Knot Theory 2: Reidemester to the Rescue
May. 20th, 2005 03:59 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowI'm going to try to explain this next bit without any pictures. That may be a mistake; the texts I've seen on the subject all come profusely supplied with diagrams. In particular, the "Reidemeister moves" I try to describe below are usually depicted rather than described. I hope I've made things clear enough; if not, it may help to play around a bit with rubber bands and bits of string. (If it still isn't clear, feel free to complain; I'm willing to try again...)
The knots and links we've been talking about sit in three-dimensional space. That's a bit of a drawback when it comes to discussing them in print, since the printed page thus far remains two-dimensional. For these purposes, then, we usually work not with the links themselves, but with two-dimensional pictures of them. These pictures consist of curves in the plane - on a sheet of paper, if you like - some of which cross each other. At the crossings, one curve passes under the other (and we'll assume that we never have three curves crossing at the same point); the usual convention is to have the curve that's passing under "break" at the crossing. These pictures are "plane representations" of the links, and on paper or on the blackboard, they are what we work with. (It may help to think of them as shadows.)
Unfortunately, this raises a new problem. The same knot or link will have many different plane representations. For example, consider the simplest possible knot, a circle without any twisting at all. (This is often called "the unknot", which is a convenient name if a misleading one - it is a knot, just as zero is a number.) The most natural plane representation is simply a circle: O. But (remember that the circle is sitting in three dimensions) we could give it a half-twist, and its shadow would look like a figure-eight: 8. (We have to decide which strand passes under the other at the intersection point, but that's no big deal.) The plane representations look rather different, but they represent the same knot. Again, consider an unlinked pair of circles. One plane representation would just be two circles, with no crossing points: O O. But another would have two crossing points, as one circle passes under the other. (If you have trouble picturing this, you might take a couple of rubber bands and lay one on top of the other.) How do we deal with this?
The first tool is the two-dimensional analog of ambient isotopy. To describe this, let me give a couple of ad hoc definitions. (These aren't standard terms; I just want to have words available.) A strand is any segment of a knot, or the curve corresponding to it, and an arc is the portion of a knot lying between two adjacent crossing points. (So the figure-eight has two arcs, and the second representation of the two-circle link has four.) Now, our two-dimensional isotopy allows us to manipulate arcs pretty much as we please, except that we mustn't change the crossing points, neither adding nor removing any of them. That's not going to be enough, as we saw above; two plane representations of the same knot or link may have different numbers of crossing points.
To make up for the gap between two-dimensional and three-dimensional isotopy, we need to introduce the "Reidemeister moves". There are three types of Reidemeister move.
Type I: If an arc begins and ends at the same crossing point, it can be straightened out. Conversely, a straight arc can have a loop added to it. (This move relates the representations O and 8; either arc of the figure-eight could be "untwisted", producing a simple loop.)
Type II: If two strands cross each other twice, with one of the strands passing under the other both times, and there are no other crossings in between (on either strand), the two strands can be separated. (This applies to the two representations of the two-circle link.)
Type III: This one's more difficult to describe. Suppose there are three strands - call them A, B, and C - which outline a triangular shape. That is, A crosses B and, at the next crossing, crosses C; meanwhile, over on one side of A, B and C cross, and that's the next crossing for both of them. Suppose also that A passes under B and C. Then A can be slid over to the other side of the B-C crossing.
Reidemeister proved the following. Suppose that L1 and L2 are two links, and that P1 and P2 are plane representations of L1 and L2 respectively. Then L1 and L2 are isotopic (in three dimensions) if and only if it's possible to transform P1 into P2 by some combination of two-dimensional isotopy and Reidemeister moves. That means we can safely work with the plane representations, without worrying that we're somehow missing some manipulation that's possible in three dimensions but isn't available in two.
Deep breath
Okay. With that in place, we can go on to the problem of constructing invariants. Stay tuned, if you're still interested.
The knots and links we've been talking about sit in three-dimensional space. That's a bit of a drawback when it comes to discussing them in print, since the printed page thus far remains two-dimensional. For these purposes, then, we usually work not with the links themselves, but with two-dimensional pictures of them. These pictures consist of curves in the plane - on a sheet of paper, if you like - some of which cross each other. At the crossings, one curve passes under the other (and we'll assume that we never have three curves crossing at the same point); the usual convention is to have the curve that's passing under "break" at the crossing. These pictures are "plane representations" of the links, and on paper or on the blackboard, they are what we work with. (It may help to think of them as shadows.)
Unfortunately, this raises a new problem. The same knot or link will have many different plane representations. For example, consider the simplest possible knot, a circle without any twisting at all. (This is often called "the unknot", which is a convenient name if a misleading one - it is a knot, just as zero is a number.) The most natural plane representation is simply a circle: O. But (remember that the circle is sitting in three dimensions) we could give it a half-twist, and its shadow would look like a figure-eight: 8. (We have to decide which strand passes under the other at the intersection point, but that's no big deal.) The plane representations look rather different, but they represent the same knot. Again, consider an unlinked pair of circles. One plane representation would just be two circles, with no crossing points: O O. But another would have two crossing points, as one circle passes under the other. (If you have trouble picturing this, you might take a couple of rubber bands and lay one on top of the other.) How do we deal with this?
The first tool is the two-dimensional analog of ambient isotopy. To describe this, let me give a couple of ad hoc definitions. (These aren't standard terms; I just want to have words available.) A strand is any segment of a knot, or the curve corresponding to it, and an arc is the portion of a knot lying between two adjacent crossing points. (So the figure-eight has two arcs, and the second representation of the two-circle link has four.) Now, our two-dimensional isotopy allows us to manipulate arcs pretty much as we please, except that we mustn't change the crossing points, neither adding nor removing any of them. That's not going to be enough, as we saw above; two plane representations of the same knot or link may have different numbers of crossing points.
To make up for the gap between two-dimensional and three-dimensional isotopy, we need to introduce the "Reidemeister moves". There are three types of Reidemeister move.
Type I: If an arc begins and ends at the same crossing point, it can be straightened out. Conversely, a straight arc can have a loop added to it. (This move relates the representations O and 8; either arc of the figure-eight could be "untwisted", producing a simple loop.)
Type II: If two strands cross each other twice, with one of the strands passing under the other both times, and there are no other crossings in between (on either strand), the two strands can be separated. (This applies to the two representations of the two-circle link.)
Type III: This one's more difficult to describe. Suppose there are three strands - call them A, B, and C - which outline a triangular shape. That is, A crosses B and, at the next crossing, crosses C; meanwhile, over on one side of A, B and C cross, and that's the next crossing for both of them. Suppose also that A passes under B and C. Then A can be slid over to the other side of the B-C crossing.
Reidemeister proved the following. Suppose that L1 and L2 are two links, and that P1 and P2 are plane representations of L1 and L2 respectively. Then L1 and L2 are isotopic (in three dimensions) if and only if it's possible to transform P1 into P2 by some combination of two-dimensional isotopy and Reidemeister moves. That means we can safely work with the plane representations, without worrying that we're somehow missing some manipulation that's possible in three dimensions but isn't available in two.
Deep breath
Okay. With that in place, we can go on to the problem of constructing invariants. Stay tuned, if you're still interested.
![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
no subject
Date: 2005-05-21 04:31 am (UTC)no subject
Date: 2005-05-21 02:47 pm (UTC)A__..__B
.....\/
...../\
.....\/
A__/\__B
where strand A passes below strand B both times. A Type II move allows us to change that to this:
A__......__B
.....\..../
......\../
....../..\
A__/....\__B
or vice versa.
As for Type III, there are three strands involved:
B..........C
..\......./
A_\___/__A
.....\../
......\/
....../\
.....C..B
and A passes below B and C both times. Then we can slide A down below the B-C crossing:
.....B..C
......\/
....../\
...../..\
A_/ __ \ _A
../.......\
B..........C
A still passes below B and C, and we don't change the B-C crossing - whichever was on top before stays on top.
I hope that's clearer - both that the asciigraphs don't get messed up, and that they do clarify!
no subject
Date: 2005-05-21 04:22 pm (UTC)no subject
Date: 2005-05-21 07:51 pm (UTC)