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Topics on Dehn surgery

University of British Columbia

Cyclic surgery on satellite knots in S³ is classified and a necessary condition is given for a knot in S³ to admit a nontrivial cyclic surgery with slope m/l, \m\ > 1. A complete classification of cyclic group actions on the Poincaré sphere with 1-dimensional fixed point sets is obtained. It is proved that the following knots have property I, i.e. the fundamental group of the manifold obtained by Dehn surgery on such a knot cannot be the binary icosahedral group I₁₂₀, the fundamental group of the Poincaré homology 3-sphere: nontrefoil torus knots, satellite knots, nontrefoil generalized double knots, periodic knots with some possible specific exceptions, amphicheiral strongly invertible knots, certain families of pretzel knots. Further the Poincaré sphere cannot be obtained by Dehn surgery on slice knots and a certain family of knots formed by band-connect sums. It is proved that if a nonsufficiently large hyperbolic knot in S³ admits two nontrivial cyclic Dehn surgeries then there is at least one nonintegral boundary slope for the knot. There are examples of such knots. Thus nonintegral boundary slopes exist.

Text

2011-03-07

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http://hdl.handle.net/2429/32117

Mathematics

University of British Columbia

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