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    Some Non-abelian Covers of Knot Complements

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    Genre
    Thesis/Dissertation
    Date
    2019
    Author
    Morris, Timothy Michael
    Advisor
    Stover, Matthew
    Committee member
    Futer, David
    Petersen, Kathleen
    Taylor, Samuel J.
    Department
    Mathematics
    Subject
    Mathematics
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/3308
    
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    DOI
    http://dx.doi.org/10.34944/dspace/3290
    Abstract
    Let K be a tame knot embedded in S³. We address the problem of finding the minimal degree non-cyclic cover p : X → S³ \ K. When K has non-trivial Alexander polynomial we construct finite non-abelian representations p : π1 (S³\ K) → G, and provide bounds for the order of G in terms of the crossing number of K, which is an improvement on a result of Broaddus in this case. Using classical covering space theory along with the theory of Alexander stratifications we establish an upper and lower bound for the first betti number of the cover Xp associated to the ker (p) of S³ \ K, consequently showing that it can be arbitrarily large, which provides an effective proof of a result involving peripheral subgroup separation. We also demonstrate that Xp contains non-peripheral homology for certain computable examples, which mirrors a famous result of Cooper, Long, and Reid when K is a knot with non-trivial Alexander polynomial.
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