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M. Izumi, V. F. R. Jones, S. Morrison, and N. Snyder: “Subfactors of index less than 5, III: Quadruple points,” Comm. Math. Phys. 316 : 2 (2012), pp. 531–554. MR 2993924 Zbl 1272.46051 ArXiv 1109.3190 article

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” \( \mathcal{Q} \) and \( \mathcal{Q}^{\prime} \) from [Morrison and Snyder 2012]. 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 \( \mathcal{Q}^{\prime} \) immediately. The second uses connections, and when combined with an additional number theoretic argument it eliminates both weeds \( \mathcal{Q} \) and \( \mathcal{Q}^{\prime} \). 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.

Noah Snyder	Related
Scott Morrison	Related
Masaki Izumi	Related

V. F. R. Jones, S. Morrison, and N. Snyder: “The classification of subfactors of index at most 5,” Bull. Am. Math. Soc. (N.S.) 51 : 2 (2014), pp. 277–327. MR 3166042 Zbl 1301.46039 ArXiv 1304.6141 article

A subfactor is an inclusion \( N\subset M \) of von Neumann algebras with trivial centers. The simplest example comes from the fixed points of a group action \( M^G \subset M \), and subfactors can be thought of as fixed points of more general group-like algebraic structures. These algebraic structures are closely related to tensor categories and have played important roles in knot theory, quantum groups, statistical mechanics, and topological quantum field theory. There is a measure of size of a subfactor, called the index. Remarkably, the values of the index below 4 are quantized, which suggests that it may be possible to classify subfactors of small index. Subfactors of index at most 4 were classified in the 1980s and early 1990s. The possible index values above 4 are not quantized, but once you exclude a certain family, it turns out that again the possibilities are quantized. Recently, the classification of subfactors has been extended up to index 5, and (outside of the infinite families) there are only 10 subfactors of index between 4 and 5. We give a summary of the key ideas in this classification and discuss what is known about these special small subfactors.

Noah Snyder	Related
Scott Morrison	Related