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)