGY->[$dim]->torsion; print $sc->HOMOLOGY->[$dim]->betti_number;

With some luck, polymake will even auto-TAB-complete these names on the command line.

[ramanath]

Indeed, that is convenient!

Is there documentation some place to see what other nice features like this is out there that I might be missing? I am primarily interested in features of simplicial complexes (topaz). I had searched for Betti numbers under documentation before but didn't find any.

Thank you.

-Ram

[gawrilow]

The reference documentation for property type HomologyGroup (linked from HOMOLOGY property) briefly lists the field names, albeit without further explanation. Admittedly, there is still room for improvement.