Talk:Wild arc

Two different versions of the wild arc?

The wild arc diagram on this wild arc page seems differently knotted to the wild knot on the wild knot page. The "wild arc" diagram on page 177 of the famous Hocking and Young "Topology" book agrees with the wild knot wikipedia page diagram, not with the wikipedia page wild arc diagram. The Hocking/Young book claims that their diagram illustrates the original Artin and Fox article.

I don't see how to continuously transform the wild arc and wild knot diagrams into each other locally. (Obviously it isn't possible globally.) But the local immersion differs in the style of crossings. on the wild arc page, each descending loop goes under/over/under the other lines, and when ascending, it goes over/under/over, in that order. But on the wild knot page, the descending loops go under/under/over and the ascending loops go over/under/under. I don't see any obvious way to continuously transform the diagrams to make them have the same crossings.

It seems at first glance that the alternating under/over/under and over/under/over diagram (wild arc) should be more "strongly knotted" in some sense. The other one seems like it could be easier to unravel in some sense.

If the two diagrams are not homotopically equivalent, that would suggest that the diagram on the wild arc page might not be an accurate version of the original Artin/Fox article because the Hocking/Young diagram is knotted like the wild knot page diagram.

Maybe the homotopy groups of the complements of these sets are equivalent in some sense. But I am also interested to know if one of the curves can be continuously transformed into the other.

Does anyone know what the facts of this case are?
--Alan U. Kennington (talk) 02:24, 18 May 2015 (UTC)

@Alan U. Kennington I looked at the original article from Artin and Fox. In it, they give not one, but five examples of wild arcs. Two of these wild arcs are described as being arcs in 3-space whose complement is not simply connected. They are called Example 1.1 and Example 1.1*. The image currently in the article is Example 1.1*. The image shown in Hocking and Young is Example 1.1, which is exactly the same except for the crossings being different. I think its fair to say that Artin and Fox's Example 1.1 is probably what most people think of as the Artin and Fox arc, so it should probably be added to this page, but I don't see why this page couldn't include more than one of their examples. Mathwriter2718 (talk) 11:47, 5 July 2024 (UTC)

@Mathwriter2718 I've forgotten the details now because it was 9 years ago that I last thought about this issue. However, what you say sounds very reasonable to me. The more examples, the better. I have no idea how to create that style of diagram though. It would take several hours of hard work to program such a diagram in MetaPost. If you have the enthusiasm to create the diagrams, I think you should do it.

There's a rough parallel here with Peano curves. What people call the Peano curve is usually the later curve by Hilbert, which is very different. But most people, including textbooks, get the naming wrong. I think wikipedia should present the historical truth, not perpetuate a historical falsehood, no matter how many textbooks commit the error.

— Alan U. Kennington (talk) 06:14, 6 July 2024 (UTC)

@Alan U. Kennington I agree that it would be best to put in both. I tried to modify the image on Wild knot using Inkscape to make the Artin-Fox Example 1.1 (just make it extend wildly in both directions instead of only one, and don't have the ends connect). However, I seem to be missing the magic touch, and my diagram looks bad. I wonder if the creator of that image knows something I don't, or if they just spent many hours toiling over Bezier curves until it looked perfect. Mathwriter2718 (talk) 12:06, 6 July 2024 (UTC)

@Mathwriter2718 Their diagram is not quite perfect. The curves on the left seem to be entirely half-circles, quarter-circle-pairs, and straight lines. But you can see where the line segments don't smoothly join into the half-circles. The right half of the diagram is created differently, with half-circles, straight lines, and possibly some Bezier curves, but maybe not. The "algorithm" on the right seems very different to the "algorithm" on the left. The right half of the curve looks like it's done by hand.

If I was doing it, I would create MetaPost continuous curves for both the left and right half. I think I would still stick with semi-circles and quarter-circles as much as possible. If you download the PDF file mpdemo.pdf.bz2 from my web site [1]http://www.geometry.org/tex/conc/mp/ you will see my attempts at wild curves in diagrams 3d31 to 3d34 on pages 225–226. These don't show the 3d structure very well. I think the gap-method shows the relative distance of curves much better. But creating the gaps in the curves is not completely trivial to do.

It would take a day's work to put together the algorithms to make the repeating pattern like in the diagram which is there right now. I'm not sure how popular the wild arc web page is. Maybe not very many people would benefit from such efforts.

— Alan U. Kennington (talk) 14:36, 6 July 2024 (UTC)