Knotty question

November 2, 2008 7:37 AM

I know zero about knot theory or topology, but here's a question that has always intrigued me: Given a pile of string on the floor, with two loose ends, can you determine whether pulling the ends will result in a knot or a straight piece of string simply by examining or counting the under/over crossings?

This question probably either falls into the category of "trivial" or "unsolved research problem".

posted by schrodycat to Science & Nature (12 answers total)

I believe what you are asking is essentially whether one can determine whether any knot is in fact the 'unknot': in which case if you solved it you would get yourself a Fields Medal (the "Nobel" for mathematics.)

Good luck. Most of the 'invariants' of Knot theory are derived essentially from counting the under and over crossings, so that's a good start.
posted by geos at 7:52 AM on November 2, 2008

It's not going to be something as simple as counting the over/under crossings. Consider a length of string with several well-spaced loops in it, some twisted in the "over" direction and some in the "under" direction. It's evident that you can put any number of "over" and "under" crossings in such a string, and it's equally evident that such a string will never have a knot in it.
posted by Johnny Assay at 8:01 AM on November 2, 2008 [2 favorites]

Response by poster: Johnny Assay: Yes, I agree you can get an arbitrary number of over or under crossings this way, but if I were to trace my finger along the string and give +1 for every overpass I come to and -1 for every underpass I come to each loop would always be zero in your example. But for a simple overhand knot, it's non-zero.
posted by schrodycat at 8:22 AM on November 2, 2008 [3 favorites]

Recent article about topology
posted by hortense at 9:37 AM on November 2, 2008

But for a simple overhand knot, it's non-zero.

It is? If I'm reading your proposal correctly, any crossing will involve one "overpass" and one "underpass", and the total will be zero for any knot. For example, look at the "stopper" knot in the Wiki diagram of the overhand knot. Starting from the "loose end" at the right-hand side of the diagram, you alternate +1, -1, +1, -1, +1, -1 — for a total of zero.

I suspect I'm misinterpreting what you have in mind by your procedure, though.
posted by Johnny Assay at 9:40 AM on November 2, 2008

Response by poster: Johnny: No, you are correct. I just miscounted. It's zero. Thanks for pointing that out.
posted by schrodycat at 9:44 AM on November 2, 2008

Best answer: Schrodycat: as Johnny Assay pointed out, the number you describe will always yield zero. There are versions that will do a bit better: if you only look at the overcrossings, and assign +1 for each overcrossing with the undercrossing going to the right, and -1 for undercrossing going to the left, then this does give a knot invariant... however, there should be many knots with this invariant zero that are not the unknot.

Honestly, if you want to learn about relatively basic knot theory, pick up a copy of Dale Rolfsen's "Knots and Links". It's a fairly easy read (assuming that you have a bit of a math background), with quite a few pictures in it. It goes over most of the topological background requried, if I recall correctly as well. My copy isn't with me at the moment to verify.

Anyhow, the gist is that the problem in question is quite hard.
posted by vernondalhart at 10:22 AM on November 2, 2008

If the two ends of the string are loose, then any mathematician will tell you that the string is not knotted. The reason is that we consider two knots to be equivalent if one can be obtained from the other by pulling, twisting, etc, as long as you don't do any cutting. If the two ends of the string are free, then you can always untie it without cutting (just start and one end and work to the other), so it will never be knotted in the mathematical sense.

So that means that you need to take your pile of string and tie the two loose ends together, and then your question makes sense. ie, given a knotted string with no loose ends, is there any way to tell from the number of over/under crossings that the knot is actually knotted? And as vernondalhart pointed out, that is a very difficult question to answer.
posted by number9dream at 1:20 PM on November 2, 2008

Of all hypothetical questions ... Have you tried knitting? It will eventually bring you to a practical situation of strong simile where will you be forced to actually figure it out. And being a little careful, you will. But only by doing, not by thinking. If the terrain and the theory doesn't match, the terrain wins.
posted by Wencke Braathen at 4:41 PM on November 2, 2008

umm.. dale rolfsen knots and links is only really understandable if you have a decent background in 3 manifold topology, which almost noone does...i think it's nice book (partly for that reason) but i think you are better off with murakami or even daniel fox (milnor's undergrafuate advisor)
posted by geos at 6:14 PM on November 2, 2008

Does it assume that much? I thought that the preliminary chapters went over most of the basics. I don't really remember, since I don't have it in front of me. Also, when I did read parts of it, we mostly skipped the earlier parts.

Well, anyhow, it's a pretty decent book. Maybe not too appropriate in this case though.
posted by vernondalhart at 7:32 PM on November 2, 2008

I took a course in Knot Theory in college during my brief tenure as a math major. It's a fascinating topic, but my brilliant prof made it hands down the best course I ever took.

We used Colin Adams' The Knot Book as our textbook. I highly recommend it. It's not even really meant as a textbook. As topology books go, it's rather light reading.
posted by spamguy at 8:13 PM on November 2, 2008