Note: This extension will only work with polymake 4.0.
This extension for polymake was developed to compute cohomology of cellular sheaves and homology of cellular cosheaves. There is a special emphasis on computing tropical (co)homology.
On a polyhedral complex one can define several sheaves and cosheaves, which are a property of the polyhedral complex named SHEAF resp. COSHEAF. A (co)sheaf associates to each face of the polyhedral complex some vector space and maps between those spaces. This information is given as CHOSEN_BASIS (the spaces assigned to the faces) and BLOCKS (the maps between the spaces). Often the spaces will not be set as this information can be part of the maps. Additionally each face of the polyhedral complex is given an orientation, and compatibility of orientations is returned with ORIENTATIONS.
To a (co)sheaf we associate a chain complex, from which we compute its homology, the Betti numbers.
We have some functions that construct some relevant sheaves for tropical cohomology: the Wp and Fp sheaves.
For example, for a polyhedral complex, we associate to each face sigma a linear space LL(sigma). This information is stored as CHOSEN_BASIS. For tau contained in sigma there are natural projection maps LL(sigma) to LL(tau), which are given by the property SIMPLE_BLOCKS. The sheaves Wp are given by the pth exterior powers of the LL(sigma) and natural maps.
The vector space associated to a face tau for the sheaf Fp are generated by \Lambda^p(LL(sigma)) for sigma adjacent to tau.
First clone the git into a directory. Then open polymake and
import_extension("/path/to/git/folder");
For examples how to compute things with this extension, please have a look at the scripts folder.
Tropical Homology - Ilia Itenberg, Ludmil Katzarkov, Grigory Mikhalkin, Ilia Zharkov
Superforms, Tropical Cohomology and Poincaré Duality - Philipp Jell, Kristin Shaw, Jascha Smacka