The exits are North South and East but are blocked by an infinite number of mutants

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:

  1. 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.
  2. 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.
  3. 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 x,y,z. The finiteness of its equivalents has to do with the Markov constant C(x,y,z)=x^2+y^2+z^2-xyz (not 3xyz 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.

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