Critical knots for minimum distance energy and complementary domains of arrangements of hypersurfaces

In this thesis, we will discuss two separate topics. First, we find a critical knot for an knot energy function. A knot is a closed curve or polygon in three space. It is possible to for a computer to simulate the flow of a knot to its minimum energy conformation. There is no guarantee, however, that a true minimizer exists near the computer's alleged minimizer. We take advantage of both the symmetry of the minimizer and the symmetry invariance of the energy function to prove that there is a critical point of the energy function near the computer's minimizer.

Second, we will discuss how to determine the number of complementary domains of arrangements of algebraic curves in 2-space and ellipsoids in 3-space. In each of these situations, we supply equations that provide an upper bound for the number of complementary domains. These upper bounds are applicable even when the exact intersections between the curves or surfaces are unknown.