Planar algebras are useful in studying subfactors, representation theory, and knot theory. We focus on taking a skein theoretic, combinatorial approach to developing and describing planar algebras. Evaluating and simplifying closed diagrams is a theme in this dissertation.

The jellyfish algorithm is an evaluation algorithm on certain planar algebras defined using generators and relations. Planar algebras defined in this way must be shown to not be the trivial planar algebra. We give a proof that the jellyfish algorithm is well-defined on the D 2n planar algebras using only the presentations.

In this dissertation we also present a new planar algebra motivated by the An and Ainfinity graph planar algebras and the notion of coloring diagrams using a graph of colors. We call this planar algebra the Pop-Switch planar algebra and give a full description including an interesting isomorphism between the monoid of the space of closed diagrams and a specific set of sequences of integers.