A combinatorial proof of the invariance of tangle floer homology

dc.contributorRoberts, Lawrence
dc.contributorTosun, Bulent
dc.contributorTrace, Bruce
dc.contributorEvans, Martin
dc.contributorHarms, Benjamin
dc.contributor.advisorRoberts, Lawrence
dc.contributor.authorHoman, Timothy Adam
dc.contributor.otherUniversity of Alabama Tuscaloosa
dc.date.accessioned2020-01-16T15:04:21Z
dc.date.available2020-01-16T15:04:21Z
dc.date.issued2019
dc.descriptionElectronic Thesis or Dissertationen_US
dc.description.abstractThe aim of this work is to take the combinatorial construction put forward by Petkova and Vértesi for tangle Floer homology and show that many of the arguments that apply to grid diagrams for knots can be applied to grid diagrams for tangles. In particular, we showed that the stabilization and commutation arguments used in combinatorial knot Floer homology can be applied mutatis mutandis to combinatorial tangle Floer homology, giving us an equivalence of chain complexes (either exactly in the case of commutations or up to the size of the grid in stabilizations). We then added a new move, the stretch move, and showed that the same arguments which work for commutations work for this move as well. We then extended these arguments to the context of A-infinity structures. We developed for our stabilization arguments a new type of algebraic notation and used this notation to demonstrate and simplify useful algebraic results. These results were then applied to produce type D and type DA equivalences between grid complexes and their stabilized counterparts. For commutation moves we proceeded more directly, constructing the needed type D homomorphisms and homotopies as needed and then showing that these give us a type D equivalence between tangle grid diagrams and their commuted counterparts. We also showed that these arguments can also be applied to our new stretch move. Finally, we showed that these grid moves are sufficient to accomplish the planar tangle moves required to establish equivalence of the tangles themselves with the exception of one move.en_US
dc.format.extent124 p.
dc.format.mediumelectronic
dc.format.mimetypeapplication/pdf
dc.identifier.otheru0015_0000001_0003469
dc.identifier.otherHoman_alatus_0004D_13884
dc.identifier.urihttp://ir.ua.edu/handle/123456789/6526
dc.languageEnglish
dc.language.isoen_US
dc.publisherUniversity of Alabama Libraries
dc.relation.hasversionborn digital
dc.relation.ispartofThe University of Alabama Electronic Theses and Dissertations
dc.relation.ispartofThe University of Alabama Libraries Digital Collections
dc.rightsAll rights reserved by the author unless otherwise indicated.en_US
dc.subjectMathematics
dc.titleA combinatorial proof of the invariance of tangle floer homologyen_US
dc.typethesis
dc.typetext
etdms.degree.departmentUniversity of Alabama. Department of Mathematics
etdms.degree.disciplineMathematics
etdms.degree.grantorThe University of Alabama
etdms.degree.leveldoctoral
etdms.degree.namePh.D.
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