# Slice ribbon conjecture, pretzel knots and mutation

 dc.contributor.advisor Gordon, Cameron, 1945- dc.creator Long, Ligang en dc.date.accessioned 2014-11-06T17:50:12Z en dc.date.issued 2014-08 en dc.date.submitted August 2014 en dc.date.updated 2014-11-06T17:50:12Z en dc.description text en dc.description.abstract In this paper we explore the slice-ribbon conjecture for some families of pretzel knots. Donaldson's diagonalization theorem provides a powerful obstruction to sliceness via the union of the double branched cover W of B⁴ over a slicing disk and a plumbing manifold P([capital gamma]). Donaldson's theorem classifies all slice 4-strand pretzel knots up to mutation. The correction term is another 3-manifold invariant defined by Ozsváth and Szabó. For a slice knot K the number of vanishing correction terms of Y[subscript K] is at least the square root of the order of H₁(Y[subscript K];Z). Donaldson's theorem and the correction term argument together give a strong condition for 5-strand pretzel knots to be slice. However, neither Donaldson's theorem nor the correction terms can distinguish 4-strand and 5-strand slice pretzel knots from their mutants. A version of the twisted Alexander polynomial proposed by Paul Kirk and Charles Livingston provides a feasible way to distinguish those 5-strand slice pretzel knots and their mutants; however the twisted Alexander polynomial fails on 4-strand slice pretzel knots. en dc.description.department Mathematics en dc.format.mimetype application/pdf en dc.identifier.uri http://hdl.handle.net/2152/27145 en dc.language.iso en en dc.subject Slice ribbon conjecture en dc.subject Pretzel knots en dc.subject Mutation en dc.subject Donaldson Theorem en dc.subject Correction terms en dc.subject Twisted Alexander polynomial en dc.title Slice ribbon conjecture, pretzel knots and mutation en dc.type Thesis en thesis.degree.department Mathematics en thesis.degree.discipline Mathematics en thesis.degree.grantor The University of Texas at Austin en thesis.degree.level Doctoral en thesis.degree.name Doctor of Philosophy en

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