/pdf/identifying-tunnel-number-one-knots-2tg1pxjq8h.pdf

Identifying tunnel number one knots

Let N C P and P C M be inclusions of Hi factors with finite Jones index. We study the composed inclusion N C P C M by computing the fusion of N-P and P-M bimodules and determine various properties of N C M in terms of the \"small\" inclusions. A nice class of such subfactors arises in the following way: let H and K be two finite groups acting properly outerly on the hyperfinite II\\ factor M and consider the inclusion M C M x K. We show that properties like irreducibility, finite depth, amenability and strong amenability (in the sense of Popa) of M^ C M x K can be expressed in terms of properties of the group G generated by H and K in OutM. In particular, the inclusion is amenable iff M is hyperfinite and the group G is amenable. We obtain many new examples of infinite depth subfactors (amenable and nonamenable ones), whose principal graphs have subexponential and/or exponential growth and can be determined explicitly. Furthermore, we construct irreducible, amenable subfactors of the hyperfinite II i factor which are not strongly amenable.

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Composition of subfactors : new examples of infinite depth subfactors

All generating pairs of all two-generator Fuchsian groups

Let f and g be elements of the isometry group Isom(H2) of the hyperbolic plane H2, and assume that one of them is orientation-reversing. We determine when the group (f,g) they generate is discrete; in particular, we obtain the classification of such groups. As an application to knot theory, we completely determine the tunnel number one Montesinos knots.

Two-generator Discrete Subgroups of Isom(H2) Containing Orientation-reversing Elements

LetPbe a polygon on hyperbolic plane H2. A Coxeter decomposition of a polygonPis a non-trivial decomposition ofPinto finitely many Coxeter polygonsFi, such that any two polygonsF1andF2having a common side are symmetric with respect to this common side. In this paper we classify the Coxeter decompositions of triangles, quadrilaterals and ideal polygons.

Coxeter Decompositions of Hyperbolic Polygons

This paper gives a short geometric algorithm for deciding the discreteness of most 2-generator subgroups of PSL(2,R), as well as a self-contained algorithmic approach to the complete classification

J. Peter Matelski

Papers

The classification of discrete 2-generator subgroups of PSL(2,R)