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Chirotope -- the chirotope (oriented matroid) of a point or vector configuration

Description

A Chirotope packages the chirotope of a configuration: for each $d$-subset $S$ of the columns, the sign of $\det A_S$, stored as a topcom-format string. Two configurations have the same chirotope iff they have the same oriented matroid -- in particular, the same set of triangulations.

Construct one with chirotope from a matrix (via topcom) or with naiveChirotope (a slower pure-Macaulay2 alternative, useful for verification). The String form of chirotope wraps a precomputed topcom string.

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 : C = chirotope A

o2 = Chirotope{cache => CacheTable{}         }
               String => 6,3:
                         -+-++-+-+++-+-++--++


o2 : Chirotope
 
i3 : C == naiveChirotope A

o3 = true

See also

Functions and methods returning an object of class Chirotope:

  • chirotope(Matrix) -- see chirotope -- compute the chirotope of a point or vector configuration
  • chirotope(String) -- see chirotope -- compute the chirotope of a point or vector configuration
  • naiveChirotope(Matrix) -- see naiveChirotope -- compute the chirotope of a configuration via determinants in Macaulay2

Methods that use an object of class Chirotope:

  • Chirotope == Chirotope (missing documentation)
  • toString(Chirotope) (missing documentation)

For the programmer

The object Chirotope is a type, with ancestor classes HashTable < Thing.

The source of this document is in Triangulations.m2:2768:0.