  Subfactors where the initial branching point of the principal graph is
3-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.
