Painlevé equations, topological type property and reconstruction by the topological recursion

Abstract: In this article we prove that Lax pairs associated with $\hbar$-dependent six Painlev\'e equations satisfy the topological type property proposed by Berg`ere, Borot and Eynard for any generic choice of the monodromy parameters. Consequently we show that one can reconstruct the formal $\hbar$-expansion of the isomonodromic $\tau$-function and of the determinantal formulas by applying the so-called topological recursion to the spectral curve attached to the Lax pair in all six Painlev\'e cases. Finally we illustrate the former results with the explicit computations of the first orders of the six $\tau$-functions.