M Theory Lesson 36

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.

1 Response so far »

  1. 1

    Doug said,

    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


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