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<div><h1>SRdeformations -- Deformations of Stanley-Reisner rings and related computations.</h1>
<div class="single"><h2>Description</h2>
<div><p><b>Overview:</b></p>
<p>The goal of this package is to provide a basic framework for deformations of Stanley-Reisner rings, the deformations polytope, its tropical subcomplex and tropical mirror symmetry in general.</p>
<p>This is work in progress, many interesting pieces still missing and more testing is necessary. The efficiency and robustness of some of the pieces has to be improved.</p>
<p>See the <i>Macaulay2</i> package repository for new versions of this file.</p>
<p>All suggestions and contributions are welcome.</p>
<p><b>Contents:</b></p>
<p>This package so far consists of the following parts:</p>
<p>The first part provides a data structure for deformations of monomial ideals (including a toric grading). The class <a href="___First__Order__Deformation.html" title="The class of all first order deformations of monomial ideals.">FirstOrderDeformation</a> stores (and computes the dimension of) a big torus graded part of the vector space of first order deformations (specified by a Laurent monomial).</p>
<p>The second part gives an implementation of (not necessarily simplicial) embedded complexes and co-complexes and their correspondence to monomial ideals. The main focus here is on a flexible implementation, computing and storing data only as soon as it is needed. See <a href="___Complex.html" title="The class of all embedded complexes.">Complex</a>, <a href="___Co__Complex.html" title="The class of all embedded co-complexes.">CoComplex</a>, <a href="___Face.html" title="The class of all faces of complexes or co-complexes.">Face</a>.</p>
<p>The third part computes all first order deformations of Stanley-Reisner ideals using part one and two via the results of Klaus Altmann and Jan Arthur Christophersen who reduce the problem to links of faces of simplicial complexes. See <a href="_deformations__Face.html" title="Compute the deformations associated to a face.">deformationsFace</a> and <a href="_deform.html" title="Compute the deformations associated to a Stanley-Reisner complex.">deform</a>.</p>
<p>The fourth part gives functions to compute the first order deformations polytope (see <a href="___P__T1.html" title="Compute the deformation polytope associated to a Stanley-Reisner complex.">PT1</a>) and its tropical subcocomplex (see <a href="_trop__Def.html" title="The co-complex of tropical faces of the deformation polytope.">tropDef</a>).</p>
<p><b>What’ new:</b></p>
<p><i>Oct 6, 2009 (Version 0.48)</i></p>
<p>Resolved a compatiblity issue with the new version of the M2 package <i>Polyhedra</i>.</p>
<p></p>
<p><i>Oct 3, 2009 (Version 0.47)</i></p>
<p>Improvements to the documentation.</p>
<p><i>Sept 20, 2009 (Version 0.46)</i></p>
<p>Added a function <a href="_hull.html" title="The positive hull complex.">hull</a> to compute the positive hull of a list of vectors.</p>
<p></p>
<p><i>Sept 8, 2009 (Version 0.45)</i></p>
<p>Adjusted the methods using ConvexInterface.m2 to version 0.25 which now indexes the vertices instead of using coordinates. This improves the performance substantially.</p>
<p>Improved the computation of the dimension of a <a href="___First__Order__Deformation.html" title="The class of all first order deformations of monomial ideals.">FirstOrderDeformation</a> using a much smaller linear system of equations.</p>
<p>Fixed a bug in <a href="_dim_lp__Face_rp.html" title="Compute the dimension of a face.">dim(Face)</a>.</p>
<p>Exported the symbol <a href="_dual__Face.html" title="Optional data of Face.">dualFace</a> which stores the dual of a face to make this key available for the user.</p>
<p><a href="_conv__Hull.html" title="The convex hull complex.">convHull</a> now stores in the faces their dimensions (see the key indices).</p>
<p>Added an <a href="../../Macaulay2Doc/html/___Option.html" title="the class of all pairs x => y">Option</a> <a href="_file.html" title="Store result of a computation in a file.">file</a> to <a href="_conv__Hull.html" title="The convex hull complex.">convHull</a> and <a href="___P__T1.html" title="Compute the deformation polytope associated to a Stanley-Reisner complex.">PT1</a> to store the result in a file. <a href="_conv__Hull.html" title="The convex hull complex.">convHull</a> applied to a <a href="../../Macaulay2Doc/html/___String.html" title="the class of all strings">String</a> reads a convex hull from a file.</p>
<p>Improved speed of <a href="_global__Sections.html" title="The global sections of a toric divisor.">globalSections</a> when dealing with ideals in the Cox ring of projective space.</p>
<p>Added an optional key deform to complexes storing the associated deformations, see <a href="_deform.html" title="Compute the deformations associated to a Stanley-Reisner complex.">deform</a>.</p>
<p>Added functions <a href="_save__Deformations.html" title="Store the deformation data of a complex in a file.">saveDeformations</a> and <a href="_load__Deformations.html" title="Read the deformation data of a complex from a file.">loadDeformations</a> to save to and load from a file the deformations associated to a <a href="___Complex.html" title="The class of all embedded complexes.">Complex</a>.</p>
<p><b>Technical note:</b></p>
<p>In the future some of the methods and keys may change their name.</p>
<p>At some point <a href="___Complex.html" title="The class of all embedded complexes.">Complex</a> and <a href="___Co__Complex.html" title="The class of all embedded co-complexes.">CoComplex</a> will get a common ancestor.</p>
<p><b>Installation:</b></p>
<p><a href="../../Macaulay2Doc/html/_install__Package.html" title="load and install a package and its documentation ">installPackage</a>("SRdeformations")</p>
<p><b>Options:</b></p>
<p>Computations with polyhedra are used in this package so far in by following functions:</p>
<p><a href="_conv__Hull.html" title="The convex hull complex.">convHull</a> -- The convex hull complex</p>
<p><a href="_global__Sections.html" title="The global sections of a toric divisor.">globalSections</a> -- The global sections of a toric divisor</p>
<p>which again are called by <a href="___P__T1.html" title="Compute the deformation polytope associated to a Stanley-Reisner complex.">PT1</a>, <a href="_deform.html" title="Compute the deformations associated to a Stanley-Reisner complex.">deform</a> and <a href="_deformations__Face.html" title="Compute the deformations associated to a face.">deformationsFace</a>. If <a href="_deform.html" title="Compute the deformations associated to a Stanley-Reisner complex.">deform</a> and <a href="_deformations__Face.html" title="Compute the deformations associated to a face.">deformationsFace</a> are used for <a href="___Complex.html" title="The class of all embedded complexes.">Complex</a>es inside a standard simplex, i.e., Stanley-Reisner ideals with the standard projective space grading, then they don’t rely on polyhedra computations.</p>
<p>There are two choices of M2 packages to be called for computations with polyhedra:</p>
<p><a href="../../Polyhedra/html/index.html" title="for computations with convex polyhedra, cones, and fans">Polyhedra</a>:</p>
<p>Works without additional configuration, but is not very fast.</p>
<p>It uses the M2 package <i>FourierMotzkin</i>.</p>
<p>To use <i>Polyhedra</i> do</p>
<p><a href="../../Macaulay2Doc/html/_load__Package.html" title="load a package">loadPackage</a>("SRdeformations",Configuration=>{"UseConvex"=>false})</p>
<p>This is the standard configuration, i.e., it is synonymous to</p>
<p><a href="../../Macaulay2Doc/html/_load__Package.html" title="load a package">loadPackage</a>("SRdeformations")</p>
<p><i>ConvexInterface</i>:</p>
<p>This package has to be installed first, see its documentation for this.</p>
<p>It calls the Maple package Convex and is faster than Polyhedra, hence the perferable choice. If you want to do non-trivial examples you have to go for it.</p>
<p>To use it type</p>
<p><a href="../../Macaulay2Doc/html/_load__Package.html" title="load a package">loadPackage</a>("SRdeformations",Configuration=>{"UseConvex"=>true})</p>
<p></p>
<p>Note that you can change the standard option by editing the file init-SRdeformations.m2 in the .Macaulay directory in your home directory (it will be overwritten when you reinstall a newer version of the package, but there will be a backup).</p>
<p></p>
<p><b>Examples:</b></p>
<p><a href="_mirror__Sphere.html" title="Example how to compute the mirror sphere.">mirrorSphere</a> uses the above functions to compute the mirror sphere of a Calabi-Yau.</p>
<div/>
</div>
</div>
<div class="single"><h2>Author</h2>
<ul><li><div class="single"><a href="http://www.math.uni-sb.de/ag/schreyer/jb/">Janko Boehm</a><span> <<a href="mailto:boehm@math.uni-sb.de">boehm@math.uni-sb.de</a>></span></div>
</li>
</ul>
</div>
<div class="single"><h2>Version</h2>
This documentation describes version <b>0.48</b> of SRdeformations.</div>
<div class="single"><h2>Source code</h2>
The source code from which this documentation is derived is in the file <a href="../../../../Macaulay2/SRdeformations.m2">SRdeformations.m2</a>.</div>
<div class="single"><h2>Exports</h2>
<ul><li><div class="single">Types<ul><li><span><a href="___Co__Complex.html" title="The class of all embedded co-complexes.">CoComplex</a> -- The class of all embedded co-complexes.</span></li>
<li><span><a href="___Complex.html" title="The class of all embedded complexes.">Complex</a> -- The class of all embedded complexes.</span></li>
<li><span><a href="___Face.html" title="The class of all faces of complexes or co-complexes.">Face</a> -- The class of all faces of complexes or co-complexes.</span></li>
<li><span><a href="___First__Order__Deformation.html" title="The class of all first order deformations of monomial ideals.">FirstOrderDeformation</a> -- The class of all first order deformations of monomial ideals.</span></li>
</ul>
</div>
</li>
<li><div class="single">Functions<ul><li><span><a href="_add__Coker__Grading.html" title="Stores a cokernel grading in a polynomial ring.">addCokerGrading</a> -- Stores a cokernel grading in a polynomial ring.</span></li>
<li><span><a href="_add__Face__Data__To__Complex.html" title="Adds to a complex face data.">addFaceDataToComplex</a> -- Adds to a complex face data.</span></li>
<li><span><a href="_add__Facet__Data__To__Complex.html" title="Adds to a complex facet data.">addFacetDataToComplex</a> -- Adds to a complex facet data.</span></li>
<li><span><a href="_big__Torus__Degree.html" title="The big torus degree of a deformation.">bigTorusDegree</a> -- The big torus degree of a deformation.</span></li>
<li><span><a href="_boundary__Cyclic__Polytope.html" title="The boundary complex of a cyclic polytope.">boundaryCyclicPolytope</a> -- The boundary complex of a cyclic polytope.</span></li>
<li><span><a href="_boundary__Of__Polytope.html" title="The boundary of a polytope.">boundaryOfPolytope</a> -- The boundary of a polytope.</span></li>
<li><span><a href="_closed__Star.html" title="The closed star of a face of a complex.">closedStar</a> -- The closed star of a face of a complex.</span></li>
<li><span><a href="_co__Complex.html" title="Make a co-complex.">coComplex</a> -- Make a co-complex.</span></li>
<li><span><a href="_co__Complex__To__Ideal.html" title="The monomial ideal associated to a CoComplex.">coComplexToIdeal</a> -- The monomial ideal associated to a CoComplex.</span></li>
<li><span><a href="_coker__Element.html" title="Gives an element in the cokernel of a matrix.">cokerElement</a> -- Gives an element in the cokernel of a matrix.</span></li>
<li><span><a href="_complex.html" title="Make a complex.">complex</a> -- Make a complex.</span></li>
<li><span><a href="_complex__From__Facets.html" title="Make a complex from its facets.">complexFromFacets</a> -- Make a complex from its facets.</span></li>
<li><span><a href="_complex__To__Ideal.html" title="The monomial ideal associated to a complex.">complexToIdeal</a> -- The monomial ideal associated to a complex.</span></li>
<li><span><a href="_conv__Hull.html" title="The convex hull complex.">convHull</a> -- The convex hull complex.</span></li>
<li><span><a href="_coordinates.html" title="The coordinates of a face.">coordinates</a> -- The coordinates of a face.</span></li>
<li><span><a href="_deform.html" title="Compute the deformations associated to a Stanley-Reisner complex.">deform</a> -- Compute the deformations associated to a Stanley-Reisner complex.</span></li>
<li><span><a href="_deformations__Face.html" title="Compute the deformations associated to a face.">deformationsFace</a> -- Compute the deformations associated to a face.</span></li>
<li><span><a href="_denominator__Monomial.html" title="The denominator monomial of a deformation.">denominatorMonomial</a> -- The denominator monomial of a deformation.</span></li>
<li><span><a href="_dual__Grading.html" title="The dual vertices of a polytope.">dualGrading</a> -- The dual vertices of a polytope.</span></li>
<li><span><a href="_dualize.html" title="The dual of a face or complex.">dualize</a> -- The dual of a face or complex.</span></li>
<li><span><a href="_edim.html" title="The embedding dimension of a complex or co-complex.">edim</a> -- The embedding dimension of a complex or co-complex.</span></li>
<li><span><a href="_embedding__Complex.html" title="The embedding complex of a complex or co-complex.">embeddingComplex</a> -- The embedding complex of a complex or co-complex.</span></li>
<li><span><a href="_euler__Characteristic.html" title="The Euler characteristic of a complex.">eulerCharacteristic</a> -- The Euler characteristic of a complex.</span></li>
<li><span><a href="_face.html" title="Generate a face.">face</a> -- Generate a face.</span></li>
<li><span><a href="_face__To__Monomial.html" title="The monomial of a face.">faceToMonomial</a> -- The monomial of a face.</span></li>
<li><span><a href="_facets.html" title="The maximal faces of a complex.">facets</a> -- The maximal faces of a complex.</span></li>
<li><span><a href="_fc.html" title="The faces of a complex.">fc</a> -- The faces of a complex.</span></li>
<li><span><a href="_first__Order__Deformation.html" title="Makes a first order deformation.">firstOrderDeformation</a> -- Makes a first order deformation.</span></li>
<li><span><a href="_full__Cyclic__Polytope.html" title="Cyclic polytope.">fullCyclicPolytope</a> -- Cyclic polytope.</span></li>
<li><span><a href="_fvector.html" title="The F-vector of a complex.">fvector</a> -- The F-vector of a complex.</span></li>
<li><span><a href="_gens__Source.html" title="Generators of the source of a deformation.">gensSource</a> -- Generators of the source of a deformation.</span></li>
<li><span><a href="_global__Sections.html" title="The global sections of a toric divisor.">globalSections</a> -- The global sections of a toric divisor.</span></li>
<li><span><a href="_grading.html" title="The grading of a first order deformation or complex or polynomial ring.">grading</a> -- The grading of a first order deformation or complex or polynomial ring.</span></li>
<li><span><a href="_hull.html" title="The positive hull complex.">hull</a> -- The positive hull complex.</span></li>
<li><span><a href="_ideal__To__Co__Complex.html" title="The co-complex associated to a reduced monomial ideal.">idealToCoComplex</a> -- The co-complex associated to a reduced monomial ideal.</span></li>
<li><span><a href="_ideal__To__Complex.html" title="The complex associated to a reduced monomial ideal.">idealToComplex</a> -- The complex associated to a reduced monomial ideal.</span></li>
<li><span><a href="_intersect__Faces.html" title="The intersection of two faces.">intersectFaces</a> -- The intersection of two faces.</span></li>
<li><span><a href="_is__Equidimensional.html" title="Check whether a complex or co-complex is equidimensional.">isEquidimensional</a> -- Check whether a complex or co-complex is equidimensional.</span></li>
<li><span><a href="_is__Nonzero.html" title="Check whether a deformation is non-zero.">isNonzero</a> -- Check whether a deformation is non-zero.</span></li>
<li><span><a href="_is__Polytope.html" title="Check whether a complex is a polytope.">isPolytope</a> -- Check whether a complex is a polytope.</span></li>
<li><span><a href="_is__Simplicial.html" title="Check whether a complex or co-complex is simplicial.">isSimplicial</a> -- Check whether a complex or co-complex is simplicial.</span></li>
<li><span><a href="_is__Subface.html" title="Checks whether a face is a subface of another face.">isSubface</a> -- Checks whether a face is a subface of another face.</span></li>
<li><span><a href="_is__Trivial.html" title="Check whether a deformation is trivial.">isTrivial</a> -- Check whether a deformation is trivial.</span></li>
<li><span><a href="_iszero.html" title="Tests whether something is zero.">iszero</a> -- Tests whether something is zero.</span></li>
<li><span><a href="_join__Vectors.html" title="Converts a list of vectors to a matrix.">joinVectors</a> -- Converts a list of vectors to a matrix.</span></li>
<li><span><a href="_laurent.html" title="Converts an exponent vector or a deformation into a Laurent monomial.">laurent</a> -- Converts an exponent vector or a deformation into a Laurent monomial.</span></li>
<li><span><a href="_link.html" title="The link of a face of a complex.">link</a> -- The link of a face of a complex.</span></li>
<li><span><a href="_load__Deformations.html" title="Read the deformation data of a complex from a file.">loadDeformations</a> -- Read the deformation data of a complex from a file.</span></li>
<li><span><a href="_minimal__Non__Faces.html" title="The minimal non-faces of a complex.">minimalNonFaces</a> -- The minimal non-faces of a complex.</span></li>
<li><span><a href="_new__Empty__Complex.html" title="Generates an empty complex.">newEmptyComplex</a> -- Generates an empty complex.</span></li>
<li><span><a href="_numerator__Monomial.html" title="The numerator monomial of a deformation.">numeratorMonomial</a> -- The numerator monomial of a deformation.</span></li>
<li><span><a href="_of__Complex.html" title="The complex of a face.">ofComplex</a> -- The complex of a face.</span></li>
<li><span><a href="_parameters.html" title="Parameters of a deformation.">parameters</a> -- Parameters of a deformation.</span></li>
<li><span><a href="_polytopal__Facets.html" title="The facets of a polytope.">polytopalFacets</a> -- The facets of a polytope.</span></li>
<li><span><a href="_possible__Denominators.html" title="Possible denominators of deformations.">possibleDenominators</a> -- Possible denominators of deformations.</span></li>
<li><span><a href="_pre__Image.html" title="Compute the preimage.">preImage</a> -- Compute the preimage.</span></li>
<li><span><a href="___P__T1.html" title="Compute the deformation polytope associated to a Stanley-Reisner complex.">PT1</a> -- Compute the deformation polytope associated to a Stanley-Reisner complex.</span></li>
<li><span><a href="_rays__P__Pn.html" title="The rays of the standard fan of projective space.">raysPPn</a> -- The rays of the standard fan of projective space.</span></li>
<li><span><a href="_relations__Coefficients.html" title="Relations between the coefficients of a deformation.">relationsCoefficients</a> -- Relations between the coefficients of a deformation.</span></li>
<li><span><a href="_save__Deformations.html" title="Store the deformation data of a complex in a file.">saveDeformations</a> -- Store the deformation data of a complex in a file.</span></li>
<li><span><a href="_simplex.html" title="Simplex in the variables of a polynomial ring.">simplex</a> -- Simplex in the variables of a polynomial ring.</span></li>
<li><span><a href="_simplex__Dim.html" title="The dimension of a face inside the vertex simplex.">simplexDim</a> -- The dimension of a face inside the vertex simplex.</span></li>
<li><span><a href="_simplex__Ring.html" title="The underlying polynomial ring of a deformation or face or complex.">simplexRing</a> -- The underlying polynomial ring of a deformation or face or complex.</span></li>
<li><span><a href="_to__Hom.html" title="Convert a first order deformation into a homomorphism.">toHom</a> -- Convert a first order deformation into a homomorphism.</span></li>
<li><span><a href="_total__Space.html" title="Total space of a deformation.">totalSpace</a> -- Total space of a deformation.</span></li>
<li><span><a href="_trivial__Deformations.html" title="Compute the trivial deformations.">trivialDeformations</a> -- Compute the trivial deformations.</span></li>
<li><span><a href="_trop__Def.html" title="The co-complex of tropical faces of the deformation polytope.">tropDef</a> -- The co-complex of tropical faces of the deformation polytope.</span></li>
<li><span><a href="_variables.html" title="The variables of a complex or co-complex.">variables</a> -- The variables of a complex or co-complex.</span></li>
<li><span><a href="_vector__To__Monomial.html" title="Converts an exponent vector into a monomial.">vectorToMonomial</a> -- Converts an exponent vector into a monomial.</span></li>
<li><span><a href="_vert.html" title="The vertices of a face or complex.">vert</a> -- The vertices of a face or complex.</span></li>
<li><span><a href="_vertices__Dual__Polytope.html" title="The dual vertices of a polytope.">verticesDualPolytope</a> -- The dual vertices of a polytope.</span></li>
</ul>
</div>
</li>
<li><div class="single">Methods<ul><li><span>addFaceDataToComplex(Complex,List), see <span><a href="_add__Face__Data__To__Complex.html" title="Adds to a complex face data.">addFaceDataToComplex</a> -- Adds to a complex face data.</span></span></li>
<li><span>addFaceDataToComplex(Complex,List,List), see <span><a href="_add__Face__Data__To__Complex.html" title="Adds to a complex face data.">addFaceDataToComplex</a> -- Adds to a complex face data.</span></span></li>
<li><span>addFacetDataToComplex(Complex,List), see <span><a href="_add__Facet__Data__To__Complex.html" title="Adds to a complex facet data.">addFacetDataToComplex</a> -- Adds to a complex facet data.</span></span></li>
<li><span>closedStar(Face,Complex), see <span><a href="_closed__Star.html" title="The closed star of a face of a complex.">closedStar</a> -- The closed star of a face of a complex.</span></span></li>
<li><span><a href="___Complex_sp_eq_eq_sp__Complex.html" title="Compare two complexes.">Complex == Complex</a> -- Compare two complexes.</span></li>
<li><span>coordinates(Face,Complex), see <span><a href="_coordinates.html" title="The coordinates of a face.">coordinates</a> -- The coordinates of a face.</span></span></li>
<li><span>deformationsFace(Face,Complex), see <span><a href="_deformations__Face.html" title="Compute the deformations associated to a face.">deformationsFace</a> -- Compute the deformations associated to a face.</span></span></li>
<li><span>deformationsFace(Face,Complex,Ideal), see <span><a href="_deformations__Face.html" title="Compute the deformations associated to a face.">deformationsFace</a> -- Compute the deformations associated to a face.</span></span></li>
<li><span><a href="_dim_lp__Face_cm__Complex_rp.html" title="Compute the dimension of a face.">dim(Face,Complex)</a> -- Compute the dimension of a face.</span></li>
<li><span><a href="_dim_lp__Face_cm__Polynomial__Ring_rp.html" title="Compute the dimension of a face.">dim(Face,PolynomialRing)</a> -- Compute the dimension of a face.</span></li>
<li><span>face(List,Complex), see <span><a href="_face.html" title="Generate a face.">face</a> -- Generate a face.</span></span></li>
<li><span>face(List,Complex,ZZ,ZZ), see <span><a href="_face.html" title="Generate a face.">face</a> -- Generate a face.</span></span></li>
<li><span><a href="___Face_sp_eq_eq_sp__Face.html" title="Compare two faces.">Face == Face</a> -- Compare two faces.</span></li>
<li><span>faceToMonomial(Face,PolynomialRing), see <span><a href="_face__To__Monomial.html" title="The monomial of a face.">faceToMonomial</a> -- The monomial of a face.</span></span></li>
<li><span>fc(Complex,ZZ), see <span><a href="_fc.html" title="The faces of a complex.">fc</a> -- The faces of a complex.</span></span></li>
<li><span><a href="___First__Order__Deformation_sp_eq_eq_sp__First__Order__Deformation.html" title="Compare two first order deformations.">FirstOrderDeformation == FirstOrderDeformation</a> -- Compare two first order deformations.</span></li>
<li><span>idealToCoComplex(Ideal,Complex), see <span><a href="_ideal__To__Co__Complex.html" title="The co-complex associated to a reduced monomial ideal.">idealToCoComplex</a> -- The co-complex associated to a reduced monomial ideal.</span></span></li>
<li><span>idealToCoComplex(MonomialIdeal,Complex), see <span><a href="_ideal__To__Co__Complex.html" title="The co-complex associated to a reduced monomial ideal.">idealToCoComplex</a> -- The co-complex associated to a reduced monomial ideal.</span></span></li>
<li><span>idealToComplex(Ideal,Complex), see <span><a href="_ideal__To__Complex.html" title="The complex associated to a reduced monomial ideal.">idealToComplex</a> -- The complex associated to a reduced monomial ideal.</span></span></li>
<li><span>idealToComplex(MonomialIdeal,Complex), see <span><a href="_ideal__To__Complex.html" title="The complex associated to a reduced monomial ideal.">idealToComplex</a> -- The complex associated to a reduced monomial ideal.</span></span></li>
<li><span>intersectFaces(Face,Face), see <span><a href="_intersect__Faces.html" title="The intersection of two faces.">intersectFaces</a> -- The intersection of two faces.</span></span></li>
<li><span>isSubface(Face,Face), see <span><a href="_is__Subface.html" title="Checks whether a face is a subface of another face.">isSubface</a> -- Checks whether a face is a subface of another face.</span></span></li>
<li><span>link(Face,Complex), see <span><a href="_link.html" title="The link of a face of a complex.">link</a> -- The link of a face of a complex.</span></span></li>
<li><span>loadDeformations(Complex,String), see <span><a href="_load__Deformations.html" title="Read the deformation data of a complex from a file.">loadDeformations</a> -- Read the deformation data of a complex from a file.</span></span></li>
<li><span>saveDeformations(Complex,String), see <span><a href="_save__Deformations.html" title="Store the deformation data of a complex in a file.">saveDeformations</a> -- Store the deformation data of a complex in a file.</span></span></li>
<li><span>totalSpace(FirstOrderDeformation,PolynomialRing), see <span><a href="_total__Space.html" title="Total space of a deformation.">totalSpace</a> -- Total space of a deformation.</span></span></li>
<li><span>tropDef(Complex,Complex), see <span><a href="_trop__Def.html" title="The co-complex of tropical faces of the deformation polytope.">tropDef</a> -- The co-complex of tropical faces of the deformation polytope.</span></span></li>
</ul>
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<li><div class="single">Symbols<ul><li><span><a href="_complement__Complex.html" title="Optional data of Complex.">complementComplex</a> -- Optional data of Complex.</span></li>
<li><span><a href="_compute__Faces.html" title="Compute faces of a simplex.">computeFaces</a> -- Compute faces of a simplex.</span></li>
<li><span><a href="_dual__Complex.html" title="Optional data of Complex.">dualComplex</a> -- Optional data of Complex.</span></li>
<li><span><a href="_dual__Face.html" title="Optional data of Face.">dualFace</a> -- Optional data of Face.</span></li>
<li><span><a href="_embedded.html" title="Use only faces of the embedding complex.">embedded</a> -- Use only faces of the embedding complex.</span></li>
<li><span><a href="_file.html" title="Store result of a computation in a file.">file</a> -- Store result of a computation in a file.</span></li>
<li><span><a href="_mirror__Sphere.html" title="Example how to compute the mirror sphere.">mirrorSphere</a> -- Example how to compute the mirror sphere.</span></li>
<li><span><a href="_no__Boundary.html" title="Optional data of Complex.">noBoundary</a> -- Optional data of Complex.</span></li>
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</li>
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