In my research, I investigate the complexities of knots. A knot is an embedded circle in the 3-sphere; one can think of a sailor`s knot but with the ends of the rope glued together. Each nontrivial knot has crossings where one strand of the knot passes over another. The “unknotting number” of a knot measures the number of crossings that must be changed in a knot in order to untie it. I work with a generalization of the unknotting number which is related to a beautiful equivalence relation on 3-manifolds due to Cochran, Gerges, and Orr. ©