Parameterized Knots

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by Lee Stemkoski (Adelphi University)

Trigonometric Knots

A trigonometric knot is a knot as defined in the introduction, with the additional condition that in the function \( f(t) = \langle \, x(t), \, y(t), \, z(t) \, \rangle \), the coordinate functions \( x\), \(y\), and \( z \) are written in terms of trigonometric functions.

Possibly the most well-known type of trigonometric knots are torus knots, knots that lie on the surface of a torus. Torus knots are uniquely identified by a pair of relatively prime integers, \( p \) and \( q \), which specify the number of times the curve wraps around the torus in the longitudinal direction and the meridianal direction, respectively; such a knot is called a \( (p, \, q) \)-torus knot. Parametric equations for such a knot are given by: \[ \begin{eqnarray*} x(t) &=& \cos(qt) \cdot (3 + \cos(pt)) \\ y(t) &=& \sin(qt) \cdot (3 + \cos(pt)) \\ z(t) &=& \sin(pt) \end{eqnarray*} \] A \( (p,q) \)-torus knot is equivalent to a \( (q,p) \)-torus knot, and so we assume (without loss of generality) that \( p > q \). Realizing that a knot is a \( (p,q) \)-torus knot yields invariant information for that knot. For example, a \( (p,q) \)-torus knot has bridge number \( q \) (see [17]) and crossing number \( p(q - 1) \) (see [14]).

Another class of trigonometric knots are Lissajous knots, introduced in 1994 [2]. A knot is called Lissajous if it has a parameterization of the form \[ \begin{eqnarray*} x(t) &=& \cos(n_x \cdot t + \phi_x) \\ y(t) &=& \cos(n_y \cdot t + \phi_y) \\ z(t) &=& \cos(n_z \cdot t + \phi_z) \\ \end{eqnarray*} \] where the frequencies \( n_x, n_y, \) and \( n_z \) are positive integers and the phase shifts \( \phi_x, \phi_y, \) and \( \phi_z \) are real numbers. In [2], a particular invariant (the Arf invariant) of such knots is calculated and used to show that knots such as the trefoil knot, the figure-eight knot, and the cinquefoil knot have no such parameterization. In the same article, Lissajous parameterizations are given for many knots with crossing number between 5 and 10.

Naturally, one wonders if equations of other knots can be obtained if each coordinate function is a linear combination of cosine functions (rather than just a single cosine function). Such knots have a parameterization of the form \[ \begin{eqnarray*} x(t) &=& A_{x,1} \cos(n_{x,1} t + \phi_{x,1}) + \cdots + A_{x,i} \cos(n_{x,i}t + \phi_{x,i}) \\ y(t) &=& A_{y,1} \cos(n_{y,1} t + \phi_{y,1}) + \cdots + A_{y,i} \cos(n_{y,i}t + \phi_{y,i}) \\ z(t) &=& A_{z,1} \cos(n_{z,1} t + \phi_{z,1}) + \cdots + A_{z,i} \cos(n_{z,i}t + \phi_{z,i}) \\ \end{eqnarray*} \] and are referred to as both Harmonic knots and Fourier-\( (i,j,k) \) knots in the mathematical literature. By results on approximating functions with Fourier series, it can be shown that for every knot-type, there is a Fourier-\( (i,j,k) \) parameterization that yields a knot of that knot-type. The minimal values of \( i, j, \) and \( k \) that yield a parameterization of a given knot-type are an invariant of that knot-type, and are related to other knot invariants, such as the crossing number and superbridge number [19], [20]. Trigonometric identities are used to convert the standard equations of a torus knot into a Fourier-(3,3,1) parameterization in [8], where the question of "simplest" Fourier parameterization is posed. This question is answered in [7] in the case of torus knots, where it is shown that a torus-\( (p,q) \) knot has a Fourier-(1,1,2) parameterization: \[ \begin{eqnarray*} x(t) &=& \cos( pt ) \\ y(t) &=& \cos \left( qt + \frac{\pi}{2p} \right) \\ z(t) &=& \cos \left( pt + \frac{\pi}{2} \right) + \cos \left( (q - p)t + \frac{\pi}{2p} - \frac{\pi}{4q} \right) \\ \end{eqnarray*} \] Since torus knots are not Lissajous knots, these are the simplest Fourier parameterizations possible. Finally, the set of known Lissajous and Fourier parameterizations has recently been greatly enlarged by the equations given in [3].

In the figures that follow in "Gallery II," we include interactive three-dimensional graphics for three different torus knots. A description of the basic applet controls are given in the earlier section "Knot Software and LiveGraphics3D".

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