  One major obstacle in extending the classification of small index subfactors
beyond 3+\sqrt{3} is the appearance of infinite families of candidate principal
graphs with 4-valent vertices (in particular, the "weeds" Q and Q' from Part 1
(arXiv:1007.1730)). Thus instead of using triple point obstructions to
eliminate candidate graphs, we need to develop new quadruple point
obstructions. In this paper we prove two quadruple point obstructions. The
first uses quadratic tangles techniques and eliminates the weed Q' immediately.
The second uses connections, and when combined with an additional number
theoretic argument it eliminates both weeds Q and Q'. Finally, we prove the
uniqueness (up to taking duals) of the 3311 Goodman-de la Harpe-Jones subfactor
using a combination of planar algebra techniques and connections.
