Talk:Wild arc

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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)[reply]

@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)[reply]

@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)[reply]

@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)[reply]

@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)[reply]

I've created a rough first draft of an imitation of the diagram in this article, but using MetaPost.

Instead of straight lines, half-circles and quarter-circles, I have used a single Bezier curve. Since this is generated from a MetaPost program, it would be fairly straightforward to generate variants. The main question now is whether this style is clear enough.

—Alan U. Kennington (talk) 09:06, 8 July 2024 (UTC)[reply]

And now I've created a PNG file of the Hocking/Young wild arc example using MetaPost in the same way. To facilitate comparison with Wild1.png, I have left/right mirror-reversed the Hocking/Young example.

Hocking/Young style wild arc. — Hocking/Young style wild arc using MetaPost.

If you follow the curve in Wild1.png from the top of the curve, moving to the right, the sequence of overpass/underpass points is over/under/over/under/over/under. But if you follow Wild3.png in the same way, the sequence is over/over/under/over/under/under. So they are topologically different.

Wild1.png is topologically equivalent to the current diagram on the "Wild arc" page. Wild3.png is topologically equivalent to the Hocking/Young book diagram.

—Alan U. Kennington (talk) 11:43, 8 July 2024 (UTC)[reply]

@Alan U. Kennington thanks, these are excellent and will really improve the article. I have three suggestions:

I think it would be more clear if both ends were shown to converge to clearly visible points instead of ending outside of the image.
It feels like a few of the crossings are a bit strange. The main one that sticks out to me is the one in the middle and slightly left towards the bottom of Wild1.png; the rest are less offensive.
Is it possible to make the thickness decrease slightly as the arc gets smaller?

Mathwriter2718 (talk) 11:56, 8 July 2024 (UTC)[reply]

@Mathwriter2718 These are all good suggestions that you make. I wanted to wait to see whether you approve of the overall style and look of the thing before putting more effort into the fine points. (It already took about 5 hours to get this far!) Since you seem to approve of the single-Bezier-curve style that I have used, I will improve the diagrams as you suggest. It's rather late here in Australia right now. So I'll work on this as soon as I can.

I'm a bit mystified by your note 2 though. I don't see anything odd about the Wild1.png crossing "in the middle and slightly left towards the bottom". Perhaps you mean that the gap is too wide? My choice of gap sizes uses intervals in the Bezier parameter space rather than in the 2d plane. So that's something for me to improve. What I implemented for the gaps for lines going "under" is a kludge. If you look at the source code, you will see that the task is not as simple as it looks.

The cut-off after about 5 loops on either side was because my algorithm produced bad results when there were 8 loops, for example. I'll work on reducing line width near the convergence points. But anything more than 8 loops on each side would be just a tiny blob with no detail anyway. The important thing is to communicate the basic idea. Mathematicians know how to imagine limit processes if they are given 5 elements of a sequence!

By the way, I don't have a copy of the Artin/Fox article and it doesn't seem to be free on the web. So perhaps you could indicate in more detail how that article differs from the wikipedia article and the Hocking/Young example. Alan U. Kennington (talk) 12:18, 8 July 2024 (UTC)[reply]

@Alan U. Kennington I looked at the MetaPost and indeed I appreciate your hard work making these. If you decide you've already put in enough effort, I would be happy to add these diagrams as-is to the page.

On my suggestion 1: I agree that it's ok if you only have 5 loops, but if you look at, say, the Hocking-Young diagram (their Fig 4-12), you will see that they have two clearly-visible points p and q capping off the arc. That's what I was trying to say.

On my suggestion 2: The crossing I am looking at has a gap much bigger on one side than on the other side. In terms of the Bezier parameter space, the gap starts a bit too early (or too late, depending on the direction of the curve). I bet it's easiest to make this look good via fiddling with your current gap drawing via the Bezier parameter space.

It is within JSTOR's policy for me to send you the Artin-Fox article, but annoyingly I don't see how to get it to you: I can't post it on a public forum and I don't want to dox myself by offering my email. If you are at a University, they might provide access.

The Artin-Fox article concerns the following examples. The examples are the main content of the article, though there are also a palmful of theorems. I find their definition of "wildly-embedded" confusing because I'm not sure if it implies topological embedding. An "arc" is just the image of a closed interval under a continuous map, and it is "simple" if it doesn't have self-intersections. Note that the complement of a tame arc is an open 3-ball.

Example 1.1: a wildly-embedded simple arc in the 3-sphere whose complement is not simply connected.
Example 1.1*: another wildly-embedded simple arc in the 3-sphere whose complement is not simply connected.
Example 1.2: a wildly-embedded simple arc in the 3-sphere whose complement is an open 3-ball. This one looks just like Example 1.1 but it only loops infinitely rightward, unlike Example 1.1 which loops infinitely in two directions.
Example 1.3: a wildly-embedded simple arc in the 3-sphere whose complement is simply connected but not an open 3-ball.
Example 1.4: a wildly-embedded arc in the 3-sphere which is the union of two tamely embedded arcs.
Example 2.1: a simple closed curve wildly-embedded in the 3-sphere which is the boundary of a disc even though the fundamental group of its complement is nonabelian.
Example 2.2: a simple closed curve wildly-embedded in the 3-sphere which is the boundary of a disc even though the fundamental group of its complement is $\mathbb {Z}$ .
Example 3.1: an embedded 2-sphere whose exterior is not simply connected (similar to the Alexander horned sphere but simpler)
Example 3.2: a wildly-embedded 2-sphere that divides the space it is embedded in into two open 3-balls.
Example 3.3: an embedded 2-sphere whose complement is simply connected but not an open 3-ball.

Mathwriter2718 (talk) 16:03, 8 July 2024 (UTC)[reply]

@Mathwriter2718 I will continue work on these diagrams to make them better in the ways you suggest. In particular, I've already been thinking about how to make the gaps more even in 2d space. It's not too difficult, but I omitted that kind of thing in my hurry to get something to show you as first drafts. I will change the gap drawing algorithm so that the half-gaps either side of the cross-over points are exactly the same, using intersections of circles with curves, etc. etc.

Indeed I did see the p and q in the Hocking/Young diagram, and I did want to include the limit points. That's what I usually do for the diagrams in my geometry book. However, the constant line thickness and the simplistic gap/redraw algorithm I used made the end points look a total mess. I will follow your suggestion about making the lines thinner near the ends. Part of my motivation to stick with constant line thickness was the fact that the current wikipedia article diagram does not have variable thickness lines. But my habit is to make line thickness variable near limit points, and then I magnify my diagrams enormously to investigate what would be visible under a microscope if the printer resolution was infinite. Of course, printer resolution is finite, and monitor resolution for wikipedia viewing is even more finite. But it's a habit which I usually follow anyway. It could be useful for people who can zoom in on Postscript images, which have infinite resolution.

Concerning email, I have a thousand email aliases. I'll allocate one for your use and then make it disappear as soon as the spam starts! Here it is: alan dot u dot kennington at mx128 dot geometry dot org.

Concerning all of those sphere examples, 3d diagrams are much more difficult. I once drew the Alexander horned sphere by hand to about 4 levels for a seminar I gave in 4th year mathematics at Uni. The audience couldn't even see the 3rd level! I'm sure I don't have the energy to try to draw such things in MetaPost. However, I could try anything which is not much more complex than what I have done already. In other words, just curves.

Anyway, after I've had my coffee and breakfast, and my 7th covid vaccination at midday, I will work on these diagrams again. Alan U. Kennington (talk) 01:01, 9 July 2024 (UTC)[reply]

@Alan U. Kennington I used a temporary email service to send you the article at that address. I hope it is useful for you. As for wild 2-spheres, they are quite interesting, but they aren't the subject of this article anyway. I don't know if I've ever seen one drawn in MetaPost! Usually I see the Alexander sphere drawn by hand. Good luck with the covid vaccine. Mathwriter2718 (talk) 01:13, 9 July 2024 (UTC)[reply]

@Mathwriter2718 If you refresh this web page, you should notice that I have updated the files Wild1.png and Wild3.png. I have made the gaps near the middle of the pictures shorter, and the gaps are now balanced with respect to the "overpass" curves because they are created by circles. By the way, the gaps in the Hocking/Young book are so tiny that I had to use a magnifying glass to see them.

Also, I have extended the loops to 14 on each side. This gives a clumsy jumble near the limit points. However, MetaPost does not have variable line width. So to kludge a variable line width would require multiple subpaths with decreasing thickness stepwise as the limit points are approached. I think it is probably best to leave the line thickness constant and let the curves become a clumsy jumble near the limit points. —Alan U. Kennington (talk) 05:05, 9 July 2024 (UTC)[reply]

@Alan U. Kennington I think they are beautiful! You can have the honors of adding them to the page. I'll do a bunch of edits afterwards. Mathwriter2718 (talk) 19:17, 9 July 2024 (UTC)[reply]

@Mathwriter2718 Okay. I have added the two diagrams. It's been about 10 years since I last added anything more substantial than typo fixes to Wikipedia. So my formatting is definitely not the best. I'll leave it to you to make it look more professional, and also to add the mathematical details about topology of the complement etc.

By the way, your email with the Fox/Artin article never arrived. However, I found it very quickly in a search engine. Someone left it lying around in their home page. So now I have it. Alan U. Kennington (talk) 02:38, 10 July 2024 (UTC)[reply]

@Alan U. Kennington thanks so much! I am getting really confused and concerned about the definitions of wildness in the literature, on this page, and on Wild knot, and whether they actually agree. I'll try to figure it out what's going on and edit this page and Wild knot accordingly, but it might take some time. Mathwriter2718 (talk) 18:05, 10 July 2024 (UTC)[reply]

@Mathwriter2718 Yes definitely it is confusing. I can't help with the theoretical aspects of wild embeddings because I'm really more into point-set topology, not combinatorial topology. It would be very valuable for Wikipedia readers to have this wild embedding issue clarified. Even the level of the 1948 Fox/Artin paper is beyond me. So I'll leave it to you! — Alan U. Kennington (talk) 01:00, 11 July 2024 (UTC)[reply]