Constructing Polynomial Knots

Shastri proved[3] that every knot can be expressed as the image of a parametric function t 7→ (x(t), y(t), z(t)), where x, y, and z are polynomials in t. However, it is difficult based on his proof to actually find a polynomial knot of a given knot type. We present an algorithm for converting a piecewise linear parameterization of a knot into a polynomial parameterization of a knot of the same type, and prove that it works in general. We also show how this algorithm can be modified to give compact polynomial knots and (in some cases) trigonometric knots. 1. Preliminaries Definition 1. Let κ(t) = (x(t), y(t)) be a continuous parametric function from R → R. A pair (a, b) is a crossing of x(t) and y(t) if a 6= b, x(a) = x(b), and y(a) = y(b). If I is an interval in the domain of κ, we say that a crossing (a, b) is in I if a, b ∈ I; similarly, if I and J are intervals, (a, b) is in (I, J) if a ∈ I and b ∈ J . Example 2. κ(t) = (t, t − t) has two crossings: (−1, 1) and (1,−1) (note that we consider these crossings distinct, even though they correspond to the same double point). κ(t) = (t, 0) has infinitely many crossings, at {(t,−t) | t ∈ R\{0}}. Lemma 3. Let κ : t 7→ (x(t), y(t), z(t)) be a continuous injective map from R → R, and let ẑ(t) be a continuous function with the property that, for all crossings (a, b) of x(t) and y(t), we have ẑ(a) > ẑ(b) if and only if z(a) > z(b). Then κ has the same knot type as κ̂(t) = (x(t), y(t), ẑ(t)). Proof. The knots have the same projection on the xy-plane; because of the conditions on ẑ, they also have the same crossings — in other words, they have the same diagram. Since a diagram determines a knot up to equivalence, κ and κ̂ have the same knot type. 2. The Näıve Algorithm In this section, we provide a first attempt at an algorithm for finding polynomial parameterizations of knots (note that we are dealing with knots whose ends go off to infinity, not compact knots). While the algorithm presented in this section is flawed and does not work in general, it contains the main ideas of the correct algorithm. At the heart of the algorithm is the following theorem, which allows us to replace a single component of a parametric function with a polynomial under certain conditions: