# LatticePolytopes -- for computations with lattice polytopes

## Description

A lattice polytope is a bounded object of the type Polyhedron. This package is focused on functions that are specific for lattice polytopes rather than general polyhedra. Examples of such methods are isCayley , cayley and randZPoly. Moreover the package contains known classifications of smooth 2-polytopes with up to 12 lattice points and smooth 3-polytopes with up to 16 lattice points. These classifications are accessible via the functions listSmooth2D and listSmooth3D

LatticePolytopes uses the Polyhedra package by René Birkner and the NormalToricVarieties package by Gregory Smith

The following is an example illustrating the main functions provided in the package.

For an introduction to polytopes, we recommend Günter M. Ziegler's Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1995.

## Version

This documentation describes version 1.0 of LatticePolytopes.

## Source code

The source code from which this documentation is derived is in the file LatticePolytopes.m2.

## Exports

• Functions and commands
• adjointPolytope -- computes the adjoint of a polytope
• ambientHalfspaces -- gives the defining halfspaces of a polytope
• areIsomorphic -- checks if two smooth polytopes are isomorphic
• cayley -- constructs the Cayley sum of polytopes
• codegree -- computes the codegree of a polytope
• degreeOfJetSeparation -- computes the degree of jetSeparation at a given point
• epsilonBounds -- computes bounds for the Seshadri constant a general point
• gaussFiber -- computes the general fiber of the Gauss map
• gaussImage -- computes the image the Gauss map
• gausskFiber -- computes the general fiber of the Gauss map of order k
• gausskImage -- computes the image of the Gauss map of order k
• isCayley -- checks if a polytope is Cayley
• isJetSpanned -- checks if the polarized toric variety associated to a set of lattice points is k-jet spanned at a given point.
• iskCayleykEdges -- Checks if a polytope is Cayley of type [P_0*P_1]^k and has every edge of length k
• jetMatrix -- construct the matrix of k-jets evaluated at a given point.
• listSmooth2D -- gives the list of all smooth 2-polytopes with up to 12 lattice points
• listSmooth3D -- gives the list of all smooth 3-polytopes with up to 16 lattice points
• randQPoly -- gives a random rational polytope
• randZPoly -- gives a random lattice polytope
• toricBlowUp -- calculates the stellar subdivision of a polytope at a given face.
• toricDiv -- constructs the toric Weil divisor associated to a polytope
• torusEmbedding -- gives the toric embedding corresponding to a set of lattice points
• Methods
• "adjointPolytope(Matrix,ZZ)" -- see adjointPolytope -- computes the adjoint of a polytope
• "adjointPolytope(Polyhedron,ZZ)" -- see adjointPolytope -- computes the adjoint of a polytope
• "ambientHalfspaces(Polyhedron)" -- see ambientHalfspaces -- gives the defining halfspaces of a polytope
• "areIsomorphic(Matrix,Matrix)" -- see areIsomorphic -- checks if two smooth polytopes are isomorphic
• "areIsomorphic(Polyhedron,Polyhedron)" -- see areIsomorphic -- checks if two smooth polytopes are isomorphic
• "cayley(List)" -- see cayley -- constructs the Cayley sum of polytopes
• "cayley(List,ZZ)" -- see cayley -- constructs the Cayley sum of polytopes
• "cayley(Matrix,Matrix)" -- see cayley -- constructs the Cayley sum of polytopes
• "cayley(Matrix,Matrix,Matrix)" -- see cayley -- constructs the Cayley sum of polytopes
• "cayley(Matrix,Matrix,Matrix,ZZ)" -- see cayley -- constructs the Cayley sum of polytopes
• "cayley(Matrix,Matrix,ZZ)" -- see cayley -- constructs the Cayley sum of polytopes
• "cayley(Polyhedron,Polyhedron)" -- see cayley -- constructs the Cayley sum of polytopes
• "cayley(Polyhedron,Polyhedron,Polyhedron)" -- see cayley -- constructs the Cayley sum of polytopes
• "cayley(Polyhedron,Polyhedron,Polyhedron,ZZ)" -- see cayley -- constructs the Cayley sum of polytopes
• "cayley(Polyhedron,Polyhedron,ZZ)" -- see cayley -- constructs the Cayley sum of polytopes
• "codegree(Matrix)" -- see codegree -- computes the codegree of a polytope
• "codegree(Polyhedron)" -- see codegree -- computes the codegree of a polytope
• "degreeOfJetSeparation(List,Matrix)" -- see degreeOfJetSeparation -- computes the degree of jetSeparation at a given point
• "degreeOfJetSeparation(Matrix,Matrix)" -- see degreeOfJetSeparation -- computes the degree of jetSeparation at a given point
• "epsilonBounds(Polyhedron,ZZ)" -- see epsilonBounds -- computes bounds for the Seshadri constant a general point
• "gaussFiber(List)" -- see gaussFiber -- computes the general fiber of the Gauss map
• "gaussFiber(Matrix)" -- see gaussFiber -- computes the general fiber of the Gauss map
• "gaussImage(List)" -- see gaussImage -- computes the image the Gauss map
• "gaussImage(Matrix)" -- see gaussImage -- computes the image the Gauss map
• "gausskFiber(List,ZZ)" -- see gausskFiber -- computes the general fiber of the Gauss map of order k
• "gausskFiber(Matrix,ZZ)" -- see gausskFiber -- computes the general fiber of the Gauss map of order k
• "gausskImage(List,ZZ)" -- see gausskImage -- computes the image of the Gauss map of order k
• "gausskImage(Matrix,ZZ)" -- see gausskImage -- computes the image of the Gauss map of order k
• "isCayley(Matrix)" -- see isCayley -- checks if a polytope is Cayley
• "isCayley(Polyhedron)" -- see isCayley -- checks if a polytope is Cayley
• "isJetSpanned(List,ZZ,Matrix)" -- see isJetSpanned -- checks if the polarized toric variety associated to a set of lattice points is k-jet spanned at a given point.
• "isJetSpanned(Matrix,ZZ,Matrix)" -- see isJetSpanned -- checks if the polarized toric variety associated to a set of lattice points is k-jet spanned at a given point.
• "iskCayleykEdges(Polyhedron)" -- see iskCayleykEdges -- Checks if a polytope is Cayley of type [P_0*P_1]^k and has every edge of length k
• "jetMatrix(List,ZZ)" -- see jetMatrix -- construct the matrix of k-jets evaluated at a given point.
• "jetMatrix(List,ZZ,Matrix)" -- see jetMatrix -- construct the matrix of k-jets evaluated at a given point.
• "jetMatrix(Matrix,ZZ)" -- see jetMatrix -- construct the matrix of k-jets evaluated at a given point.
• "jetMatrix(Matrix,ZZ,Matrix)" -- see jetMatrix -- construct the matrix of k-jets evaluated at a given point.
• "randQPoly(ZZ,ZZ)" -- see randQPoly -- gives a random rational polytope
• "randZPoly(ZZ,ZZ)" -- see randZPoly -- gives a random lattice polytope
• "randZPoly(ZZ,ZZ,ZZ)" -- see randZPoly -- gives a random lattice polytope
• "toricBlowUp(Matrix,Matrix)" -- see toricBlowUp -- calculates the stellar subdivision of a polytope at a given face.
• "toricBlowUp(Matrix,Matrix,ZZ)" -- see toricBlowUp -- calculates the stellar subdivision of a polytope at a given face.
• "toricBlowUp(Polyhedron,Polyhedron)" -- see toricBlowUp -- calculates the stellar subdivision of a polytope at a given face.
• "toricBlowUp(Polyhedron,Polyhedron,ZZ)" -- see toricBlowUp -- calculates the stellar subdivision of a polytope at a given face.
• "toricDiv(Matrix)" -- see toricDiv -- constructs the toric Weil divisor associated to a polytope
• "toricDiv(Polyhedron)" -- see toricDiv -- constructs the toric Weil divisor associated to a polytope
• "torusEmbedding(List)" -- see torusEmbedding -- gives the toric embedding corresponding to a set of lattice points
• "torusEmbedding(Matrix)" -- see torusEmbedding -- gives the toric embedding corresponding to a set of lattice points
• Symbols
• smoothTest (missing documentation)

## For the programmer

The object LatticePolytopes is .