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orientedCircuits -- compute the oriented circuits of an oriented matroid or point or vector configuration

Synopsis

Description

An oriented circuit is determined by a linear relationship on the columns of (the augmented matrix of) $A$, of minimal support. The circuit is the pair of lists of indices of the columns where the coefficients is positive (respectively negative).

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 : om = naiveChirotopeString A

o2 = 7,4:
     ---0++-+----+-++++++--+-+++++++-+++
 
i3 : netList orientedCircuits om

     +------------+---------+
o3 = |{0}         |{1, 2, 6}|
     +------------+---------+
     |{0, 2, 4, 6}|{3}      |
     +------------+---------+
     |{0, 3}      |{2, 5, 6}|
     +------------+---------+
     |{0, 3, 5}   |{1, 4}   |
     +------------+---------+
     |{0, 4}      |{1, 2, 3}|
     +------------+---------+
     |{0, 4}      |{2, 3, 5}|
     +------------+---------+
     |{0, 4}      |{2, 5, 6}|
     +------------+---------+
     |{0, 4, 6}   |{1, 3}   |
     +------------+---------+
     |{0, 4, 6}   |{3, 5}   |
     +------------+---------+
     |{0, 5}      |{1, 2, 3}|
     +------------+---------+
     |{0, 5}      |{1, 2, 4}|
     +------------+---------+
     |{0, 5, 6}   |{1, 3}   |
     +------------+---------+
     |{0, 5, 6}   |{1, 4}   |
     +------------+---------+
     |{1, 2, 4, 6}|{3}      |
     +------------+---------+
     |{1, 3}      |{2, 5, 6}|
     +------------+---------+
     |{1, 4}      |{2, 3, 5}|
     +------------+---------+
     |{1, 4}      |{2, 5, 6}|
     +------------+---------+
     |{1, 4, 6}   |{3, 5}   |
     +------------+---------+
     |{2, 4, 5, 6}|{3}      |
     +------------+---------+

Let's look at the linear relation giving rise to $\{\{0,3\}, \{ 2, 5, 6\}\}$.

i4 : Ahomog = A || matrix{{7:1}}

o4 = | 0 -1 2  3   4  -5  6    |
     | 0 1  -4 9   16 25  36   |
     | 0 1  8  -27 64 125 -216 |
     | 1 1  1  1   1  1   1    |

              4       7
o4 : Matrix ZZ  <-- ZZ
 
i5 : Ahomog_{0,3,2,5,6}

o5 = | 0 3   2  -5  6    |
     | 0 9   -4 25  36   |
     | 0 -27 8  125 -216 |
     | 1 1   1  1   1    |

              4       5
o5 : Matrix ZZ  <-- ZZ
 
i6 : syz oo

o6 = | -242 |
     | -440 |
     | 495  |
     | 72   |
     | 115  |

              5       1
o6 : Matrix ZZ  <-- ZZ

See also

Ways to use orientedCircuits:

For the programmer

The object orientedCircuits is a method function with options.