Subfactors of Index Less Than 5, Part 2: Triple Points

Abstract

We summarize the known obstructions to subfactors with principal graphs which begin with a triple point. One is based on Jones's quadratic tangles techniques, although we apply it in a novel way. The other two are based on connections techniques; one due to Ocneanu, and the other previously unpublished, although likely known to Haagerup.We then apply these obstructions to the classification of subfactors with index below 5. In particular, we eliminate three of the five families of possible principal graphs called "weeds" in the classification from [MS10].

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

“…This is the third paper in a series of four papers (along with [MS10,MPPS10,PT10]) in which we extend the previously known classification of subfactors of index less than 3 + √ 3 [Haa94, Bis98, AH99, AY09, BMPS09] up to index 5. Theorem 1.1.…”

“…Combining the results of the first two papers we have Theorem 1.2 (From [MS10,MPPS10]). The principal graph of any subfactor of index between 4 and 5 is a translate of one of an explicit finite list of graph pairs, which we call the vines, or is a translated extension of one of the following graph pairs, which we call the weeds.…”

“…We will assume that readers are familiar with planar algebras [Jon99] and with connections [Ocn88,EK98]; see the background section to [MPPS10] and [BMPS09, §2.1] for more details. In Section 2 we also assume familiarity with the main ideas of [Jon03] and in Section 4 we assume familarity with the main ideas of [CMS11].…”

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 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).…”