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Introduction to Knots Knot Theory Is a Huge Subject Withing Topology. We'll Sample Some of the Easier, Fun Parts. Let's Take

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Introduction to Knots Knot Theory Is a Huge Subject Withing Topology. We'll Sample Some of the Easier, Fun Parts. Let's Take Introduction to Knots Knot theory is a huge subject withing topology. We'll sample some of the easier, fun parts. Let's take a look at knots. What exactly is a knot? Well, informally it's a string that might wind around itself. But we have to be careful. If we leave the ends free, they can move around and the knot will untie itself. So we will define a knot to be a continuous image of a circle in three-dimensional space. We have to be careful that our continuous functions are nice. We don't want any infinite crazy things going on. We will only deal with \tame" knots, which are ones we could build with a finite number of straight sticks if we had to. Equivalently, we could require that the knot be made out of a tube of fixed finite diameter, instead of being infinitely thin. Avoid wild knots like this! We will study knots by looking at knot diagrams which are basically pictures of knots. These are going to be curves in a plane, except at a finite number of places the curve is broken to indicate that another part of the curve goes \over" it at that point. The above wild knot is shown as a knot diagram, and it should be pretty clear what's going on. To be specific, the rules are: 1) a knot diagram consists of a finite number of curves in the plane, each of which has two ends; 2) ends can only appear in pairs near each other, with a strand of the knot passing between them|that is, the only parts of a diagram that are not just curves are crossings which look like the picture below. You are not allowed to have another strand of the knot near a crossing, because we do not wish to be confused by what piece goes where. We also need to know some rules so that we can tell when two different pictures really represent the same knot. For instance, the picture below is \obviously" an unknot, a circle which is not really knotted at all. 1 An unknot Two knots will be considered the same if they are ambient isotopic which is a really fancy way of saying that you can stretch and rotate space so that the diagram of one becomes the diagram of the other. There's a lot of fancy defintions behind the scenes here. We're just going to concentrate on taking a string tied into one of the knots (with its ends fused together) and manipulting it so that it turns into the other knot. The rules we will use apply to knot diagrams, so that if two knots are the same, you can turn a diagram for one into a diagram for the other by a finite number of applications of these rules. These rules are called the Reidemeister moves and each changes part of a knot diagram. You can do (or undo) any of these moves and you have another diagram for the same knot. 2 For instance, to turn the diagram above into an obvious unknot, \unpoke" the bottom left, then \untwist" the top right. The third move, which re-arranges the crossings in a diagram by sliding one strand of a diagram across a crossing, is not necessary in this particular case. Try your luck on the diagram below, which was discovered by Kenneth Millett. It is a diagram for an unknot, but cannot be simplified (in terms of how many crossings there are, anyway) without first making it more complicated! Sometimes, we will consider not just knots, but links which are sets of knots, which may or may not intertwine with each other. These guys can also be represented with knot diagrams, and are also the same if two diagrams are connected by a sequence of Reidemeister moves. We might also give our knots or links an orientation, by drawing arrows on the diagram to indicate which direction we want to go around the knot (or each piece of a link). In general, it is very hard to tell from a pair of diagrams whether or not they represent the same knot. The above unknot is one example of this. As another, can you determine whether the simple trefoil knot is equivalent to its mirror image? How about the Figure-of-eight knot? If we orient them, is either equivalent to its reverse? 3.
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