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    Derivatives of Genus One and Three Knots

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    Author
    Park, Junghwan
    Date
    2017-04-20
    Advisor
    Harvey, Shelly
    Degree
    Doctor of Philosophy
    Abstract
    A derivative L of an algebraically slice knot K is an oriented link disjointly embedded in a Seifert surface of K such that its homology class forms a basis for a metabolizer H of K. For genus one knots, we produce a new example of a smoothly slice knot with non-slice derivatives. Such examples were first discovered by Cochran and Davis. In order to do so, we define an operation on a homology B^4 that we call an n-twist annulus modification. Further, we give a new construction of smoothly slice knots and exotically slice knots via n-twist annulus modifications. For genus three knots, we show that the set S_{K,H} ={ mu_L(123) - mu_L'(123) | L,L' are derivatives associated with a metabolizer H} contains n · Z, where n is an integer determined by a Seifert form of K and a metabolizer H. As a corollary, we show that it is possible to realize any integer as the Milnor's triple linking number of a derivative of the unknot on a fixed Seifert surface and with a fixed metabolizer.
    Keyword
    Knot concordance
    Citation
    Park, Junghwan. "Derivatives of Genus One and Three Knots." (2017) Diss., Rice University. https://hdl.handle.net/1911/96122.
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    Home | FAQ | Contact Us | Privacy Notice | Accessibility Statement
    Managed by the Digital Scholarship Services at Fondren Library, Rice University
    Physical Address: 6100 Main Street, Houston, Texas 77005
    Mailing Address: MS-44, P.O.BOX 1892, Houston, Texas 77251-1892
    Site Map