The Plain Weaving Theorem again: The Vending Machine Algorithm

I am again writing about the Plain Weaving Theorem that Akleman, Chen, Xing, and Gross demonstrated by a topological proof in 2009. In essence the PWT states that every connected map specifies an enantiomorphic pair of plain-woven baskets. By plain-woven it is meant that every weaving element goes over-and-under in strict alternation—which is not exactly the way the term is commonly understood in the fiber arts. In knot theory terms, a plain-woven basket is an alternating link whose projection is properly embedded in a surface.

A connected map can be thought of as a connected drawing of lines and vertices on a surface such that, firstly, lines do not cross (other than at vertices;) and, secondly, cutting along all of the lines would cut the surface up into simply connected regions called faces. In other words, none of the cut-out pieces of surface would contain a hole or handle. Thinking about possible drawings on the surface of a tea cup (one with a handle) may clarify the above definition.

The PWT establishes an incredible ubiquity for weaving. There is left no firm foundation for considering maps, triangulations, dessins d'enfants, or any other familiar mathematical bricks, to be more fundamental building blocks of surfaces than weaving.

THE VENDING MACHINE ALGORITHM

Hoping to describe the practical application of the Plain Weaving Theorem in a memorable way, I introduce the Vending Machine algorithm for converting a map into a weave pattern:

Given a map drawn on a surface, place a vending machine midway between the two ends of each edge. Pedestrians will now fully short-cut the corners of the original faces. The pedestrian paths show the paths of the weaving elements. The pesky over-and-under weaving business at the vending machines is settled by choosing a left- or right-handed wood screw. Insert the screw in the middle of each face and incline the nearest weaving elements going around the screw to conform with the inclination of its threads. That being done, the over-and-under business takes care of itself, and you've got a basket.