Playing around with simple trigonometric curves like

c(t)=((cos(2t)-a)cos(t), (cos(2t)+a)sin(t))

can be rewarding. The quatrefoil below is obtained for a=.2, and has a knotted curve as a lift.

The Björling surfaces happily spiral around these knots.

Now, here is a challenge: Does every knot occur as a curve on a minimal surface of finite total curvature? My guess is that probably yes, and it would certainly be more interesting if the answer is no.

##### Resources

Mathematica Notebook

PoVRay Source