Abstract

          A knot is an embedding of the unit circle into real three-space. Harmonic knots are defined as trigonometric polynomial parameterizations of the unit circle. In this poster we construct harmonic nine crossing knots. To construct harmonic knots we created a program in Mathematica that uses truncated Fourier series to approximate linear functions representing the paths of polygonal knots. Harmonic representations of knots are used to define knot invariants including the harmonic index.  The harmonic index of a knot is the minimum integer n such that there is an nth degree harmonic parameterization of the same knot type. Previous research found the harmonic parameterizations of knots types up through eight crossing knots. The parameterizations presented in this poster are used to determine upper bounds for the harmonic index of each nine crossing knot. These parameterizations also aid in the research of new knot invariants by providing more examples of harmonic representations.