For example, the oriented matroid CUBE has $40$ signed circuits. Its contraction to the vertex $8$ has $34$ signed circuits.
What is a smart way to list out all the circuits without missing out or double-counting any?
2026-02-22 19:36:41.1771789001
How to list all circuits of an oriented matroid and all circuits of its contraction of a vertex by hand?
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Taking by hand a little loosely, one could use a computer.
The software TOPCOM can compute the signed circuits of a point configuration. Both Sage and Macaulay2 have packages for working with matroids (see here and here, respectively). Either could compute all of the circuits of the underlying unoriented matroid, leaving you the work of signing them.
Computing the circuits without a computer necessitates good bookkeeping. Writing down the circuits lexicographically can be helpful when checking that nothing was missed.