Thursday, April 11, 2013

Universality of peyote stitch (and linking, and looping, and knotted looping)

Peyote stitch
Knotted Looping

The map operation medial converts any map into a map that is 4-regular and chess-colorable. Once chess-colored, all the faces in a particular color, say black, are connected via their corners. Therefore, a map defined by placing a vertex in the middle of every black-colored face, with connecting edges iff the two faces touch at the corners, is connected. (Note that this 'new' map is either the original map or its dual, depending on the color choice.) Since this new map is connected, it has at least one spanning tree. If we circumnavigate this spanning tree while staying on the edges of the chess-colored map, the result is an Euler non-crossing circuit in the chess-colored map. This circuit is a possible working order to realize the surface in peyote stitch (or linking, or looping, or knotted looping.)

Every spanning tree in the primal map (or its dual) directs an Euler non-crossing circuit in the medial; each of these, in turn, directs a Hamilton circuit in the truncate of the medial (a.k.a., the bevel of the primal map, which is the same map as the bevel of the dual.)

Medial map (chess-colored)

Dual map
Triangular truchet tile for linking.

Triangular truchet tiles for looping.
If the surface is genus zero, the peyote stitch working order is represented by an undip word, since the beads (or links, or loops, or knots) are simply photon edges.

Primal map
Thus, multiple ways to weave, link, loop, or knot a given surface are directly derivable by map operations from any of its maps.

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