The HOMFLY polynomial of a link is a two-variable link invariant which was introduced in the 1980s. It can be defined recursively using a skein relation and specializes to other link invariants such as the Alexander polynomial and Jones polynomial. In this talk, we will begin by describing how to associate a link to a plabic graph. Then, we will describe how to recursively compute the HOMFLY polynomial of these links in the case where the quiver Q_G associated to the plabic graph is an orientation of a forest. Finally, we will discuss a closed form expression for the HOMFLY polynomial of such links where the coefficients are expressed in terms of independent sets of the quiver.