In the paper From Dominoes to Hexagons
, D. Thurston proves a nice little theorem. That is, consider a disc with $2n$ endpoints marked on the boundary, such as we see in planar operad diagrams. Then all $n!$ pairings of points marked either in
(-) or out
(+) are described by a loop free diagram of triple points
, which are 6-valent intersections of three strands. Take a look at Thurston's pictures of planar tilings made from diamonds.
Thurston was looking at how planar algebras require more generators than the Temperley-Lieb
algebra, which they contain.