The untwisting number of a knot

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dc.contributor.advisorPutman, Andrew
dc.creatorInce, Kenan A
dc.date.accessioned2017-08-03T14:23:06Z
dc.date.available2017-08-03T14:23:06Z
dc.date.created2016-05
dc.date.issued2016-04-14
dc.date.submittedMay 2016
dc.date.updated2017-08-03T14:23:07Z
dc.description.abstractThe unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. The algebraic unknotting number is the minimum number of crossing changes needed to transform a knot into an Alexander polynomial-one knot. We work with a generalization of unknotting number due to Mathieu-Domergue, which we call the untwisting number. The p-untwisting number is the minimum number (over all diagrams of a knot) of full twists on at most 2p strands of a knot, with half of the strands oriented in each direction, necessary to transform that knot into the unknot. First, we show that the algebraic untwisting number is equal to the algebraic unknotting number. However, we also exhibit several families of knots for which the difference between the unknotting and untwisting numbers is arbitrarily large, even when we only allow twists on a fixed number of strands or fewer. Second, we show that a common route for obstructing low unknotting number, the Montesinos trick, does not generalize to the untwisting number. However, we use a different approach to get conditions on the Heegaard Floer correction terms of the branched double cover of a knot with untwisting number one. This allows us to show that several 10-and 11-crossing knots cannot be unknotted by a single positive or negative generalized crossing change. We also use the Ozsváth-Szabó tau invariant and the Rasmussen s invariant to differentiate between the p- and q-untwisting numbers for certain p and q.
dc.format.mimetypeapplication/pdf
dc.identifier.citationInce, Kenan A. "The untwisting number of a knot." (2016) Diss., Rice University. <a href="https://hdl.handle.net/1911/96523">https://hdl.handle.net/1911/96523</a>.
dc.identifier.urihttps://hdl.handle.net/1911/96523
dc.language.isoeng
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.
dc.subjectknot theory
dc.subjectunknotting number
dc.subjectuntwisting number
dc.subjectgeometric topology
dc.subjectHeegaard Floer homology
dc.subjectgeneralized crossing change
dc.subjectcrossing change
dc.titleThe untwisting number of a knot
dc.typeThesis
dc.type.materialText
thesis.degree.departmentMathematics
thesis.degree.disciplineNatural Sciences
thesis.degree.grantorRice University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy
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