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Parametrization of torus knots

JE Jeffrey C. Everts
MR Miha Ravnik
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The sphere and torus are parametrized in the standard ways. A (p, q)-torus knot with tube radius a can be parametrized by R(u, v) = γ(u) + n(u) (a cos v) + b(u) (a sin v), with 0 ≤ u, v ≤ 2π, where γ(u) is the parametric curve

and n(u) and b(u) are normal and binormal unit vectors of γ(u), respectively. Inspection of Eq. (5) shows that a (p, q)-torus knot winds p times around the axis of rotational symmetry of a torus with inner radius r and outer radius R, and q times around a circle in the interior of this torus. The genus of a (p, q)-torus knot is given by g = (p − 1) (q − 1)/2, which is defined for a knot as the minimal genus of a Seifert surface of the knot (with a Seifert surface being a surface which has the knot, in this case γ(u), as a boundary).

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