Subfactors of Index Less Than 5, Part 3: Quadruple Points

Abstract

Abstract. One major obstacle in extending the classification of small index subfactors beyond 3 + √ 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 [MS10]). 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.

Singly generated planar algebras of small dimension, Part III

“…Recently the classification has been extended up to index 5, and even beyond. See [JMS14,MS12,MPPS12,IJMS12,PT12]. Such classifications would not be possible without the reduction of the subfactor problem to an essentially combinatorial one.…”

A rotational approach to triple point obstructions

“…For example, a triple point obstruction due to Ocneanu shows that as long as the index is at least 4 the initial triple point must be at odd depth. These triple point obstructions play a crucial role in the classification of small index subfactors [Haa94,MS10,MPPS10,IJMS11,PT10].…”

“…The main result of this paper is a mutual generalization of these two triple point obstructions which proves the stronger conclusion using only the weaker assumptions. As one might expect, this paper uses a mix of connections and planar algebras following [IJMS11]. Furthermore, one can think of this argument as giving an alternate proof of the triple point obstruction from [Jon03].…”

“…Since rotational eigenvalues must be roots of unity this gives highly nontrivial constraints on candidate connections. We assume that the reader is familiar with both planar algebras and connections, referring the reader to [IJMS11] for more detail.…”

A Planar Calculus for Infinite Index Subfactors

“…This combinatorial data was axiomatized in three slightly different structures: paragroups [Ocn88], λ-lattices [Pop95], and planar algebras [Jon99]. When combined, these viewpoints produce strong results, e.g., standard invariants with index in (4,5) are completely classified, excluding the A ∞ standard invariant at each index value [Pop93] (see [MS11,MPPS11,IJMS11,PT11] for more details).…”