Geometry of tropical mutation surfaces with a single mutation

Recently, Escobar, Harada, and Manon introduced the theory of polyptych lattices. This theory gives a general framework for constructing projective varieties from polytopes in a polyptych lattice. When all the mutations of the polyptych lattice are linear isomorphisms, this framework recovers the classical theory of toric varieties. In this article, we study rank two polyptych lattices with a single mutation. We prove that the associated projective surface $X$ is a $\mathbb{G}_m$-surface that admits an equivariant $1$-complement $B\in |-K_X|$ such that $B$ supports an effective ample divisor. Conversely, we show that a $\mathbb{G}_m$-surface $X$ that admits an equivariant $1$-complement $B\in |-K_X|$ supporting an effective ample divisor comes from a polyptych lattice polytope. Finally, we compute the complexity of the pair $(X,B)$ in terms of the data of the polyptych lattice, we describe the Cox ring of $X$, and study its toric degenerations.