  We calculate the refined topological string partition function of the
Calabi-Yau threefold which is the total space of the canonical bundle on
$\mathbb{P}^2$ (the local $\mathbb{P}^2$). The refined topological vertex
formalism can not be directly applied to local $\mathbb{P}^2$ therefore we use
the properties of the refined Hopf link to define a new two legged vertex which
together with the refined vertex gives the partition function of the local
$\mathbb{P}^2$.
