Harmonic knots
In this thesis we define harmonic knots and examine some of their properties and applications. A harmonic knot is a knot that can be expressed parametrically in $\Re\sp3$ as (x(t), y(t), z(t)) where each coordinate function is a trigonometric polynomial. We show every knot is ambient isotopic to a harmonic knot, so we are able to define the harmonic index of a knot type as the smallest degree of any harmonic knot in the knot class. The harmonic index of a knot is related to its superbridge index and crossing number. In particular, we show the superbridge index of a knot is bounded by the knot's harmonic index and we show there is a bound on the harmonic index of a knot computed from the knot's crossing number, and vice versa. Hence only finitely many knot types occur with any given harmonic index. We provide explicit harmonic parametrizations of all prime knots through eight crossings and discuss how the harmonic index of these knots is related to their knot energy. The parametrizations were found using the mathematical computing language Maple. We conclude by showing how closed, circular DNA can be modeled using harmonic knots
Downloadable Article, English, 1995