Degeneration of topological string partition functions and mirror curves of the Calabi-Yau threefolds X${}_{N,M}$

In this article we study certain degenerations of the mirror curves associated with the Calabi-Yau threefolds X${}_{N,M}$, and the effect of these degenerations on the refined topological string partition function of X${}_{N,M}$. We show that when the mirror curve degenerates and become the union of the lower genus curves the corresponding partition function factorizes into pieces corresponding to the components of the degenerate mirror curve. Moreover we show that using degeneration of a generalised mirror curve it is possible to obtain the partition function corresponding to X${}_{N,M-1}$ from X${}_{N,M}$.