First Order Signatures and Knot Concordance

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dc.contributor.advisorCochran, Tim D.en_US
dc.contributor.committeeMemberHarvey, Shellyen_US
dc.contributor.committeeMemberBorcea, Lilianaen_US
dc.creatorDavis, Christopheren_US
dc.date.accessioned2012-09-05T23:58:18Zen_US
dc.date.accessioned2012-09-05T23:58:20Zen_US
dc.date.available2012-09-05T23:58:18Zen_US
dc.date.available2012-09-05T23:58:20Zen_US
dc.date.created2012-05en_US
dc.date.issued2012-09-05en_US
dc.date.submittedMay 2012en_US
dc.date.updated2012-09-05T23:58:21Zen_US
dc.description.abstractInvariants of knots coming from twisted signatures have played a central role in the study of knot concordance. Unfortunately, except in the simplest of cases, these signature invariants have proven exceedingly difficult to compute. As a consequence, many knots which presumably can be detected by these invariants are not a well understood as they should be. We study a family of signature invariants of knots and show that they provide concordance information. Significantly, we provide a tractable means for computing these signatures. Once armed with these tools we use them first to study the knot concordance group generated by the twist knots which are of order 2 in the algebraic concordance group. With our computational tools we can show that with only finitely many exceptions, they form a linearly independent set in the concordance group. We go on to study a procedure given by Cochran-Harvey-Leidy which produces infinite rank subgroups of the knot concordance group which, in some sense are extremely subtle and difficult to detect. The construction they give has an inherent ambiguity due to the difficulty of computing some signature invariants. This ambiguity prevents their construction from yielding an actual linearly independent set. Using the tools we develop we make progress to removing this ambiguity from their procedure.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.citationDavis, Christopher. "First Order Signatures and Knot Concordance." (2012) Diss., Rice University. <a href="https://hdl.handle.net/1911/64621">https://hdl.handle.net/1911/64621</a>.en_US
dc.identifier.slug123456789/ETD-2012-05-65en_US
dc.identifier.urihttps://hdl.handle.net/1911/64621en_US
dc.language.isoengen_US
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.en_US
dc.subjectKnot concordanceen_US
dc.subjectL2 Homologyen_US
dc.subjectTwisted signaturesen_US
dc.subjectRho invariantsen_US
dc.titleFirst Order Signatures and Knot Concordanceen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentMathematicsen_US
thesis.degree.disciplineNatural Sciencesen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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