bistellarFlip -- perform a bistellar flip on a triangulation

Usage:

T' = bistellarFlip(T, c)

tri' = bistellarFlip(tri, c)
Inputs:
- T, an instance of the type Triangulation,
- tri, a list, a triangulation, given as a list of lists of column indices
- c, a list, a pair $\{neg, pos\}$ of disjoint lists of column indices: the signed parts of an affine circuit, typically an entry of flipCandidates
Outputs:
- a thing, a Triangulation (or, in the (List, List) form, a List) obtained from $T$ by flipping along $c$, or null if the circuit is not flippable in $T$

Description

A circuit on $d{+}1$ points admits two triangulations of its convex hull: $T_-$, omitting each vertex listed in neg, and $T_+$, omitting each vertex listed in pos. Exactly one of these (say $T_-$) sits inside $T$, supported on a common link $L$; the bistellar flip replaces $L \star T_-$ with $L \star T_+$.

If neither side has a common link in $T$ -- for example, the simplices of one side appear with mismatched links, or one side is only partially present -- the flip is undefined and this returns null.

i1 : A = transpose matrix {{0,3},{0,1},{-1,-1},{1,-1},{-4,-2},{4,-2}}

o1 = | 0 0 -1 1  -4 4  |
     | 3 1 -1 -1 -2 -2 |

              2       6
o1 : Matrix ZZ  <-- ZZ

i2 : T = regularFineTriangulation A

o2 = triangulation {{0, 1, 2}, {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3}, {2, 3, 4}, {3, 4, 5}}

o2 : Triangulation

i3 : cs = flipCandidates T

o3 = {{{1}, {0, 2, 3}}, {{1, 4}, {0, 2}}, {{1, 5}, {0, 3}}, {{2}, {0, 3, 4}},
     ------------------------------------------------------------------------
     {{2}, {1, 3, 4}}, {{3}, {0, 4, 5}}, {{3, 4}, {2, 5}}}

o3 : List

i4 : T' = bistellarFlip(T, cs#0)

o4 = triangulation {{0, 2, 3}, {0, 2, 4}, {0, 3, 5}, {2, 3, 4}, {3, 4, 5}}

o4 : Triangulation

i5 : max T'

o5 = {{0, 2, 3}, {0, 2, 4}, {0, 3, 5}, {2, 3, 4}, {3, 4, 5}}

o5 : List

i6 : isWellDefined T'

o6 = true

Many candidate circuits returned by flipCandidates are not flippable in any given $T$; those return null.

i7 : for c in cs list bistellarFlip(T, c)

o7 = {triangulation {{0, 2, 3}, {0, 2, 4}, {0, 3, 5}, {2, 3, 4}, {3, 4, 5}},
     ------------------------------------------------------------------------
     triangulation {{0, 1, 3}, {0, 1, 4}, {0, 3, 5}, {1, 2, 3}, {1, 2, 4},
     ------------------------------------------------------------------------
     {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, 4},
     ------------------------------------------------------------------------
     {1, 2, 3}, {1, 3, 5}, {2, 3, 4}, {3, 4, 5}}, , , , triangulation {{0, 1,
     ------------------------------------------------------------------------
     2}, {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3}, {2, 3, 5}, {2, 4, 5}}}

o7 : List

Ways to use bistellarFlip:

bistellarFlip(List,List)
bistellarFlip(Triangulation,List)

For the programmer

The object bistellarFlip is a method function.

The source of this document is in Triangulations.m2:2103:0.

bistellarFlip -- perform a bistellar flip on a triangulation

Description

See also

Ways to use bistellarFlip:

For the programmer