Siefert Surface for Trefoil

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This is a Seifert surface for the trefoil knot defined by the equation in spherical coordinates: 4 r cos(p) = 4 cos(p) + sin(3t) sin(p), with the restriction: | sin(3t/2) tan(p) | < 1/2 Here, r (rho) is the distance from the origin, p (phi) is the angle from the positive z-axis (latitude), and t (theta) is the angle in the xy-plane from the positive x-axis measured counterclockwise. I'd never seen a closed-form equation for a Seifert surface before, so I decided it would be fun to figure one out. How I Created This I used Mathematica to create this thing. While I had an equation for the surface, Mathematica's plotting algorithm works best with a parametrization. The surface is not isomorphic to any part of the plane, so a single parametrization wouldn't be possible. I ended up parametrizing the surface in five parts. The top and bottom patches were parametrized by solving for r as a function of p and t. The three twists were parametrized by solving for t as a function of r and p. The five patches smoothly connected. The domain for t and p.