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

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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” \({\mathcal{Q}}\) and \({\mathcal{Q}'}\) from Part 1 (Morrison and Snyder in Commun. Math. Phys., doi:10.1007/s00220-012-1426-y, 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 \({\mathcal{Q}'}\) immediately. The second uses connections, and when combined with an additional number theoretic argument it eliminates both weeds \({\mathcal{Q}}\) and \({\mathcal{Q}'}\) . 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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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto, 606-8502, Japan

    Masaki Izumi

  2. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Vaughan F. R. Jones & Scott Morrison

  3. Department of Mathematics, Columbia University, New York, NY, 10026, USA

    Noah Snyder

Authors
  1. Masaki Izumi
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  2. Vaughan F. R. Jones
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  3. Scott Morrison
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  4. Noah Snyder
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Correspondence to Scott Morrison.

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Communicated by Y. Kawahigashi

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Izumi, M., Jones, V.F.R., Morrison, S. et al. Subfactors of Index Less Than 5, Part 3: Quadruple Points. Commun. Math. Phys. 316, 531–554 (2012). https://doi.org/10.1007/s00220-012-1472-5

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