Symmetric union of knots, double branched covers and Dehn surgery

This dissertation studies symmetric unions of knots, a classical construction in knot theory introduced in the 1950s by Kinoshita and Terasaka. Because of the flexibility in their construction and the fact that they are ribbon, hence smoothly slice, symmetric unions appear quite frequently in the literature. In this dissertation we focus on two aspects of these knots: Firstly, we study symmetric unions in the context of problems regarding knot invariants. Recently, a class of symmetric unions were proposed to construct nontrivial knots with trivial Jones polynomial. We show, however, that such a knot is always trivial and hence this construction cannot be used to answer the open question asking whether the Jones polynomial detects the unknot. We then discuss why symmetric union is a valuable construction to study knot invariants and why our result provides strong evidence for the non-existence of nontrivial knots with trivial Jones polynomial. Secondly, it is still unknown whether there exists a ribbon knot which cannot be presented as a symmetric union. Thus, similar to the Slice-Ribbon conjecture, one may ask whether every ribbon knot is a symmetric union. We use double branched covers and surgeries to give obstructions for a ribbon knot to admit certain symmetric union diagrams. We classify the simplest type of symmetric unions that are composite, two-bridge, Montesinos, or amphichiral, and in doing so, give infinite families of ribbon knots that cannot admit the simplest type of symmetric union diagrams. One of the most important aspects of this dissertation is a conjecture about composite symmetric unions, whose resolution would give rise to the first examples of ribbon knots that cannot be presented as symmetric unions.Mathematic