Abstract: Plabic graphs were introduced by Postnikov in order to study a stratification of the totally nonnegative Grassmannian. To each plabic graph, one can associate a quiver, which is a directed graph that encodes a cluster algebra, as well as a link. In this thesis, we study invariants of these links, called plabic links, and their connections to the plabic graphs' quivers.

We focus primarily on forest quivers, which are quivers whose underlying graphs are forests. We define the HOMFLY polynomial of a forest quiver and show that it agrees with the HOMFLY polynomial of any plabic link coming from a connected plabic graph whose quiver is that forest quiver. We define this polynomial recursively and also prove a closed formula for it. We will also comment on a way to extend the definition of part of this polynomial to some other acyclic quivers. Finally, we discuss the Khovanov-Rozansky homology of certain plabic links associated to forest quivers and describe how to recursively compute it for a subset of these links.