  The goal of this article is to prove that the determinantal formulas of the
Painlev'e 2 system identify with the correlation functions computed from the
topological recursion on their spectral curve for an arbitrary non-zero
monodromy parameter. The result is established for two different Lax pairs
associated to the Painlev'e 2 system, namely the Jimbo-Miwa Lax pair and the
Harnad-Tracy-Widom Lax pair, whose spectral curves are not connected by any
symplectic transformation. We provide a new method to prove the topological
type property without using the insertion operators. In the process, taking the
time parameter t to infinity gives that the symplectic invariants F(g) computed
from the Hermite-Weber curve and the Bessel curve are equal to respectively.
This result generalizes similar results obtained from random matrix theory in
the special case where {\theta} = 0. We believe that this approach should apply
for all 6 Painlev'e equations with arbitrary monodromy parameters. Explicit
computations up to g = 3 are provided along the paper as an illustration of the
results.
