stoutfellow (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
stoutfellow) wrote2005-05-15 08:42 pm
stoutfellow) wrote2005-05-15 08:42 pm
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The Knot That Can Be Untied Is Not The True Knot
A while back, ![[livejournal.com profile]](https://www.dreamwidth.org/img/external/lj-userinfo.gif)
ozarque mentioned knot theory on her LJ. By coincidence, a week or so later I attended a presentation on the subject. It's not my area, but I'd like to try to consolidate my knowledge, and one of the best ways of doing that is, of course, to try to explain it to someone else. Hence, this post. I'm not going to go into technical detail; I'd just like to describe some of the underlying ideas - which sets of tools have been brought to bear, and why.
There are several reasons why mathematicians have become interested in knots. There are significant applications in physics, and also in the geometry of three-dimensional objects; but more than that ("more" in the eyes of any pure mathematician, that is), they are beautiful and intriguingly complex in their own right - that is, they deserve study as ends, not merely as means. Now, one of the first questions one asks is, "When are two of these things 'the same', and how can we tell?" (Okay, that's two questions. Whatever.) I'll talk about the first question in this post, and a little bit about the second in another.
So, what do we mean when we say that two knots are the same? Obviously, the actual size of the knot doesn't matter. For that matter, the relative proportions don't matter either - if you stretch the knot in one direction but leave it alone in another, it's still essentially the same knot. But there's more; if you tie two strands together in a square knot and then pull one of the strands straight, you get something called a "lark's head", which looks quite a bit different, but it's really, basically, the same knot as before. Tweaking and tugging the knot, pulling and stretching it - these shouldn't change the underlying character of the knot. To a mathematician's eye, this sort of consideration immediately suggests that topology is the way to go to study knots. (Technically speaking, the sort of equivalence we're going to use is called "ambient isotopy" - "isotopy" for the kind of manipulations we're allowing, "ambient" because we're doing it all within ordinary three-dimensional space, rather than in some abstract context.)
But there's a hitch. The sorts of manipulations we're discussing include the ones we'd use to untie a knot; in other words, an ordinary shoelace knot is equivalent to a single unknotted strand; other knots can be reworked into two strands, or perhaps more. That's not very interesting. We resolve the problem by making the following demand: the loose ends of a knot should be thought of as spliced together. In other words, our knots are made up, not of strands, but of loops; to a mathematician, they are twisted circles, not twisted line segments.
There's another complication we have to add. A knot may consist of more than one of these twisted circles, and they may be hooked together - like the links of a chain, or of chain mail - or separate. Terminology varies, but I'll refer to these multi-piece objects as "links", and reserve the word "knot" for single loops. Whatever words one uses, though, knot theory discusses objects of both types. (Some of the theoretical reasons for making this last extension will come out in the next post. For now, though, just consider the fact that it may be hard to tell, looking at a knot, just how many strands - or how many loops - it involves.)
One last comment: a knot, as defined above, is thought of as embedded in three-dimensional space. Might we not try putting them in four dimensions, or even more? We could, but it turns out that nothing new and interesting would be gained by doing that. There's a theorem which says, roughly, that if an object is n dimensional and it's embedded in m-dimensional space where m is greater than 2n+1, then there's an embedding in 2n+1 dimensional space which has essentially the same properties. Since a circle (twisted or not) is a one-dimensional object, everything interesting that could happen will happen in three dimensions. (Why is a circle one-dimensional? Pick a point on the circle, and a direction - clockwise or counterclockwise. Then you can identify any other point on the circle by giving one number - it's so many units away from the base point, in the chosen direction. Similarly, the surface of the Earth is two-dimensional, because we can identify any point on it by giving two numbers, e.g., its latitude and longitude.)
Next time, I'll talk about knot invariants. (If you've read any of my earlier mathematical posts, you knew that invariants would be on the menu, didn't you?)
ozarque mentioned knot theory on her LJ. By coincidence, a week or so later I attended a presentation on the subject. It's not my area, but I'd like to try to consolidate my knowledge, and one of the best ways of doing that is, of course, to try to explain it to someone else. Hence, this post. I'm not going to go into technical detail; I'd just like to describe some of the underlying ideas - which sets of tools have been brought to bear, and why.There are several reasons why mathematicians have become interested in knots. There are significant applications in physics, and also in the geometry of three-dimensional objects; but more than that ("more" in the eyes of any pure mathematician, that is), they are beautiful and intriguingly complex in their own right - that is, they deserve study as ends, not merely as means. Now, one of the first questions one asks is, "When are two of these things 'the same', and how can we tell?" (Okay, that's two questions. Whatever.) I'll talk about the first question in this post, and a little bit about the second in another.
So, what do we mean when we say that two knots are the same? Obviously, the actual size of the knot doesn't matter. For that matter, the relative proportions don't matter either - if you stretch the knot in one direction but leave it alone in another, it's still essentially the same knot. But there's more; if you tie two strands together in a square knot and then pull one of the strands straight, you get something called a "lark's head", which looks quite a bit different, but it's really, basically, the same knot as before. Tweaking and tugging the knot, pulling and stretching it - these shouldn't change the underlying character of the knot. To a mathematician's eye, this sort of consideration immediately suggests that topology is the way to go to study knots. (Technically speaking, the sort of equivalence we're going to use is called "ambient isotopy" - "isotopy" for the kind of manipulations we're allowing, "ambient" because we're doing it all within ordinary three-dimensional space, rather than in some abstract context.)
But there's a hitch. The sorts of manipulations we're discussing include the ones we'd use to untie a knot; in other words, an ordinary shoelace knot is equivalent to a single unknotted strand; other knots can be reworked into two strands, or perhaps more. That's not very interesting. We resolve the problem by making the following demand: the loose ends of a knot should be thought of as spliced together. In other words, our knots are made up, not of strands, but of loops; to a mathematician, they are twisted circles, not twisted line segments.
There's another complication we have to add. A knot may consist of more than one of these twisted circles, and they may be hooked together - like the links of a chain, or of chain mail - or separate. Terminology varies, but I'll refer to these multi-piece objects as "links", and reserve the word "knot" for single loops. Whatever words one uses, though, knot theory discusses objects of both types. (Some of the theoretical reasons for making this last extension will come out in the next post. For now, though, just consider the fact that it may be hard to tell, looking at a knot, just how many strands - or how many loops - it involves.)
One last comment: a knot, as defined above, is thought of as embedded in three-dimensional space. Might we not try putting them in four dimensions, or even more? We could, but it turns out that nothing new and interesting would be gained by doing that. There's a theorem which says, roughly, that if an object is n dimensional and it's embedded in m-dimensional space where m is greater than 2n+1, then there's an embedding in 2n+1 dimensional space which has essentially the same properties. Since a circle (twisted or not) is a one-dimensional object, everything interesting that could happen will happen in three dimensions. (Why is a circle one-dimensional? Pick a point on the circle, and a direction - clockwise or counterclockwise. Then you can identify any other point on the circle by giving one number - it's so many units away from the base point, in the chosen direction. Similarly, the surface of the Earth is two-dimensional, because we can identify any point on it by giving two numbers, e.g., its latitude and longitude.)
Next time, I'll talk about knot invariants. (If you've read any of my earlier mathematical posts, you knew that invariants would be on the menu, didn't you?)
Just barely following along (maybe)
Re: Just barely following along (maybe)
Re: Just barely following along (maybe)
no subject
no subject
(Italics mine.) Based on this, my suspicion - and it's no more than a suspicion; applied math is not my area - is that such studies are ongoing, but that as yet they're in their infancy. (The Kauffman book was published in 1991, and a lot can happen in a decade and a half, though.)