Harvey, Shelly2017-08-012017-08-012017-052017-04-20May 2017Park, Junghwan. "Derivatives of Genus One and Three Knots." (2017) Diss., Rice University. <a href="https://hdl.handle.net/1911/96122">https://hdl.handle.net/1911/96122</a>.https://hdl.handle.net/1911/96122A 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.application/pdfengCopyright 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.Knot concordanceThesis2017-08-01