A polynomial parametrization of torus knots

TL;DR

This paper presents an explicit polynomial parametrization of torus knots $K_{2,N}$ for odd integers $N$, using advanced mathematical tools like Stieltjes series and Padé approximants, with precise crossing point control.

Contribution

It introduces a novel explicit polynomial curve construction for torus knots $K_{2,N}$ with specific degree properties and crossing point configurations.

Findings

Provides explicit polynomial parametrizations for torus knots.

Uses Stieltjes series and Padé approximants in the proof.

Achieves control over crossing points in the polynomial curve.

Abstract

For every odd integer $N$ we give an explicit construction of a polynomial curve $\cC(t) = (x(t), y (t))$, where $\deg x = 3$, $\deg y = N + 1 + 2\pent N4$ that has exactly $N$ crossing points $\cC(t_i)= \cC(s_i)$ whose parameters satisfy $s_1 < ... < s_{N} < t_1 < ... < t_{N}$. Our proof makes use of the theory of Stieltjes series and Pad\'e approximants. This allows us an explicit polynomial parametrization of the torus knot $K_{2,N}$.