Metabelian techniques in knot concordance

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dc.contributor.advisorGordon, Cameron, 1945-
dc.contributor.committeeMemberGompf, Robert
dc.contributor.committeeMemberLuecke, John
dc.contributor.committeeMemberReid, Alan
dc.contributor.committeeMemberLivingston, Charles
dc.creatorMiller, Allison Northey
dc.creator.orcid0000-0002-4056-5474
dc.date.accessioned2018-08-08T14:47:36Z
dc.date.available2018-08-08T14:47:36Z
dc.date.created2018-05
dc.date.issued2018-04-18
dc.date.submittedMay 2018
dc.date.updated2018-08-08T14:47:36Z
dc.description.abstractThis dissertation lies in the field of knot concordance, the study of 4-dimensional properties of knots. We give four distinct results, which are united by their mutual reliance on concordance invariants associated to metabelian covers of certain 3-manifolds. First, we give some examples of 2-bridge knots for which twisted Alexander polynomials but not Casson-Gordon signatures obstruct sliceness. We then use Casson-Gordon signatures to give a complete characterization of the topologically slice odd 3-strand pretzel knots, and an almost complete characterization of the topologically slice even 3-strand pretzel knots. Next, we describe large infinite families of 4-strand pretzel knots which are not even topologically slice, despite being positive mutants of ribbon knots. We conclude by proving that given any patterns P and Q of opposite winding number, for any number n ≥ 0 there exists a knot K such that the minimal genus of a cobordism between P(K) and Q(K) is at least n. This completes the argument, partially established by Cochran-Harvey, that two patterns are a finite distance apart in their action on concordance if and only if they have the same algebraic winding number.
dc.description.departmentMathematics
dc.format.mimetypeapplication/pdf
dc.identifierdoi:10.15781/T2RV0DJ33
dc.identifier.urihttp://hdl.handle.net/2152/66021
dc.language.isoen
dc.subjectKnot theory
dc.subjectConcordance
dc.titleMetabelian techniques in knot concordance
dc.typeThesis
dc.type.materialtext
thesis.degree.departmentMathematics
thesis.degree.disciplineMathematics
thesis.degree.grantorThe University of Texas at Austin
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy

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