Math, Symmetry, and Weaving: the 3 dual pairs of hypermap representations

As described in a previous post, the classical hypermap representations come in three dual-pairs. Using the nomenclature of that earlier post, the dual-pairs are: Belyi-James, Walsh-Chess, and Cori-Quad. I am going to refer to these pairs of hypermap representations as: Symmetry, Math, and Weaving. Here is why.

The 'Symmetry' pair: Belyi and James

Because they expose the full symmetry of hypermaps--hypervertices, hyperfaces, and hyperedges are interchangeable roles--these two representations are undoubtedly the most fundamental, but perhaps the least useful. In the Belyi (a.k.a., the canonical triangulation) and the James representations the six Lins trialities are just the permutations of three colors. That is too much symmetry be saying anything useful about something with less symmetry.

The 'Math' dual-pair: Walsh and Chess

Math is a monochromatic world. Color and color names are avoided if possible. These two hypermap representations, Walsh and Chess, get by with just two colors (black and white) by associating specific graph elements (faces, vertices, respectively) exclusively with the third color--which therefore never needs naming. Walsh and Chess are in fact the most common hypermap representations in the math literature. See Bernardi and Fusy for a use of Chess.

The 'Weaving' dual-pair: Cori and Quad

These two hypermap representations, Cori and Quad, are 4-regular (respectively, on vertices, and on faces) and are readable as weaving diagrams.