Plumbers' knots and unstable Vassiliev theory

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dc.contributor.authorGiusti, Chad David, 1978-
dc.date.accessioned2010-12-03T22:07:48Z
dc.date.available2010-12-03T22:07:48Z
dc.date.issued2010-06
dc.descriptionviii, 57 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.en_US
dc.description.abstractWe introduce a new finite-complexity knot theory, the theory of plumbers' knots, as a model for classical knot theory. The spaces of plumbers' curves admit a combinatorial cell structure, which we exploit to algorithmically solve the classification problem for plumbers' knots of a fixed complexity. We describe cellular subdivision maps on the spaces of plumbers' curves which consistently make the spaces of plumbers' knots and their discriminants into directed systems. In this context, we revisit the construction of the Vassiliev spectral sequence. We construct homotopical resolutions of the discriminants of the spaces of plumbers knots and describe how their cell structures lift to these resolutions. Next, we introduce an inverse system of unstable Vassiliev spectral sequences whose limit includes, on its E ∞ - page, the classical finite-type invariants. Finally, we extend the definition of the Vassiliev derivative to all singularity types of plumbers' curves and use it to construct canonical chain representatives of the resolution of the Alexander dual for any invariant of plumbers' knots.en_US
dc.description.sponsorshipCommittee in charge: Dev Sinha, Chairperson, Mathematics; Hal Sadofsky, Member, Mathematics; Arkady Berenstein, Member, Mathematics; Daniel Dugger, Member, Mathematics; Andrzej Proskurowski, Outside Member, Computer & Information Scienceen_US
dc.identifier.urihttps://hdl.handle.net/1794/10869
dc.language.isoen_USen_US
dc.publisherUniversity of Oregonen_US
dc.relation.ispartofseriesUniversity of Oregon theses, Dept. of Mathematics, Ph. D., 2010;
dc.subjectPlumbers' knotsen_US
dc.subjectVassiliev derivativesen_US
dc.subjectFinite-complexity knotsen_US
dc.subjectSpectral sequencesen_US
dc.subjectAlexander dualen_US
dc.subjectCanonical chainsen_US
dc.subjectMathematicsen_US
dc.subjectTheoretical mathematicsen_US
dc.titlePlumbers' knots and unstable Vassiliev theoryen_US
dc.typeThesisen_US

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