Mutation of Fano Simplices and Markov type equations
It is well known that there is a bijective correspondence between the set of positive integer solutions to the Markov equation and the set of Fano triangles mutation equivalent to the Fano triangle of $\mathbb{P}^2$. In this paper, we establish a higher dimensional generalization of this correspondence for arbitrary Fano simplices of any dimension. On the polyhedral side, we introduce a distinguished class of facets, called admissible facets, and show that their number is preserved under facet mutation. As a consequence, facet mutation classes of Fano simplices carry natural exchange graph structures whose valency is equal to the number of admissible facets. On the arithmetic side, we associate to each Fano simplex a weighted Markov-type equation together with a distinguished positive integer solution, and show that the corresponding arithmetic mutations, given by Vieta involutions, are compatible with facet mutations. More precisely, the assignment from Fano simplices to Diophantine data intertwines combinatorial mutations with arithmetic mutations, thereby relating the mutation dynamics of Fano simplices to the arithmetic dynamics of positive integer solutions. Finally, we introduce a piecewise linear transformation on dual polytopes, called a sliding operator, which realizes combinatorial mutation in the dual picture. As applications, we obtain a volume formula for dual simplices in terms of the associated Diophantine data and recover the multiplicity change formula under mutation.