Thursday, December 4, 2014

Basket maps and duality

Definition: a map that is 4-valent and chess-colorable is termed a basket map.
A basket map can unambiguously direct the weaving of a basket when it has been chess-colored in two colors named "left-handed helicity" and "right-handed helicity." In weaving the basket, the four-valent vertices are crossings, the edges are the weaving elements, the faces are fabric openings, and the question,"Does this go over or under?" is settled by reference to the specified handedness of the fabric openings. 
More concisely, a map weaving strategy is: place a right-handed fabric opening at every vertex and place a left-handed fabric opening in the center of every face—or vice versa. 
An attribute more commonly discussed in the graph theory literature than chess-colorability is the closely related attribute of bipartite-ness. The dual of a basket map is a bipartite quad-faced map (BQFM) and vice versa. Given a suitably inclusive definition of quadrangulation, BQFM is synonymous with bipartite quadrangulation.

A map and its dual are just the two "parts" of a bipartite quad-faced map. We can generate the BQFM by applying the map operation radial to either the map or its dual. The dual pair can be recovered from the BQFM by applying inverse radial—a map operation that only operates on BQFM's. The action of inverse radial is to collapse quad faces into edges, conserving only one color of vertices in the process. The two possible choices of conserved color generate the dual pair.

Recall that the dual of any BQFM is a basket map. So, when we look at an arbitrary map we are looking at half of the holes in a basket!

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