Interpreting vertices of Hom(p, q)

[Jack Love]

I'll pose this question using an example: I give Polymake a 2-cube ($p=cube(2)) and a line segment ($q=new Polytope<Rational>(POINTS=>new Matrix<Rational>([[1, 0], [0, 1]])), and then build the hom-polytope $hom=mapping_polytope($p, $q) and ask for its vertices. How can I determine which vertices correspond to which maps? For example, one of the vertices corresponds to the map that sends the entire cube to [0, 1]. How can I determine which vertex corresponds to that map?

[joswig]

You can force polymake to write out the labels:

Code: Select all

> $p=cube(2); > $q=new Polytope<Rational>(POINTS=>new Matrix<Rational>([[1, 0], [1, 1]]); > $hom=mapping_polytope($p, $q, relabel=>1); > print rows_labeled($hom->VERTICES,$hom->VERTEX_LABELS); v0*F1,v1*F1,v2*F0,v3*F0:1 1/2 0 1/2 v0*F1,v1*F1,v2*F1,v3*F1:1 0 0 0 v0*F1,v1*F0,v2*F1,v3*F0:1 1/2 1/2 0 v0*F0,v1*F0,v2*F1,v3*F1:1 1/2 0 -1/2 v0*F0,v1*F1,v2*F0,v3*F1:1 1/2 -1/2 0 v0*F0,v1*F0,v2*F0,v3*F0:1 1 0 0

Please note that in your code $q is not a line segment.

[Jack Love]

This is great, thank you! Yes, $p is not a line segment; that was a typo on my part, as was the "[0, 1]" later in the post.