I used the word mutation in the previous post because of the (implicit) connection to quiver mutation. The quiver mutation is easy to define: take an oriented graph (quiver) where multiple edges are allowed, but must have consistent orientation (i.e., no 2-edge oriented cycles are allowed). Mutation at vertex v is done in three steps:
- v is removed and each oriented path of length two through v is contracted into an edge. That is, the stopover at v is eliminated.
- Step 1 may create some 2-edge oriented cycles, which must be deleted. That is, we cancel the pairs of arrows going in opposite directions.
- The replacement vertex v’ is inserted, connected to the rest of the graph in the same way that v was, but with opposite orientation. In practice, one simply reuses v for this purpose.
Some quivers have a finite set of mutation equivalent ones; others an infinite one. Perhaps the simplest nontrivial case is the oriented 3-cycle with edges of multiplicities . The finiteness of its equivalents has to do with the Markov constant (not this time), which is invariant under mutation. This is investigated in the paper “Cluster-Cyclic Quivers with Three Vertices and the Markov Equation” by Beineke, Brűstle and Hille. The appendix by Kerner relates the Markov constant to Hochschild cohomology, which I take as a clue for me to finish this post.
So I’ll leave you to play with the mutation applet linked in the embedded tweet below.