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.
Arcadian Functor 
Doug said,
April 13, 2007 @ 12:25 am
Hi Kea,
Ribbons and braids may be more powerful when non-planar or at least 3D.
The Protein Data Bank [PDB] has several examples available, including a molecule of the month. Various molecules are assigned a 4_alphanumeric code and viewable at different sites with different resolutions.
http://www.pdb.org/pdb/home/home.do
Mt favorite is Jmol which often uses NMR data. An example:
http://molvis.sdsc.edu/fgij/fg.htm?mol=1xi4