Macaulay2 » Documentation
Packages » Topcom :: topcomRegularTriangulationWeights(Matrix,List)
next | previous | forward | backward | up | index | toc

topcomRegularTriangulationWeights(Matrix,List) -- find a list of heights of a regular triangulation

Synopsis

Description

If the given triangulation is regular, this returns the corresponding list of weights. If we lift each point to this height (hieights can be negative too), then the lower convex hull of these lifted points is the given triangulation. This can be checked via the routine regularSubdivision in the package Polyhedra.

i1 : A = matrix {
         {0, -1, 2, 3, 4, -5, 6},
         {0, 1, -4, 9, 16, 25, 36},
         {0, 1, 8, -27, 64, 125, -216}}

o1 = | 0 -1 2  3   4  -5  6    |
     | 0 1  -4 9   16 25  36   |
     | 0 1  8  -27 64 125 -216 |

              3       7
o1 : Matrix ZZ  <-- ZZ
 
i2 : Ts = topcomAllTriangulations A;
 
i3 : #Ts == 25

o3 = true
 
i4 : netList Ts

     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
o4 = |{0, 1, 2, 3}|{0, 1, 3, 6}|{0, 2, 3, 6}|{1, 2, 3, 4}|{1, 2, 4, 5}|{1, 3, 4, 5}|{1, 3, 5, 6}|{2, 3, 4, 6}|{3, 4, 5, 6}|            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 3}|{0, 1, 3, 6}|{0, 2, 3, 6}|{1, 2, 3, 4}|{1, 2, 4, 5}|{1, 3, 4, 6}|{1, 4, 5, 6}|{2, 3, 4, 6}|            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 3}|{0, 1, 3, 6}|{0, 2, 3, 6}|{1, 2, 3, 5}|{1, 3, 5, 6}|{2, 3, 4, 5}|{2, 3, 4, 6}|{3, 4, 5, 6}|            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 4}|{0, 1, 3, 4}|{0, 1, 3, 6}|{0, 2, 3, 4}|{0, 2, 3, 6}|{1, 2, 4, 5}|{1, 3, 4, 5}|{1, 3, 5, 6}|{2, 3, 4, 6}|{3, 4, 5, 6}|            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{1, 2, 3, 4}|{1, 2, 3, 6}|{1, 2, 4, 5}|{1, 3, 4, 5}|{1, 3, 5, 6}|{2, 3, 4, 6}|{3, 4, 5, 6}|            |            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{1, 2, 3, 4}|{1, 2, 3, 6}|{1, 2, 4, 5}|{1, 3, 4, 6}|{1, 4, 5, 6}|{2, 3, 4, 6}|            |            |            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{1, 2, 3, 5}|{1, 2, 3, 6}|{1, 3, 5, 6}|{2, 3, 4, 5}|{2, 3, 4, 6}|{3, 4, 5, 6}|            |            |            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 5}|{0, 1, 3, 4}|{0, 1, 3, 6}|{0, 1, 4, 5}|{0, 2, 3, 4}|{0, 2, 3, 6}|{0, 2, 4, 5}|{1, 3, 4, 5}|{1, 3, 5, 6}|{2, 3, 4, 6}|{3, 4, 5, 6}|
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 4}|{0, 1, 3, 4}|{0, 1, 3, 6}|{0, 2, 3, 4}|{0, 2, 3, 6}|{1, 2, 4, 5}|{1, 3, 4, 6}|{1, 4, 5, 6}|{2, 3, 4, 6}|            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 5}|{0, 1, 3, 5}|{0, 1, 3, 6}|{0, 2, 3, 5}|{0, 2, 3, 6}|{1, 3, 5, 6}|{2, 3, 4, 5}|{2, 3, 4, 6}|{3, 4, 5, 6}|            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 5}|{0, 1, 3, 5}|{0, 1, 3, 6}|{0, 2, 3, 4}|{0, 2, 3, 6}|{0, 2, 4, 5}|{0, 3, 4, 5}|{1, 3, 5, 6}|{2, 3, 4, 6}|{3, 4, 5, 6}|            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 5}|{0, 1, 5, 6}|{0, 2, 3, 5}|{0, 2, 3, 6}|{0, 3, 5, 6}|{2, 3, 4, 5}|{2, 3, 4, 6}|{3, 4, 5, 6}|            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 4}|{0, 1, 4, 6}|{0, 2, 3, 4}|{0, 2, 3, 6}|{0, 3, 4, 6}|{1, 2, 4, 5}|{1, 4, 5, 6}|{2, 3, 4, 6}|            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 5}|{0, 1, 3, 4}|{0, 1, 3, 6}|{0, 1, 4, 5}|{0, 2, 3, 4}|{0, 2, 3, 6}|{0, 2, 4, 5}|{1, 3, 4, 6}|{1, 4, 5, 6}|{2, 3, 4, 6}|            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{1, 2, 5, 6}|{2, 3, 4, 5}|{2, 3, 4, 6}|{2, 3, 5, 6}|{3, 4, 5, 6}|            |            |            |            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{1, 2, 4, 5}|{1, 2, 4, 6}|{1, 4, 5, 6}|            |            |            |            |            |            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{1, 2, 5, 6}|{2, 4, 5, 6}|            |            |            |            |            |            |            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 5}|{0, 1, 5, 6}|{0, 2, 5, 6}|{2, 3, 4, 5}|{2, 3, 4, 6}|{2, 3, 5, 6}|{3, 4, 5, 6}|            |            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 5}|{0, 1, 4, 5}|{0, 1, 4, 6}|{0, 2, 3, 4}|{0, 2, 3, 6}|{0, 2, 4, 5}|{0, 3, 4, 6}|{1, 4, 5, 6}|{2, 3, 4, 6}|            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 4}|{0, 1, 4, 6}|{0, 2, 4, 6}|{1, 2, 4, 5}|{1, 4, 5, 6}|            |            |            |            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 5}|{0, 1, 5, 6}|{0, 2, 3, 4}|{0, 2, 3, 6}|{0, 2, 4, 5}|{0, 3, 4, 5}|{0, 3, 5, 6}|{2, 3, 4, 6}|{3, 4, 5, 6}|            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 5}|{0, 1, 5, 6}|{0, 2, 3, 4}|{0, 2, 3, 6}|{0, 2, 4, 5}|{0, 3, 4, 6}|{0, 4, 5, 6}|{2, 3, 4, 6}|            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 5}|{0, 1, 4, 5}|{0, 1, 4, 6}|{0, 2, 4, 5}|{0, 2, 4, 6}|{1, 4, 5, 6}|            |            |            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 5}|{0, 1, 5, 6}|{0, 2, 5, 6}|{2, 4, 5, 6}|            |            |            |            |            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
     |{0, 1, 2, 5}|{0, 1, 5, 6}|{0, 2, 4, 5}|{0, 2, 4, 6}|{0, 4, 5, 6}|            |            |            |            |            |            |
     +------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+------------+
 
i5 : Ts/(tri -> topcomIsRegularTriangulation(A, tri))

o5 = {true, true, true, true, true, true, true, true, true, true, true, true,
     ------------------------------------------------------------------------
     true, true, true, true, true, true, true, true, true, true, true, true,
     ------------------------------------------------------------------------
     true}

o5 : List
 
i6 : weights = topcomRegularTriangulationWeights(A, Ts#0)

       24235     77731    17659
o6 = {-------, --------, -------, 0, 0, 0, 0}
      2934162  10059984  1676664

o6 : List
 
i7 : needsPackage "Polyhedra"

o7 = Polyhedra

o7 : Package
 
i8 : Ts#0

o8 = {{0, 1, 2, 3}, {0, 1, 3, 6}, {0, 2, 3, 6}, {1, 2, 3, 4}, {1, 2, 4, 5},
     ------------------------------------------------------------------------
     {1, 3, 4, 5}, {1, 3, 5, 6}, {2, 3, 4, 6}, {3, 4, 5, 6}}

o8 : List
 
i9 : tri0 = regularSubdivision(A, matrix{weights})

o9 = {{0, 1, 2, 3}, {0, 1, 3, 6}, {0, 2, 3, 6}, {1, 2, 3, 4}, {1, 2, 4, 5},
     ------------------------------------------------------------------------
     {1, 3, 4, 5}, {1, 3, 5, 6}, {2, 3, 4, 6}, {3, 4, 5, 6}}

o9 : List
 
i10 : tri0 = tri0//sort/sort

o10 = {{0, 1, 2, 3}, {0, 1, 3, 6}, {0, 2, 3, 6}, {1, 2, 3, 4}, {1, 2, 4, 5},
      -----------------------------------------------------------------------
      {1, 3, 4, 5}, {1, 3, 5, 6}, {2, 3, 4, 6}, {3, 4, 5, 6}}

o10 : List
 
i11 : assert(tri0 == Ts#0)

See also

Ways to use this method: