application: fan

This application deals with polyhedral fans. You can define a fan, e.g. via its RAYS and MAXIMAL_CONES and compute several properties like HASSE_DIAGRAM and F_VECTOR.

imports from: common, graph, polytope
uses: topaz

Objects

PolyhedralFan
A polyhedral fan
Properties of PolyhedralFan

GORENSTEIN: common::Bool
A fan is GORENSTEIN if it is Q_GORENSTEIN with Q_GORENSTEIN_INDEX equal to one

Q_GORENSTEIN: common::Bool
A fan is Q_GORENSTEIN if each maximal cone is Q_GORENTEIN

Q_GORENSTEIN_INDEX: common::Int
If a fan is Q_GORENSTEIN, then its Q_GORENSTEIN_INDEX is the least common multiple of the Q-Gorenstein indices of its maximal cones. Otherwise Q_GORENSTEIN_INDEX is undefined.

SMOOTH_FAN: common::Bool
A fan is smooth if all cones of the fan are smooth.

Basic properties

AMBIENT_DIM: common::Int
Dimension of the space which contains the polyhedral fan.

COMBINATORIAL_DIM: common::Int
the combinatorial dimension, that is, the dimension of the fan modulo its LINEALITY_SPACE.

CONES: common::Array<IncidenceMatrix<NonSymmetric>>
List of all cones of the fan of each dimension. Indices refer to RAYS.

DIM: common::Int
Dimension of the polyhedral fan.

FACET_NORMALS: common::Matrix<Scalar>
The possible facet normals of all maximal cones.

FULL_DIM: common::Bool
DIM and AMBIENT_DIM coincide.

INPUT_CONES: common::Array<Set<Int>>
Maybe redundant list of not necessarily maximal cones. Indices refer to INPUT_RAYS.

INPUT_LINEALITY: common::Matrix<Scalar>
Vectors whose linear span defines a subset of the lineality space of the fan; redundancies are allowed.
Input section only. Ask for LINEALITY_SPACE if you want to know the lineality space.

INPUT_RAYS: common::Matrix<Scalar>
Rays from which the cones are formed. May be redundant.

LINEALITY_DIM: common::Int
Dimension of LINEALITY_SPACE.

LINEALITY_SPACE: common::Matrix<Scalar>
Since we do not require our cones to be pointed: a basis of the lineality space of the fan.

LINEAR_SPAN_NORMALS: common::Matrix<Scalar>
The possible linear span normals of all maximal cones.

MAXIMAL_CONES: common::IncidenceMatrix<NonSymmetric>
Non redundant list of maximal cones. Indices refer to RAYS.

MAXIMAL_CONES_DIMS: common::Array<Int>
The dimensions of the maximal cones.

MAXIMAL_CONES_FACETS: common::SparseMatrix<Int, NonSymmetric>
Tells for each maximal cone what are its facets. A negative number means that the corresponding row of FACET_NORMALS has to be negated.

MAXIMAL_CONES_INCIDENCES: common::Array<IncidenceMatrix<NonSymmetric>>
array of incidence matrices of all maximal cones

MAXIMAL_CONES_LINEAR_SPAN_NORMALS: common::IncidenceMatrix<NonSymmetric>
Tells for each maximal cone what is its linear span. Indices refer to LINEAR_SPAN_NORMALS. Rows correspond to MAXIMAL_CONES_FACETS

N_FACET_NORMALS: common::Int
The number of possible facet normals of all maximal cones.

N_MAXIMAL_CONES: common::Int
Number of MAXIMAL_CONES.

N_RAYS: common::Int
Number of RAYS.

ORTH_LINEALITY_SPACE: common::Matrix<Scalar>
A basis of the orthogonal complement to LINEALITY_SPACE.

POINTED: common::Bool
LINEALITY_SPACE is trivial.

RAYS: common::Matrix<Scalar>
Rays from which the cones are formed. Non-redundant.

RAY_LABELS: common::Array<String>
Unique names assigned to the RAYS. If specified, they are shown by visualization tools instead of vertex indices.
For a polyhedral fan built from scratch, you should create this property by yourself, either manually in a text editor, or with a client program.
Combinatorics
COMPLETE: common::Bool
The polyhedral fan is complete if its suport is the whole space.
Due to undecibilty issues this is checked heuristically only. See the remarks on SPHERE for details.
DUAL_GRAPH: graph::Graph<Undirected>
The graph whose nodes are the maximal cones which are connected if they share a common facet.
F2_VECTOR: common::Matrix<Integer>
f_ik is the number of incident pairs of i-dimensional cones and k-dimensional cones; the main diagonal contains the F_VECTOR.
F_VECTOR: common::Vector<Integer>
f_k is the number of k-dimensional cones.
GRAPH: graph::Graph<Undirected>
The graph of the cone intersected with a sphere, that is, the vertices are the rays which are connected if they are contained in a common two-dimensional cone.
HASSE_DIAGRAM: graph::FaceLattice
The poset of subcones of the polyhedral fan organized as a directed graph. Each node corresponds to some proper subcone of the fan. The nodes corresponding to the rays and maximal cones appear in the same order as the elements of RAYS and MAXIMAL_CONES properties.
One special node represents the origin.
PURE: common::Bool
The polyhedral fan is pure if all maximal cones are of the same dimension.
SIMPLICIAL: common::Bool
The polyhedral fan is simplicial if all maximal cones are simplicial.

REGULAR: common::Bool

True if the fan is the normal fan of a polyhedron.

HOMOLOGY: topaz::Array<HomologyGroup>

The homology of the intersection of the fan with the unit sphere.

INTERSECTION_COMPLEX: topaz::SimplicialComplex

If the fan is SIMPLICIAL the simplicial complex obtained by intersection the fan with the unit sphere. If the fan is not SIMPLICIAL the crosscut complex of the insersection.

cone ()

Gives the ith maximal cone.

VISUAL ()

Visualizes the fan, intersected with the unit ball. return: Visual::PolyhedralFan

FIXME: use combinatorics via MAXIMAL_CONES_FACETS

Options

option list:

Visual::Polygons::decorations

ConesPerm

UNDOCUMENTED

Properties of ConesPerm

PERMUTATION: common::Array<Int>
transforming MAXIMAL_CONES from this into basic object

RaysPerm

UNDOCUMENTED

Properties of RaysPerm

PERMUTATION: common::Array<Int>
transforming RAYS from this into basic object

SymmetricFan

A PolyhedralFan with a symmetry group acting on the coordinates.

derived from: PolyhedralFan

Properties of SymmetricFan

Basic properties
CONES_ORBIT_SIZES: common::Array<Array<Int>>
Number of CONES_REPS in each orbit.
CONES_REPS: common::Array<Array<Set<Int>>>
List of all cones of all dimensions of the fan, one from each orbit. Indices refer to RAYS.
INPUT_CONES_REPS: common::Array<Set<Int>>
Maybe redundant list of not necessarily maximal cones, one from each orbit. Indices refer to INPUT_RAYS.
MAXIMAL_CONES_IN_ORBITS: common::Array<Set<Int>>
Tells which maximal cone is in the orbit of which representative, indices refers to rows of MAXIMAL_CONES.
MAXIMAL_CONES_ORBIT_SIZES: common::Array<Int>
Number of MAXIMAL_CONES_REPS in each orbit.
MAXIMAL_CONES_REPS: common::Array<Set<Int>>
Non redundant list of maximal cones, one from each orbit. Indices refer to RAYS.
MAXIMAL_CONES_REPS_DIMS: common::Array<Int>
The dimensions of MAXIMAL_CONES_REPS.
MAXIMAL_CONES_REPS_FACETS: common::SparseMatrix<Int, NonSymmetric>
Tells for each maximal cone representative what are its facets. A negative number means that the corresponding row of REPS_FACET_NORMALS has to be negated.
MAXIMAL_CONES_REPS_LINEAR_SPAN_NORMALS: common::IncidenceMatrix<NonSymmetric>
Tells for each maximal cone representative what is its linear span. Indices refer to REPS_LINEAR_SPAN_NORMALS. Rows correspond to MAXIMAL_CONES_REPS_FACETS
N_MAXIMAL_CONE_ORBITS: common::Int
Number of orbits of MAXIMAL_CONES.
N_RAY_ORBITS: common::Int
Number of orbits of RAYS.
RAYS_IMAGES: common::Array<Array<Int>>
Each row contains the image of all RAYS under one element of the symmetry group.
RAYS_IN_ORBITS: common::Array<Set<Int>>
Tells which ray is in the orbit of which representative, indices refers to rows of RAYS.
RAYS_ORBIT_SIZES: common::Array<Int>
Number of elements of each orbit of RAYS.
RAYS_REPS: common::Matrix<Scalar>
One ray from each orbit. Non-redundant.
RAYS_REPS_LABELS: common::Array<String>
Unique names assigned to the RAYS_REPS.
REPS_FACET_NORMALS: common::Matrix<Scalar>
The possible facet normals of all maximal cone representatives.
REPS_LINEAR_SPAN_NORMALS: common::Matrix<Scalar>
The possible linear span normals of all maximal cone representatives.
SYMMETRY_GENERATORS: common::Array<Array<Int>>
Each element of the array is a generator of the subgroup of the symmetric group acting on the coordintates. Each generator is represented by an Array whose ith entry is the image of ith coordinate.
SYMMETRY_GROUP: common::Array<Array<Int>>
Each element of the array is an element of the symmetry group.

N_SYMMETRIES: common::Int

Number of elements of the symmetry group.

ORBITS_F_VECTOR: common::Vector<Integer>

f_k is the number of k-dimensional cones up to symmetry.

cone_representative ()

Gives the ith maximal cone representative.

TropicalVariety

UNDOCUMENTED

derived from: PolyhedralFan

Properties of TropicalVariety

Tropical varieties
MULTIPLICITIES: common::Vector<Scalar>
Multiplicities of the MAXIMAL_CONES.

common_refinement (f1, f2) → PolyhedralFan

Computes the common refinement of two complete fans.

Parameters

PolyhedralFan	f1
PolyhedralFan	f2

Returns

PolyhedralFan

check_fan (rays, cones) → PolyhedralFan

Checks whether a given set of rays together with a list cones defines a polyhedral fan. If this is the case, the ouput is the PolyhedralFan defined by rays as INPUT_RAYS, cones as INPUT_CONES, lineality_space as LINEALITY_SPACE if this option is given.

Parameters

Matrix	rays
Array< Set<int> >	cones

Options

Matrix	lineality_space	Common lineality space for the cones.
Bool	verbose	prints information about the check.

Returns

PolyhedralFan

check_fan_objects (C ...) → PolyhedralFan

Checks whether the polytope::Cone objects form a polyhedral fan. If this is the case, returns that PolyhedralFan.

Parameters

Cone

C ...

Options

Bool

verbose

prints information about the check.

Returns

PolyhedralFan

hyperplane_arrangement (H)

Compute the fan given by a bunch of hyperplanes H.

Parameters

Matrix

normal_fan (p)

Computes the normal fan of p.

Parameters

Polytope

Navigation

application: fan

Objects

PolyhedralFan

Properties of PolyhedralFan

Basic properties

Combinatorics

Others

Topology

User Methods of PolyhedralFan

Visualization

Options

Permutations of PolyhedralFan

Properties of ConesPerm

Properties of RaysPerm

SymmetricFan

Properties of SymmetricFan

Basic properties

Basic properties.

Combinatorics

User Methods of SymmetricFan

TropicalVariety

Properties of TropicalVariety

Tropical varieties

User Functions

Parameters

Returns

Consistency check

Parameters

Options

Returns

Parameters

Options

Returns

Constructing a fan

Parameters

Parameters