Harvey, Shelly2022-09-262022-09-262022-052022-04-22May 2022Williams, Shawn. "Extensions of the Fox-Milnor Condition." (2022) Diss., Rice University. <a href="https://hdl.handle.net/1911/113356">https://hdl.handle.net/1911/113356</a>.https://hdl.handle.net/1911/113356The search for slice knots is an important task in low dimensional topology. In the 1960s, Fox and Milnor proved a theorem stating that the Alexander polynomial of a slice knot satisfies a special factorization. A decade later, Kawauchi extended this theorem for the multivariable Alexander polynomial of slice links. This factorization, known as the Fox-Milnor condition, has been used and generalized many times as an obstruction to a link being slice. In this defense, we will see two more extensions of this condition, first to the multivariable Alexander polynomial of 1-solvable links, and then for the first order Alexander polynomial of ribbon knots.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.knotslinksslicesolvabilityFox-MilnorfactorizationlocalizationAlexander moduleAlexander polynomialThesis2022-09-26