Metabelian techniques in knot concordance




Miller, Allison Northey

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This dissertation lies in the field of knot concordance, the study of 4-dimensional properties of knots. We give four distinct results, which are united by their mutual reliance on concordance invariants associated to metabelian covers of certain 3-manifolds. First, we give some examples of 2-bridge knots for which twisted Alexander polynomials but not Casson-Gordon signatures obstruct sliceness. We then use Casson-Gordon signatures to give a complete characterization of the topologically slice odd 3-strand pretzel knots, and an almost complete characterization of the topologically slice even 3-strand pretzel knots. Next, we describe large infinite families of 4-strand pretzel knots which are not even topologically slice, despite being positive mutants of ribbon knots. We conclude by proving that given any patterns P and Q of opposite winding number, for any number n ≥ 0 there exists a knot K such that the minimal genus of a cobordism between P(K) and Q(K) is at least n. This completes the argument, partially established by Cochran-Harvey, that two patterns are a finite distance apart in their action on concordance if and only if they have the same algebraic winding number.



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