Morrison, Scott2014-04-142014-04-140273-0979http://hdl.handle.net/1885/11573We find a new obstruction to the principal graphs of subfactors. It shows that in a certain family of 3-supertransitive principal graphs, there must be a cycle by depth 6, with one exception, the principal graph of the Haagerup subfactor.Scott Morrison is supported by a Discovery Early Career Research Award (DE120100232) from the Australian Research Council. This work was also supported by the ARC Discovery Project Grant ‘Subfactors and symmetries’ (DP140100732), and the DARPA grant ‘Quantum symmetries: Planar Algebras and Free Probability’ (HR0011-12-1-0009).8 pages© The Author; http://www.sherpa.ac.uk/romeo/issn/0273-0979/ author can archive pre-print (ie pre-refereeing); author can archive post-print (ie final draft post-refereeing); author can archive publisher's version/PDFOperator Algebras (math.OA)Quantum Algebra (math.QA)201410.1112/blms/bdu0092015-12-11