Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Let's reconsider the graphs that we viewed at the beginning of this unit. We saw that by counting the number of faces, vertices, and edges, we could find the Euler characteristic of a particular graph. Furthermore, we found that for any graph that we can draw on a plane, the Euler characteristic is 2. What about the following graph, though?
This graph obviously can be drawn on a flat piece of paper, and yet we are going to have a tough time finding its Euler number. Counting edges and vertices is easy, but counting the faces presents a challenge. This difficulty is due to the fact that there are edges that intersect one another. On the flat piece of paper, this simply looks like one edge overlaying another edge. We can't actually find the Euler characteristic of this graph, because there is no way to draw it on the plane without edges intersecting each other. This graph is actually non-planar; that is, it can't be embedded in the plane.
Remember the Flatland explorer? After completing her explorations, she discovered that the red thread and the blue thread never crossed each other. The reason for this was because her surface had a hole in it. We can use this property of the torus to embed our non-planar graph without any intersections:
The problem we encountered before was one of embedding. On the plane, this graph can't exist without edges crossing one another, but on the surface of a torus, it can. Notice in the image above that the connections that make up the graph have not changed; the only difference is the surface upon which the graph is drawn.
Embedding refers to how a topological object—a graph, surface, or manifold—is positioned in space. The concept of embedding is central to the idea of an extrinsic view of topology simply because we cannot view something from the outside unless it is somehow situated in some larger, or higher-dimensional, space. Otherwise, from where would we be viewing it? Furthermore, there can be many different ways to embed an object in that larger space.
As with our graph above, we occasionally encounter objects that cannot be embedded in our space without a self-intersection, a fact that technically means they can't be embedded in our space at all. Such structures are called "non-orientable surfaces," and we will learn more about them in the next section. For now, let's look more closely at how an object with the same intrinsic topology (i.e., the same connections) can have different embeddings.
Let's take another look at the subway loop from our earlier example.
We are treating our subway as a one-dimensional manifold. Remember, this means that when we ride the train, we perceive only forward or backward motion, even though we know that our subway is a loop because we keep coming back to the same stop. From our intrinsic perspective, the actual subway map could be any of the embeddings shown above.
The subway map represents an embedding of our 1-manifold in a two-dimensional plane. By looking at the map, we view the manifold extrinsically. The designer of the map has many choices as to how to draw it, provided that the order of the stops remains the same. Intrinsically, all of these possibilities are the same, but each version of the map is different extrinsically. Some of these maps can be turned into one another by bending and stretching, but some of them can't.
This version of the map is unlike the others. It is plain to see that no matter how much we manipulate it, we cannot transform it into a circle without making a cut and re-gluing the ends. However, recall that, experienced intrinsically, this configuration is no different than a circle. Obviously, from an extrinsic view, this equivalence no longer holds.
This configuration is an example of a knot. A knot, to a topologist, is simply a particular embedding of a circle in 3 dimensional space—also known as 3-space. It may appear that these knots are embedded in the plane, but recall that in the plane there is no such notion as "above" or "below." Clearly, we need these directional concepts in order to have knots. All knots, when viewed intrinsically, are the same; they become interesting, really, only when we look at them extrinsically.
Some knots are easily undone, such as the one shown in image A. Sliding the overlying side to either the right or the left creates what can, topologically, be considered a circle. Other knots, such as that shown in image B, are a bit more difficult, though not impossible, to undo. In this case, sliding the bottom overlying half-loop down a bit, then sliding the middle overlying part to the right creates what looks like a circle within a circle. Finally, a mere twist of the remaining overlying part again creates a topological circle.
Unfortunately, if we try to perform the same types of manipulations, called "Reidemeister moves," on knot C, we will be out of luck. A little mental projection should convince you that clearing up one part of the knot will only make things worse in other parts. This type of knot, known as a "trefoil knot," cannot be undone in the extrinsic view of topology. However, as we saw before, in the intrinsic view, this is really no different topologically than a circle. The only way to undo this knot would be to un-embed it, that is, take it out of our space, untangle it, and then re-embed it in our space. A four-dimensional being would have little trouble doing this, but we'll save our examination of the exploits of four-dimensional beings for a later unit. For now, all that is important is that we cannot undo it in 3-space.
Central to this study of knots is the concept of isotopy. Isotopy is a form of equivalence in which one topological object can be transformed into another while maintaining the property of being an embedding. Although one needs to be careful in defining it, it is a precise way to capture the notion of deforming without crossing. This is what we are doing when we use our Reidemeister moves to undo knots. Hence, we would say that knot A is isotopic to knot B and that both are isotopic to a circle. Knot C, however, is isotopic to none of these things, because we would have to un-embed it to undo it.
The mathematical study of knots has applications to the scientific study of DNA. DNA is the genetic material that encodes the information that is the blueprint for living livings. DNA is basically a very long strand of alternating pieces of genetic material called "nucleotides." Information is encoded in the DNA molecule by the specific ordering of these nucleotides.
When biologists and geneticists are trying to define the specific sequence of nucleotides in a strand of DNA, they first must break the molecule up into smaller pieces. These pieces often form loops and knots similar to what you see here:
This structure resembles the kinds of knots that we were studying earlier. This DNA knot is considered to be "packed" in a form unsuitable for replication. Before the DNA can be copied, it must be "unpacked" by helper molecules known as enzymes. This process proceeds in a fashion similar to the undoing of mathematical knots. In fact, we can use concepts from knot theory to understand and make predictions about DNA packing and unpacking. This, in turn, enables us to make predictions about how certain enzymes will function.
This structure resembles the kinds of knots that we were studying earlier. This DNA knot is considered to be "packed" in a form unsuitable for replication. Before the DNA can be copied, it must be "unpacked" by helper molecules known as enzymes. This process proceeds in a fashion similar to the undoing of mathematical knots. In fact, we can use concepts from knot theory to understand and make predictions about DNA packing and unpacking. This, in turn, enables us to make predictions about how certain enzymes will function.
The picture above shows various DNA knots. Any place where a knot crosses over itself is called a double point. The number of double points is known as a knot diagram's "crossing number." What's more, each double point is classified as either positive or negative, depending on which way the overlying strand must be turned so that it lines up with the underlying strand. If a clockwise turn of less than 180° will bring about an alignment, then the double point is considered "positive"; conversely, if a counterclockwise turn of less than 180° is sufficient to bring the strands into alignment, then the double point is considered "negative."
With each double point "worth 1" (either +1 or -1, as just discussed), the sum of all the values of a knot's double points is called its "writhe." Certain enzymes are able to reverse the sign of particular double points, thereby allowing the knot to be undone, that is, the DNA to be unpacked (as shown in part B of the following diagram).
By comparing the crossing numbers and writhes of the same DNA knot after successive applications of the enzyme gyrase, genetic researchers were able to conclude that gyrase systematically reverses the signs of double points in a DNA molecule.
The application of principles of knot theory, itself a subset of extrinsic topology, to DNA enzyme analysis represents an interesting example of a branch of mathematics that was originally studied for its own sake, as topology mostly is, having unexpected applications in another field. We will explore some other applications of topology, specifically intrinsic topology, a little later in this unit. Before we proceed, however, we must take a look at an entire class of topological objects that we have not yet discussed. These strange objects, in which the concepts of left and right are meaningless, are the non-orientable surfaces.
Next: 4.6 Non-Orientability