Polytope

Polytope ¶

Polytope(points, labels=None, backend=None, deterministic_glsm_basis=False)

A lattice polytope.

This class handles all computations relating to lattice polytopes, such as the computation of lattice points and faces. When using reflexive polytopes, it also allows the computation of topological properties of the arising Calabi-Yau hypersurfaces that only depend on the polytope.

The polytope is the convex hull of the input points. All lattice points of the polytope are computed on construction and are labelled. Input points keep the labels given in labels (if any), and every other lattice point receives an integer label that does not clash with the given ones. Most methods accept and return labels, and the helpers points_to_labels and points_to_indices convert between points, labels, and indices.

The Polytope class is also exported at the root of the package, so it can be imported from either cytools.polytope or cytools.

Parameters:

points (ArrayLike) –

The lattice points whose convex hull defines the polytope. CYTools only supports lattice polytopes, so the points must have integer coordinates. They must all be distinct, but they do not need to be vertices.
labels (ArrayLike | None, default: None ) –

Labels for the input points, i.e. points[i] is labelled labels[i]. The labels must be distinct and hashable, and there must be one per input point. If not specified, all points are given integer labels following the default point ordering (see points).
backend (str | None, default: None ) –

The backend used to construct the convex hull. The options are "ppl", "qhull", or "palp". If not specified, PPL is used for polytopes of dimension 1 to 4, and PALP otherwise. 0-dimensional polytopes always use PALP.
deterministic_glsm_basis (bool, default: False ) –

Whether to choose the GLSM (and thus divisor) basis in a deterministic manner, so that it is consistent across different machines. The basis chosen with this flag may differ from the default one.

Raises:

ValueError –

If points is a string, if the points or labels are not unique, if the number of labels does not match the number of points, or if backend is not one of the allowed options.

Examples:

We construct two polytopes from lists of points.

>>> from cytools import Polytope
>>> p1 = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,-1,-1,-1]])
>>> print(p1)
A 4-dimensional reflexive lattice polytope in ZZ^4
>>> p2 = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[-1,-1,-1,0]])
>>> print(p2)
A 3-dimensional reflexive lattice polytope in ZZ^4

Custom labels can be given to the input points.

>>> p3 = Polytope([[1,0],[0,1],[-1,-1]], labels=["a","b","c"])
>>> p3.labels
(0, 'c', 'b', 'a')

Methods:

all_triangulations –

Compute all triangulations of the polytope using TOPCOM.
ambient_dimension –

Return the dimension of the ambient lattice.
automorphisms –

Return the \(GL(d,\mathbb{Z})\) matrices that leave the polytope invariant.
chi –

Compute the Euler characteristic of the associated Calabi-Yau.
clear_cache –

Clear the cached results of any previous computation.
dimension –

Return the dimension of the polytope.
dual_polytope –

Return the dual polytope (also called polar polytope).
faces –

Compute the faces of the polytope.
facets –

Return the facets (codimension-1 faces) of the polytope.
find_2d_reflexive_subpolytopes –

Find the 2-dimensional reflexive subpolytopes of a 4D polytope.
glsm_basis –

Compute a basis of columns of the GLSM charge matrix.
glsm_charge_matrix –

Compute the GLSM charge matrix of the theory arising from the polytope.
glsm_linear_relations –

Compute the linear relations of the columns of the GLSM charge matrix.
h11 –

Return the Hodge number \(h^{1,1}\) of the associated Calabi-Yau.
h12 –

Return the Hodge number \(h^{1,2}\) of the associated Calabi-Yau.
h13 –

Return the Hodge number \(h^{1,3}\) of the associated Calabi-Yau.
h22 –

Return the Hodge number \(h^{2,2}\) of the associated Calabi-Yau.
hpq –

Return the Hodge number \(h^{p,q}\) of the associated Calabi-Yau.
inequalities –

Return the inequalities giving the hyperplane representation.
inequivalent_Z2_actions –

Enumerate the inequivalent toric \(\mathbb{Z}_2\) actions.
is_affinely_equivalent –

Return whether the polytopes are related by an integral affine map.
is_favorable –

Return whether the associated Calabi-Yau hypersurface is favorable.
is_linearly_equivalent –

Return whether the polytopes are related by a \(GL(d,\mathbb{Z})\) map.
is_reflexive –

Return whether the polytope is reflexive.
is_solid –

Return whether the polytope is solid (i.e. full-dimensional).
is_trilayer –

Return whether the polytope is "trilayer".
minkowski_sum –

Return the Minkowski sum of the two polytopes.
nef_partitions –

Compute the nef partitions of the polytope using PALP.
normal_form –

Return the normal form of the polytope as defined by Kreuzer-Skarke.
points –

Return the lattice points of the polytope.
points_to_indices –

Return the indices corresponding to the given points.
points_to_labels –

Return the labels corresponding to the given points.
pts_bdry –

Return the boundary lattice points of the polytope.
pts_codim2 –

Return the boundary lattice points not interior to facets.
pts_facet –

Return the lattice points interior to facets.
pts_int –

Return the interior lattice points of the polytope.
pts_not_facets –

Return the lattice points not interior to facets.
random_triangulations_fair –

Construct pseudorandom fine, regular triangulations of the polytope.
random_triangulations_fast –

Construct pseudorandom regular triangulations of the polytope.
random_triangulations_gnn –

Construct random NTFE FR(S)Ts of a reflexive 4-dimensional polytope.
triangulate –

Return a single regular triangulation of the polytope.
vertices –

Return the vertices of the polytope.
volume –

Return the normalized volume of the polytope.

Attributes:

backend (str) –

Return the backend used to compute the convex hull.
label_origin –

Return the label of the origin.
labels –

Return the labels of all lattice points, in order.
labels_bdry –

Return the labels of the boundary points.
labels_codim2 –

Return the labels of the boundary points not interior to facets.
labels_facet –

Return the labels of the points interior to facets.
labels_int –

Return the labels of the interior points.
labels_not_facet –

Return the labels of the points not interior to facets.
labels_vertices –

Return the labels of the vertices, sorted.

Polytope

Polytope ¶

backend property ¶

label_origin property ¶

labels property ¶

labels_bdry property ¶

labels_codim2 property ¶

labels_facet property ¶

labels_int property ¶

labels_not_facet property ¶

labels_vertices property ¶

all_triangulations ¶

ambient_dimension ¶

automorphisms ¶

chi ¶

clear_cache ¶

dimension ¶

dual_polytope ¶

faces ¶

facets ¶

find_2d_reflexive_subpolytopes ¶

glsm_basis ¶

glsm_charge_matrix ¶

glsm_linear_relations ¶

h11 ¶

h12 ¶

h13 ¶

h22 ¶

hpq ¶

inequalities ¶

inequivalent_Z2_actions ¶

is_affinely_equivalent ¶

is_favorable ¶

is_linearly_equivalent ¶

is_reflexive ¶

is_solid ¶

is_trilayer ¶

minkowski_sum ¶

nef_partitions ¶

normal_form ¶

points ¶

points_to_indices ¶

points_to_labels ¶

pts_bdry ¶

pts_codim2 ¶

pts_facet ¶

pts_int ¶

pts_not_facets ¶

random_triangulations_fair ¶

random_triangulations_fast ¶

random_triangulations_gnn ¶

triangulate ¶

vertices ¶

volume ¶

backend `property` ¶

label_origin `property` ¶

labels `property` ¶

labels_bdry `property` ¶

labels_codim2 `property` ¶

labels_facet `property` ¶

labels_int `property` ¶

labels_not_facet `property` ¶

labels_vertices `property` ¶