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<head><title>Complex -- The class of all embedded complexes.</title>
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<div><h1>Complex -- The class of all embedded complexes.</h1>
<div class="single"><h2>Description</h2>
<div><p>The class of all embedded complexes, not necessarily simplicial or compact or equidimensional. These are complexes with coordinates assigned to their vertices.</p>
<p><b>Creating complexes:</b></p>
<p>The following functions return complexes:</p>
<p><a href="_simplex.html" title="Simplex in the variables of a polynomial ring.">simplex</a> -- Simplex in the variables of a polynomial ring</p>
<p><a href="_boundary__Cyclic__Polytope.html" title="The boundary complex of a cyclic polytope.">boundaryCyclicPolytope</a> -- The boundary complex of a cyclic polytope with standard projective space vertices</p>
<p><a href="_full__Cyclic__Polytope.html" title="Cyclic polytope.">fullCyclicPolytope</a> -- The full cyclic polytope with moment curve vertices</p>
<p><a href="_conv__Hull.html" title="The convex hull complex.">convHull</a> -- The convex hull</p>
<p><a href="_hull.html" title="The positive hull complex.">hull</a> -- The positive hull</p>
<p><a href="_boundary__Of__Polytope.html" title="The boundary of a polytope.">boundaryOfPolytope</a> -- The boundary of a polytope</p>
<p><a href="_new__Empty__Complex.html" title="Generates an empty complex.">newEmptyComplex</a> -- Generates an empty complex.</p>
<p><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</p>
<p><a href="_dualize.html" title="The dual of a face or complex.">dualize</a> -- The dual of a co-complex.</p>
<p><a href="../../Macaulay2Doc/html/_complement_lp__Matrix_rp.html" title="find the minimal generators for cokernel of a matrix (low level form)">complement</a> -- The complement of a co-complex.</p>
<p><a href="_complex.html" title="Make a complex.">complex</a> -- Make a complex from a list of faces</p>
<p><a href="_complex__From__Facets.html" title="Make a complex from its facets.">complexFromFacets</a> -- Make a complex from a list of facets</p>
<p><a href="_embedding__Complex.html" title="The embedding complex of a complex or co-complex.">embeddingComplex</a> -- The complex containing a subcomplex</p>
<p>For examples see the documentation of these functions.</p>
<p></p>
<p><b>The data stored in a complex C:</b></p>
<p><i>C.simplexRing</i>, the polynomial ring of vertices of C.</p>
<p><i>C.grading</i>, is C.simplexRing.grading, a matrix with the coordinates of the vertices of C in its rows.</p>
<p><i>C.facets</i>, a list with the facets of C sorted into lists by dimension.</p>
<p><i>C.edim</i>, the embedding dimension of C, i.e., <a href="../../Macaulay2Doc/html/_rank.html" title="compute the rank">rank</a> <a href="../../Macaulay2Doc/html/_source.html" title="source of a map">source</a> C.grading.</p>
<p><i>C.dim</i>, the dimension of the complex.</p>
<p><i>C.isSimplicial</i>, a <a href="../../Macaulay2Doc/html/___Boolean.html" title="the class of Boolean values">Boolean</a> indicating whether C is simplicial.</p>
<p><i>C.isEquidimensional</i>, a <a href="../../Macaulay2Doc/html/___Boolean.html" title="the class of Boolean values">Boolean</a> indicating whether C is equidimensional.</p>
<p>If not just the facets but the faces of C a known (e.g., after computed with <a href="_fc.html" title="The faces of a complex.">fc</a>) then the following data is present:</p>
<p><i>C.fc</i>, a <a href="../../Macaulay2Doc/html/___Scripted__Functor.html" title="the class of all scripted functors">ScriptedFunctor</a> with the faces of C sorted and indexed by dimension.</p>
<p><i>C.fvector</i>, a <a href="../../Macaulay2Doc/html/___List.html" title="the class of all lists -- {...}">List</a> with the F-vector of C.</p>
<p>The following may be present (if known due to creation of C or due to calling some function):</p>
<p><i>C.dualComplex</i>, the dual co-complex of C in the sense of dual faces of a polytope. See <a href="_dualize.html" title="The dual of a face or complex.">dualize</a>.</p>
<p><i>C.isPolytope</i>, a <a href="../../Macaulay2Doc/html/___Boolean.html" title="the class of Boolean values">Boolean</a> indicating whether C is a polytope.</p>
<p><i>C.polytopalFacets</i>, a <a href="../../Macaulay2Doc/html/___List.html" title="the class of all lists -- {...}">List</a> with the boundary faces of the polytope C.</p>
<p><i>C.complementComplex</i>, the complement co-complex of C (if C is a subcomplex of a simplex). See <a href="../../Macaulay2Doc/html/_complement_lp__Matrix_rp.html" title="find the minimal generators for cokernel of a matrix (low level form)">complement</a>.</p>
<div/>
<table class="examples"><tr><td><pre>i1 : R=QQ[x_0..x_5]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : C=boundaryCyclicPolytope(3,R)
o2 = 2: x x x x x x x x x x x x x x x x x x x x x x x x
0 1 2 0 2 3 0 3 4 0 1 5 1 2 5 2 3 5 0 4 5 3 4 5
o2 : complex of dim 2 embedded in dim 5 (printing facets)
equidimensional, simplicial, F-vector {1, 6, 12, 8, 0, 0, 0}, Euler = 1</pre>
</td></tr>
<tr><td><pre>i3 : C.simplexRing
o3 = R
o3 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i4 : C.grading
o4 = | -1 -1 -1 -1 -1 |
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 1 0 0 |
| 0 0 0 1 0 |
| 0 0 0 0 1 |
6 5
o4 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i5 : C.fc_2
o5 = {x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x }
0 1 2 0 2 3 0 3 4 0 1 5 1 2 5 2 3 5 0 4 5 3 4 5
o5 : List</pre>
</td></tr>
<tr><td><pre>i6 : C.facets
o6 = {{}, {}, {}, {x x x , x x x , x x x , x x x , x x x , x x x , x x x ,
0 1 2 0 2 3 0 3 4 0 1 5 1 2 5 2 3 5 0 4 5
------------------------------------------------------------------------
x x x }, {}, {}, {}}
3 4 5
o6 : List</pre>
</td></tr>
<tr><td><pre>i7 : dualize C
o7 = 2: v v v v v v v v v v v v v v v v v v v v v v v v
0 1 2 0 1 4 0 3 4 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5
o7 : co-complex of dim 2 embedded in dim 5 (printing facets)
equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1</pre>
</td></tr>
<tr><td><pre>i8 : complement C
o8 = 2: x x x x x x x x x x x x x x x x x x x x x x x x
3 4 5 1 4 5 2 1 5 2 3 4 0 3 4 1 0 4 2 1 3 2 1 0
o8 : co-complex of dim 2 embedded in dim 5 (printing facets)
equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1</pre>
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<p></p>
<div/>
<table class="examples"><tr><td><pre>i9 : R=QQ[x_0..x_5]
o9 = R
o9 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i10 : C=simplex R
o10 = 5: x x x x x x
0 1 2 3 4 5
o10 : complex of dim 5 embedded in dim 5 (printing facets)
equidimensional, simplicial, F-vector {1, 6, 15, 20, 15, 6, 1}, Euler = 0</pre>
</td></tr>
<tr><td><pre>i11 : C.isPolytope
o11 = true</pre>
</td></tr>
<tr><td><pre>i12 : C.polytopalFacets
o12 = {x x x x x , x x x x x , x x x x x , x x x x x , x x x x x ,
0 1 2 3 4 0 1 2 3 5 0 1 2 4 5 0 1 3 4 5 0 2 3 4 5
-----------------------------------------------------------------------
x x x x x }
1 2 3 4 5
o12 : List</pre>
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<div class="single"><h2>See also</h2>
<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="___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="___H__H_sp__Complex.html" title="Compute the homology of a complex.">HH Complex</a> -- Compute the homology of a complex.</span></li>
</ul>
</div>
<div class="waystouse"><h2>Types of embedded complex :</h2>
<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>
</ul>
<h2>Methods that use an embedded complex :</h2>
<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>boundaryOfPolytope(Complex), see <span><a href="_boundary__Of__Polytope.html" title="The boundary of a polytope.">boundaryOfPolytope</a> -- The boundary of a polytope.</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="_complement_lp__Complex_rp.html" title="Compute the complement CoComplex.">complement(Complex)</a> -- Compute the complement CoComplex.</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>complexToIdeal(Complex), see <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></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>deform(Complex), see <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></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__Complex_rp.html" title="Compute the dimension of a complex or co-complex.">dim(Complex)</a> -- Compute the dimension of a complex or co-complex.</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>dualGrading(Complex), see <span><a href="_dual__Grading.html" title="The dual vertices of a polytope.">dualGrading</a> -- The dual vertices of a polytope.</span></span></li>
<li><span>dualize(Complex), see <span><a href="_dualize.html" title="The dual of a face or complex.">dualize</a> -- The dual of a face or complex.</span></span></li>
<li><span>edim(Complex), see <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></span></li>
<li><span>embeddingComplex(Complex), see <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></span></li>
<li><span>eulerCharacteristic(Complex), see <span><a href="_euler__Characteristic.html" title="The Euler characteristic of a complex.">eulerCharacteristic</a> -- The Euler characteristic of a complex.</span></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>facets(Complex), see <span><a href="_facets.html" title="The maximal faces of a complex.">facets</a> -- The maximal faces of a complex.</span></span></li>
<li><span>fc(Complex), see <span><a href="_fc.html" title="The faces of a complex.">fc</a> -- The faces of a complex.</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>fvector(Complex), see <span><a href="_fvector.html" title="The F-vector of a complex.">fvector</a> -- The F-vector of a complex.</span></span></li>
<li><span><a href="_grading_lp__Complex_rp.html" title="The grading of a complex.">grading(Complex)</a> -- The grading of a complex.</span></li>
<li><span><a href="___H__H_sp__Complex.html" title="Compute the homology of a complex.">HH Complex</a> -- Compute the homology of a complex.</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>isEquidimensional(Complex), see <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></span></li>
<li><span>isPolytope(Complex), see <span><a href="_is__Polytope.html" title="Check whether a complex is a polytope.">isPolytope</a> -- Check whether a complex is a polytope.</span></span></li>
<li><span>isSimplicial(Complex), see <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></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>minimalNonFaces(Complex), see <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></span></li>
<li><span><a href="_net_lp__Complex_rp.html" title="Printing complexes.">net(Complex)</a> -- Printing complexes.</span></li>
<li><span>polytopalFacets(Complex), see <span><a href="_polytopal__Facets.html" title="The facets of a polytope.">polytopalFacets</a> -- The facets of a polytope.</span></span></li>
<li><span>PT1(Complex), see <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></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><a href="_simplex__Ring_lp__Complex_rp.html" title="The underlying polynomial ring of a complex.">simplexRing(Complex)</a> -- The underlying polynomial ring of a complex.</span></li>
<li><span>trivialDeformations(Complex), see <span><a href="_trivial__Deformations.html" title="Compute the trivial deformations.">trivialDeformations</a> -- Compute the trivial deformations.</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>
<li><span>variables(Complex), see <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></span></li>
<li><span>vert(Complex), see <span><a href="_vert.html" title="The vertices of a face or complex.">vert</a> -- The vertices of a face or complex.</span></span></li>
<li><span>verticesDualPolytope(Complex), see <span><a href="_vertices__Dual__Polytope.html" title="The dual vertices of a polytope.">verticesDualPolytope</a> -- The dual vertices of a polytope.</span></span></li>
</ul>
</div>
<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Complex.html" title="The class of all embedded complexes.">Complex</a> is <span>a <a href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a href="../../Macaulay2Doc/html/___Mutable__Hash__Table.html" title="the class of all mutable hash tables">MutableHashTable</a> < <a href="../../Macaulay2Doc/html/___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="../../Macaulay2Doc/html/___Thing.html" title="the class of all things">Thing</a>.</p>
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