A knot energy is a real-valued function on a space of curves which in some sense assigns higher energy values to more complicated curves. The key property of any knot energy is self-repulsiveness: for a sequence of curves approaching a self-intersection, the energy blows up to infinity. While the study of optimally embedded curves as minimizers of energy among a given knot class has been well-documented, this thesis investigates the existence of optimally immersed self-intersecting curves. Because any self-intersecting curve will have infinite knot energy, parameter-dependent renormalizations of the energy remove the singular behavior of the curve. This process allows for the careful selection of an optimally immersed curve.