A rotational approach to triple point obstructions

Snyder, Noah

doi:10.2140/apde.2013.6.1923

Cited by 6 publications

(8 citation statements)

References 7 publications

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“…A table of the possible chiralities for the irreducible subfactors with index at most 4 with initial triple points is given in Example 3.12, as is a proof that E 7 and D 2k−1 (3 ≤ k < ∞) are not principal graphs of subfactors. Interestingly, all possible chiralities actually occur.We now recover[26, Theorem 3] which generalizes[12, Theorem 5.1.11].…”

mentioning

confidence: 65%

“…In particular, we use a quadratic relation due to Liu (Lemma 2.12) which is a clever variant of Wenzl's relation [27]. Such relations restrict the structure of the principal graph via the skein theory of the planar algebra, giving strong consequences.…”

Section: Corollary Under the Hypotheses Of Ocneanu's Obstruction Formentioning

confidence: 99%

“…We will then state a generalization from which Liu's relation quickly follows. [27].) Let f (k) ∈ T L k,± be the k-th Jones-Wenzl projection.…”

Section: Liu's Relationmentioning

confidence: 99%

“…First, the principal graphs begin as claimed by [ (1) Snyder can produce the equations in Theorem 3.10 using his technique in [26], but his technique does not generalize beyond the * 11 case. (2) Examples of principal graphs of subfactor planar algebras as in (1) and (2) of Theorem 3.10 are given in Examples 3.13 and 3.14 respectively.…”

Section: Theorem 35 Suppose Thatmentioning

confidence: 99%

“…Several examples of the latter type of obstruction for triple points include Ocneanu's triple point obstruction [7], Jones' quadratic tangles obstruction [12], the triple-single obstruction [20], and Snyder's singly valent obstruction [26], which mutually generalizes the quadratic tangles and triple-single obstructions.…”

Section: Introductionmentioning

confidence: 99%

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Chirality and principal graph obstructions

Penneys

2015

Advances in Mathematics

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Determining which bipartite graphs can be principal graphs of subfactors is an important and difficult question in subfactor theory. Using only planar algebra techniques, we prove a triple point obstruction which generalizes all known initial triple point obstructions to possible principal graphs. We also prove a similar quadruple point obstruction with the same technique. Using our obstructions, we eliminate some infinite families of possible principal graphs with initial triple and quadruple points which were a major hurdle in extending subfactor classification results above index 5.

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mentioning

confidence: 65%

Section: Corollary Under the Hypotheses Of Ocneanu's Obstruction Formentioning

confidence: 99%

“…We will then state a generalization from which Liu's relation quickly follows. [27].) Let f (k) ∈ T L k,± be the k-th Jones-Wenzl projection.…”

Section: Liu's Relationmentioning

confidence: 99%

Section: Theorem 35 Suppose Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Chirality and principal graph obstructions

Penneys

2015

Advances in Mathematics

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The Classification of Subfactors with Index at Most 5\frac{1}4

Narjess¹,

Morrison²,

Penneys³

2023

Memoirs of the AMS

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Subfactor standard invariants encode quantum symmetries. The small index subfactor classification program has been a rich source of interesting quantum symmetries. We give the complete classification of subfactor standard invariants to index 5 1 4 5\frac {1}{4} , which includes 3 + 5 3+\sqrt {5} , the first interesting composite index.

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The classification of subfactors of index at most 5

Jones

Morrison

Snyder

2013

Bull. Amer. Math. Soc.

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Abstract. A subfactor is an inclusion N ⊂ M of von Neumann algebras with trivial centers. The simplest example comes from the fixed points of a group action M G ⊂ 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's 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 '80s and early '90s. 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.

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A rotational approach to triple point obstructions

Cited by 6 publications

References 7 publications

Chirality and principal graph obstructions

Chirality and principal graph obstructions

The Classification of Subfactors with Index at Most 5\frac{1}4

The classification of subfactors of index at most 5

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