Properties of commensurability classes of hyperbolic knot complements

This thesis investigates the topology and geometry of hyperbolic knot complements that are commensurable with other knot complements. In chapter 3, we provide an infinite family examples of hyperbolic knot complements commensurable with exactly two other knot complements. In chapter 4, we exhibit an obstruction to knot complements admitting exceptional surgeries in conjunction with hidden symmetries. Finally, in chapter 5, we discuss the role of surfaces embedded in 3-orbifolds as it relates to hidden symmetries.