  Determining which bipartite graphs can be principal graphs of subfactors is
an important and difficult question in subfactor theory. Using only planar
algebra techniques, we prove a triple point obstruction which generalizes all
known initial triple point obstructions to possible principal graphs. We also
prove a similar quadruple point obstruction with the same technique. Using our
obstructions, we eliminate some infinite families of possible principal graphs
with initial triple and quadruple points which were a major hurdle in extending
subfactor classification results above index 5.
