The results presented in this thesis pertain to two distinct areas of low-dimensional topology. First, we give a classification of small Seifert fibered surgeries on hyperbolic pretzel knots, as well as a near-classification of small Seifert fibered surgeries on hyperbolic Montesinos knots. Along with recent results of Ichihara-Masai [IM13], these results complete the classification of all exceptional Dehn surgeries on arborescent knots. Second, we exhibit an infinite family of smoothly slice knots that are topologically doubly slice, but not smoothly doubly slice. A subfamily of these knots is then used to show that the subgroup of the smooth double concordance group consisting of topologically doubly slice knots is infinitely generated. One corollary of these results is that there exist infinitely many rational homology 3-spheres (with nontrivial first homology) that embed topologically, but not smoothly, into the 4-sphere. / text

Invariants of knots coming from twisted signatures have played a central role in the study of knot concordance. Unfortunately, except in the simplest of cases, these signature invariants have proven exceedingly difficult to compute. As a consequence, many knots which presumably can be detected by these invariants are not a well understood as they should be. We study a family of signature invariants of knots and show that they provide concordance information. Significantly, we provide a tractable means for computing these signatures. Once armed with these tools we use them first to study the knot concordance group generated by the twist knots which are of order 2 in the algebraic concordance group. With our computational tools we can show that with only finitely many exceptions, they form a linearly independent set in the concordance group. We go on to study a procedure given by Cochran-Harvey-Leidy which produces infinite rank subgroups of the knot concordance group which, in some sense are extremely subtle and difficult to detect. The construction they give has an inherent ambiguity due to the difficulty of computing some signature invariants. This ambiguity prevents their construction from yielding an actual linearly independent set. Using the tools we develop we make progress to removing this ambiguity from their procedure.