Subfactors where the initial branching point of the principal graph is
-valent
are subject to strong constraints called triple point obstructions. Since more
complicated initial branches increase the index of the subfactor, triple point
obstructions play a key role in the classification of small index subfactors. There are
two strong triple point obstructions, called the triple-single obstruction and the
quadratic tangles obstruction. Although these obstructions are very closely related,
neither is strictly stronger. In this paper we give a more general triple point
obstruction which subsumes both. The techniques are a mix of planar algebraic and
connection-theoretic techniques with the key role played by the rotation
operator.
