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Subfactors of index less than 5, part 3: quadruple points

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Masaki Izumi, Vaughan F. R. Jones, Scott Morrison, Noah Snyder

Operator Algebras · Submitted 14 Sep 2011

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID1109.3190 (version 1)
Pith IDOPO6LRT3, Pith's permanent record of this paper and its review
CitationMasaki Izumi, Noah Snyder, Scott Morrison, Vaughan F. R. Jones. "Subfactors of index less than 5, part 3: quadruple points." arXiv:1109.3190 (2011). https://pith.science/paper/OPO6LRT3
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arXiv listing abstract v1. Results and the main claim are underlined; .

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.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 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)). background 0.96 0.09 0.03 asserted 1.00
s1 Thus instead of using triple point obstructions to eliminate candidate graphs, we need to develop new quadruple point obstructions. gap 0.42 0.17 0.05 asserted 0.98
s2 In this paper we prove two quadruple point obstructions. contribution 0.88 0.96 0.28 asserted 1.00
s3 The first uses quadratic tangles techniques and eliminates the weed Q' immediately. contribution 0.47 0.93 0.09 asserted 1.00
s4 The second uses connections, and when combined with an additional number theoretic argument it eliminates both weeds Q and Q'. result 0.53 0.94 0.17 asserted 1.00
s5 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. result 0.87 0.95 0.34 asserted 1.00

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Original abstract

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.

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