  In this paper we investigate the relation between conformal blocks of
Liouville CFT and the topological string partition functions of the rank one
trinion theory $T_2$. The partition functions exhibit jumps when passing from
one chamber in the parameter space to another. Such jumps can be attributed to
a change of the integration contour in the free field representation of
Liouville conformal blocks. We compare the partition functions of the $T_2$
theories representing trifundamental half hypermultiplets in $N=2$, $d=4$ field
theories to the partition functions associated to bifundamental
hypermultiplets. We find that both are related to the same Liouville conformal
blocks up to inessential factors. In order to establish this picture we combine
and compare results obtained using topological vertex techniques, matrix models
and topological recursion. We furthermore check that the partition functions
obtained by gluing two $T_2$ vertices can be represented in terms of a four
point Liouville conformal block. Our results indicate that the $T_2$ vertex
offers a useful starting point for developing an analog of the instanton
calculus for SUSY gauge theories with trifundamental hypermultiplets.
