flipCandidates -- candidate affine circuits for bistellar flips of a triangulation

Usage:

cs = flipCandidates T

cs = flipCandidates(A, tri)
Inputs:
- T, an instance of the type Triangulation,
- A, a matrix, whose columns are the points of the configuration, either of size $d \times n$ (homogenized, with last row all 1's) or $(d{-}1) \times n$ (which is auto-homogenized internally)
- tri, a list, a triangulation of the columns of $A$
Outputs:
- a list, of pairs $\{neg, pos\}$ of disjoint column-index lists, the negative and positive parts of an affine circuit supported on a $(d{+}1)$-set of columns relevant to flipping $T$

Description

Two kinds of $(d{+}1)$-sets are considered:

$\bullet$ codim-2 walls of $T$ -- $(d{+}1)$-sets formed as the union of two adjacent maximal simplices of $T$. These support the standard "fine" flips that preserve the vertex set.

$\bullet$ $\sigma \cup \{v\}$ for each maximal simplex $\sigma$ of $T$ and each column $v$ not used in $T$. These are not walls of $T$, but they are the supports of bistellar flips that insert $v$; in non-acyclic configurations (e.g., complete simplicial fans) such insertion flips can be the only non-fine neighbors of $T$.

For each $(d{+}1)$-set $c$, the function returns the signed kernel partition of those columns as a pair $\{neg, pos\}$ of column indices.

Each returned circuit is a candidate input for bistellarFlip, but not all candidates yield a valid flip in $T$; see bistellarFlip. In contrast, orientedCircuits returns all circuits of the underlying point configuration (most of which are not supported on walls of $T$), and flips -- a wrapper around topcomFlips -- returns only those circuits that are actually flippable in $T$.

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 : 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

Ways to use flipCandidates:

flipCandidates(Matrix,List)
flipCandidates(Triangulation)

For the programmer

The object flipCandidates is a method function.

The source of this document is in Triangulations.m2:2161:0.

flipCandidates -- candidate affine circuits for bistellar flips of a triangulation

Description

See also

Ways to use flipCandidates:

For the programmer