Finding Harmonic Nine Crossing Knots

Abstract

A knot is an embedding of the unit circle into real three-space. Harmonic knots are de fined as trigonometric polynomial parameterizations ofthe unit circle. In this paper 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. Previous research found the harmonic parameterizations of knots types up through eight crossing knots. The parameterizations presented in this paper 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.