===== Properties and Methods defined for lattice polytopes ===== This page summarizes properties and methods defined for lattice polytopes in polymake. For an introductory example see [[user_guide:tutorials:lattice_polytopes_tutorial|here]]. Up to release 2.14, a lattice polytope in polymake used to be a subclass of a rational polytope, and all basic properties were derived from that. Starting with release 2.15 (or 3.0, not decided yet), there is no distinct type LatticePolytope anymore. A lattice polytope is an instance of rational polytope having the property LATTICE=1. All properties and methods described below have not changed. As facet inequalities are standardized automatically, they usually do not appear as integer vectors. ''polymake'' provides the ''primitive(Vector)'' to transform a rational or integral vector into an integral vector such that the gcd of all entries is 1 (as a return vector, the input remains unchanged). ''primitive(Matrix)'' applies this function to all rows of the matrix. ==== Properties for Polytope and LatticePolytope ==== ^ Name ^ Description ^ | ''LATTICE'' | The polytope has integer vertices | | ''LATTICE_POINTS'' | List of lattice points in the polytope | | ''N_LATTICE_POINTS'' | Number of lattice points in the polytope | | ''INTERIOR_LATTICE_POINTS'' | List of lattice points in the interior of the polytope | | ''N_INTERIOR_LATTICE_POINTS'' | Number of lattice points in the interior of the polytope | | ''BOUNDARY_LATTICE_POINTS'' | List of lattice points on the boundary of the polytope | | ''N_BOUNDARY_LATTICE_POINTS'' | Number of lattice points on the boundary of the polytope | | ''HILBERT_BASIS'' | The Hilbert Basis of the cone spanned by ''P x {1}'' | | ''N_HILBERT_BASIS'' | Number of elements of the Hilbert Basis | | ''MINKOWSKI_SUMMAND_CONE'' | the cone of all Minkowski summands of ''P''. This has connected user_methods ''MINKOWSKI_SUMMAND_COEFF'' and ''MINKOWSKI_SUMMAND_POINT''. The first returns the polytope correspondong to a point in the cone given by coefficients in the rays, the second for a point given in the coordiantes of the ambient space. | ==== Properties for LatticePolytope ==== ^ Name ^ Description ^ | ''REFLEXIVE'' | The polytope and it's dual have integral vertices | | ''GORENSTEIN'' | A dilation of the polytope is reflexive up to translation | | ''GORENSTEIN_INDEX'' | If P is Gorenstein, then this is the integer k such that the dilation k*P is reflexive | | ''GORENSTEIN_VECTOR'' | If P is Gorenstein, then this is the unique interior lattice point in k*P | | ''CANONICAL'' | There is exactly one interior lattice point | | ''TERMINAL'' | There is exactly one interior lattice point and all other lattice points are vertices | | ''LATTICE_EMPTY'' | The polytope contains no lattice points other than the vertices | | ''LATTICE_VOLUME'' | The normalized volume of the polytope ''VOLUME * dim!'' | | ''LATTICE_DEGREE'' | The degree of the h*-polynomial or Ehrhart polynomial| | ''LATTICE_CODEGREE'' | ''dim+1-degree'' or the smallest integer k such that k*P has an interior lattice point | | ''EHRHART_POLYNOMIAL_COEFF'' | The coefficients of the Ehrhart polynomial starting at the constant coefficient | | ''H_STAR_VECTOR'' | The coefficients of the h*-polynomial starting at the constant coefficient | | ''NORMAL'' | The cone spanned by ''P x {1}'' is generated in height 1 | | ''FACET_WIDTHS'' | The integral width of the polytope with respect to each facet normal | | ''FACET_WIDTH'' | The maximal integral width of the polytope with respect to the facet normals | | '' FACET_VERTEX_LATTICE_DISTANCES'' | The matrix of lattice distances between facets and vertices, rows: facets, columns: vertices | | ''COMPRESSED'' | FACET_WIDTH is 1 | | ''SMOOTH'' | The associated projective variety is smooth; the determinant of the edge directions is ''+/-1'' at every vertex | | ''VERY_AMPLE'' | The Hilbert Basis of the cone spanned by the edge-directions of any vertex lies inside the polytope | | ''GRAPH.LATTICE_EDGE_LENGTHS'' | the lattice lengths of the edges of the graph | ==== User Methods for LatticePolytope ==== ^ Name ^ Description ^ | ''N_LATTICE_POINTS_IN_DILATION(n)'' | The number of lattice points in the n-th dilation of the polytope | | ''FACET_POINT_LATTICE_DISTANCES(Vector)'' | The lattice distance of an integral vector from the facets of the polytope | ==== User Functions for LatticePolytope ==== ^ Name ^ Description ^ | ''ambient_lattice_normalization(Polytope)'' | transforms a ''LatticePolytope'' into a full dimensional polytope in the lattice given by the intersection of Zn and the affine hull | | ''vertex_lattice_normalization(Polytope)'' | transforms a ''LatticePolytope'' into a full dimensional polytope in the lattice spanned by the vertices of the polytope | | ''induced_lattice(Polytope)'' | returns a basis of the lattice spanned by the vertices of the polytope | | ''lattice_pyramid(Polytope)'' | returns a pyramid over the polytope where the apex sits at a lattice point at height one above the base | | ''lattice_bypyramid(Polytope)'' | returns a lattice bipyramid over the polytope with apices at height one above the base (this does not work for a simplex without interior lattice points) | | ''transportation_polytope(row_margin,column_margin)'' | returns the transportation polytope defined by the given row and column margins | | ''lattice_isomorphic_smooth_polytopes(Polytope,Polytope)'' | checks for two smooth polytopes whether they are lattice equivalent in the ambient lattice | ==== Properties for Cone ==== ^ Name ^ Description ^ | ''HILBERT_BASIS'' | The Hilbert Basis of the cone ''C'' | | ''N_HILBERT_BASIS'' | Number of elements of the Hilbert Basis | | ''SMOOTH_CONE'' | true, if the primitive generators of the cone are a basis of ''Z^n'' | | ''Q_GORENSTEIN_CONE'' | true, if all primitive generators of the cone lie in an affine hyperplane spanned by a lattice functional in the dual cone (but not in the lineality space of the dual cone| | '' Q_GORENSTEIN_INDEX '' | the height of the primitive generators, if the cone is ''Q_GORENSTEIN'' | | ''GORENSTEIN'' | true, if ''Q_GORENSTEIN'' with index 1 | | ''HOMOGENEOUS'' | true if the primitive generators of the rays lie on a hyperplane | ==== Properties for PolyhedralFan ==== ^ Name ^ Description ^ | ''SMOOTH_FAN'' | true, if all cones are smooth | | ''Q_GORENSTEIN_CONE'' | true, if all maximal cones are ''Q_GORENSTEIN'' | | '' Q_GORENSTEIN_INDEX '' | the lcm of the indices of all maixmal cones | | ''GORENSTEIN'' | true, if ''Q_GORENSTEIN'' with index 1 |