Generation of rank-3 Laurent mutation invariants

Prove that the Laurent mutation invariants of every rank-3 cluster algebra with an irreducible sign-equivalent exchange matrix are generated by the Markov mutation invariant and its variant defined in Equations (5.3) and (5.8), respectively.

Background

The paper studies two rank-3 cluster algebras with irreducible sign-equivalent exchange matrices: the once-punctured-torus cluster algebra A_P and Lampe’s cluster algebra A_L. Their principal Laurent mutation invariants are the Markov mutation invariant in Equation (5.3) and the variant of the Markov mutation invariant in Equation (5.8).

The authors establish Diophantine consequences for these two invariants, including classifications of positive integer points and mutation-generation results for associated equations. They leave unresolved whether these invariants generate all Laurent mutation invariants for the full class of rank-3 cluster algebras with irreducible sign-equivalent exchange matrices.