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Slice ribbon conjecture, pretzel knots and mutation

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. / text

Identiferoai:union.ndltd.org:UTEXAS/oai:repositories.lib.utexas.edu:2152/27145
Date06 November 2014
CreatorsLong, Ligang
Source SetsUniversity of Texas
LanguageEnglish
Detected LanguageEnglish
TypeThesis
Formatapplication/pdf

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