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makeFine -- walk to a fine triangulation by support-increasing bistellar flips

Description

At each step, this function inspects neighbors$(T, Fine\, =>\, false)$ and follows the first neighbor whose support (set of column indices used by its maximal simplices) is a strict superset of $T$'s support. It repeats until $T$ is fine, or errors when no support-increasing neighbor is available at the current $T$.

The greedy choice is cheap to compute and usually reaches a fine triangulation quickly, but it is not exhaustive: it never takes a flip that keeps the support the same size, so it can stall at a non-fine $T$ even when a fine triangulation is reachable via a longer sequence of flips that mixes support-increasing and equal-support steps. Failure of this function therefore means "the support-monotone walk got stuck," not necessarily "no fine triangulation exists."

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

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

              4       6
o1 : Matrix ZZ  <-- ZZ
 
i2 : t0 = triangulation(A, {{0,1,2,3},{0,1,2,4},{0,1,3,4},{0,2,3,4},{1,2,3,4}})

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

o2 : Triangulation
 
i3 : isFine t0

o3 = false
 
i4 : T = makeFine t0

o4 = triangulation {{0, 1, 2, 4}, {0, 1, 2, 5}, {0, 1, 4, 5}, {0, 2, 4, 5}, {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 3, 4, 5}, {2, 3, 4, 5}}

o4 : Triangulation
 
i5 : isFine T

o5 = true

See also

Ways to use makeFine:

  • makeFine(Triangulation)

For the programmer

The object makeFine is a method function.

The source of this document is in Triangulations.m2:1387:0.