In the mathematical world, ideal knots are drawn with an infinitely thin line. Such lines can still fill a sphere (a la Thurston) but monkey knot curves with crossings are more interesting in the context of M theoretic quantum information, and it would take some (kind of) infinite number of crossings to properly fill out a sphere. But basically, the monkey knot is a set of Borromean rings in three dimensions (or Borromean ribbons). The rings form a 6 crossing planar diagram. Note that if the outer 3 crossings are smoothed, one obtains a trefoil knot from the centre of the rings (along with a separate unknotted loop). I can't help wondering what this means.