  We study the geometry of fully augmented link complements in $S^3$ by looking
at their link diagrams. We extend the method introduced by Thistlethwaite and
Tsvietkova to fully augmented links and define a system of algebraic equations
in terms of parameters coming from edges and crossings of the link diagrams.
Combining it with the work of Purcell, we show that the solutions to these
algebraic equations are related to the cusp shapes of fully augmented link
complements. As an application we use the cusp shapes to study the
commensurability classes of fully augmented links.
