<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- --> <!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html --> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head><title>link -- The link of a face of a complex.</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_load__Deformations.html">next</a> | <a href="_laurent.html">previous</a> | <a href="_load__Deformations.html">forward</a> | <a href="_laurent.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div> </td> </tr> </table> <hr/> <div><h1>link -- The link of a face of a complex.</h1> <div class="single"><h2>Synopsis</h2> <ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>link(F)</tt></div> </dd></dl> </div> </li> <li><div class="single">Inputs:<ul><li><span><tt>F</tt>, <span>a <a href="___Face.html">face</a></span></span></li> <li><span><tt>C</tt>, <span>an <a href="___Complex.html">embedded complex</a></span></span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>an <a href="___Complex.html">embedded complex</a></span></span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div><p>The link of the face F of the complex C.</p> <div/> <table class="examples"><tr><td><pre>i1 : R=QQ[x_0..x_4] o1 = R o1 : PolynomialRing</pre> </td></tr> <tr><td><pre>i2 : C=boundaryOfPolytope simplex(R) o2 = 3: x x x x x x x x x x x x x x x x x x x x 0 1 2 3 0 1 2 4 0 1 3 4 0 2 3 4 1 2 3 4 o2 : complex of dim 3 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 0}, Euler = -1</pre> </td></tr> <tr><td><pre>i3 : F=C.fc_0_0 o3 = x 0 o3 : face with 1 vertex</pre> </td></tr> <tr><td><pre>i4 : link(F,C) o4 = 2: x x x x x x x x x x x x 1 2 3 1 2 4 1 3 4 2 3 4 o4 : complex of dim 2 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 4, 6, 4, 0, 0}, Euler = 1</pre> </td></tr> <tr><td><pre>i5 : closedStar(F,C) o5 = 3: x x x x x x x x x x x x x x x x 0 1 2 3 0 1 2 4 0 1 3 4 0 2 3 4 o5 : complex of dim 3 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 5, 10, 10, 4, 0}, Euler = 0</pre> </td></tr> <tr><td><pre>i6 : F=C.fc_1_0 o6 = x x 0 1 o6 : face with 2 vertices</pre> </td></tr> <tr><td><pre>i7 : link(F,C) o7 = 1: x x x x x x 2 3 2 4 3 4 o7 : complex of dim 1 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 3, 3, 0, 0, 0}, Euler = -1</pre> </td></tr> <tr><td><pre>i8 : closedStar(F,C) o8 = 3: x x x x x x x x x x x x 0 1 2 3 0 1 2 4 0 1 3 4 o8 : complex of dim 3 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 5, 10, 9, 3, 0}, Euler = 0</pre> </td></tr> </table> <p></p> <div/> <table class="examples"><tr><td><pre>i9 : R=QQ[x_0..x_4] o9 = R o9 : PolynomialRing</pre> </td></tr> <tr><td><pre>i10 : I=ideal(x_0*x_1,x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_0) o10 = ideal (x x , x x , x x , x x , x x ) 0 1 1 2 2 3 3 4 0 4 o10 : Ideal of R</pre> </td></tr> <tr><td><pre>i11 : C=idealToComplex I o11 = 1: x x x x x x x x x x 0 2 0 3 1 3 1 4 2 4 o11 : complex of dim 1 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 5, 5, 0, 0, 0}, Euler = -1</pre> </td></tr> <tr><td><pre>i12 : F=C.fc_0_0 o12 = x 0 o12 : face with 1 vertex</pre> </td></tr> <tr><td><pre>i13 : link(F,C) o13 = 0: x x 2 3 o13 : complex of dim 0 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 2, 0, 0, 0, 0}, Euler = 1</pre> </td></tr> <tr><td><pre>i14 : closedStar(F,C) o14 = 1: x x x x 0 2 0 3 o14 : complex of dim 1 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 3, 2, 0, 0, 0}, Euler = 0</pre> </td></tr> <tr><td><pre>i15 : F=C.fc_1_0 o15 = x x 0 2 o15 : face with 2 vertices</pre> </td></tr> <tr><td><pre>i16 : link(F,C) o16 = -1: {} o16 : complex of dim -1 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 0, 0, 0, 0, 0}, Euler = -1</pre> </td></tr> <tr><td><pre>i17 : closedStar(F,C) o17 = 1: x x 0 2 o17 : complex of dim 1 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 2, 1, 0, 0, 0}, Euler = 0</pre> </td></tr> </table> </div> </div> <div class="single"><h2>See also</h2> <ul><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> </ul> </div> <div class="waystouse"><h2>Ways to use <tt>link</tt> :</h2> <ul><li>link(Face,Complex)</li> </ul> </div> </div> </body> </html>