Playing around with simple trigonometric curves like
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.