On Sun, Jul 21, 2019 at 4:28 AM Brad Klee <bradklee@gmail.com> wrote:
Instead require that B2 is bisected by the inner Y edge.
Shouldn't one of the ravenous proof-checkers debate this point? Define W0 = B2Lg2Y1G1B1g1Y2G2B2R, the word associated to the usual Mobius action, bisected at B2. Depending on which yellow edge collides with blue, there are in fact two alternatives. Here is a depiction of the other: https://0x0.st/zfX4.JPG . (It looks kind of ragged, but at least the thread covers the rust and mangle. Sry, can't afford much more.) It is characterized by two non-inverting cycles: B2Lg2Y1 and B2RG2Y2. I did find the following Bridges article with nice drawings: https://pdfs.semanticscholar.org/4d8e/4d5b384e82808b0a87b2afddeab7e97ebfc6.p... The model with two non-inverting cycles is about the same as Figure 1b., after obstruction and YB-reconnection. It appears that changing the angle from 0 to -2*Pi, would reproduce the earlier obstruction in a slightly less contorted figure-eight form, yet homotopy-Eq. to the earlier geometry (after YB-reconnect). Either topology can be drawn in summary as a graph on four nodes. Both figure-eight surfaces are subsets of the 3D-immersed Klein bottle, this time with YB-reconnection along the entire self-intersection loop. (colored computer graphs anyone?) I talked to a Math Grad student studying for Quals from Allen Hatcher. He was quite surprised, and couldn't give a proper name. For now, I am content to call the geometries: "Self-obstructed Möbius strip of the first kind" and "Self-obstructed Möbius strip of the second kind" LOL. Even better, instead of saying "of the first kind": "Self-obstructed Möbius strip with three shortcuts". Under a slightly different integration, the three inverting cycle lengths are 3*Pi, 3*Pi, and 2*Pi, relative to total length 4*Pi. (prob. better to report the length metric in this symmetry.) And just for fun . . . If a character is hoping for a modicum of anonymity, this latest trash-wire, cheap-thread creation can also be worn as a mask: https://0x0.st/zfXJ.JPG https://0x0.st/zfXy.JPG This scene "ici" was shot today in Joëlle Storet's art studio. The paintings and mannequins are her work also. Cheers, Brad