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We prove existence of subfactors of finite depth of the hyperfinite II_1factor with indices (5+sqrt{13})/2= 4.302... and (5+sqrt{17})/2=4.561.... Theexistence of the former was announced by the second named author in 1993 andthat of the latter has been conjectured since then. These are the only knownsubfactors with finite depth which do not arise from classical groups, quantumgroups or rational conformal field theory.Comment: 70 pages, Latex, using epic.sty, eepic.sty, epsf.sty, here.st

Exotic Subfactors of Finite Depth with Jones Indices $(5+\sqrt{13})/2$ and $(5+\sqrt{17})/2$