Archive for July, 2009

Invariants

July 14, 2009

How can we tell if two tangles (or links, or knots) are different? That we cannot move the strings around as we are allowed and get from one tangle to another? We find invariants which can tell the difference. The best way to explain what an invariant is, is to give an example. The component number of a tangle is the number of strings in the tangle. Clearly if two tangles have different number of strings then they are not the same. For example the trefoil knot has component number 1 and the Hopf link has component number 2.

Trefoil Knot

Trefoil Knot

Hopf Link

Hopf Link

An invariant is some mathematical object, like a number or a polynomial that we can associate to tangles (or links, or knots) that depends only on the tangle-type. For instance the component number of a tangle doesn’t change when the strings move about or are stretched. Therefore, it is an invariant.

The component number is rather a blunt invariant. What if we want to tell the difference between tangles with the same component number? Let’s define an invariant for links with component number 2. We will call it the linking number. The linking number is actually an invariant for “oriented” links with component number 2. Oriented means that each string in the tangle comes with a preferred direction. We indicate this in a diagram by drawing an arrow on each string.

Oriented Hopf Link

Oriented Hopf Link

Whenever two different strings cross we can use the right hand rule to assign a positive or negative value to the crossing. Put your thumb in the direction (according to the orientation) of the over-strand and your fingers in the direction of the under-strand. If your palm is facing up (away from the screen) then it is a positive crossing and if your palm is facing down (towards the screen) then it is a negative crossing.

Signs of Oriented Crossings

Signs of Oriented Crossings

Now think of our link as having components (strings) called A and B. The linking number Lk(A,B) is the sum of the signs of the crossings in which A crosses over B. In order to see that the linking number is an invariant we need to analyze its behavior under Reidemeister moves.

Reidemeister 1

Reidemeister 1

Consider the first Reidemeister move. The left part of the equation has a crossing, but it comes from only 1 component, so it contributes 0 to the linking number. The same applies to the right part of the equation. The middle part of the equation has no crossings and so it contributes 0 to the linking number. Thus the linking number is invariant under Reidemeister 1.

Reidemeister 2

Reidemeister 2

Consider the second Reidemeister move. There are two cases, either the strands come from the same component or different components. In the first case, the left side of the equation contributes 0 to the linking number. In the second case, no matter which orientation there is on the strands, the two crossings have opposite signs and so contributes 0 to the linking number. In either case, the right side of the equation has no crossings and so contributes 0 to the linking number. Thus the linking number is invariant under Reidemeister 2.

Reidemeister 3

Reidemeister 3

Consider Reidemeister 3. Notice that each pair of strands cross in the same way but in different places on each side of the equation. Thus, no matter which components the strands belong to, nor which orientation we give them, each side contributes the same to the linking number. Thus the linking number is invariant under Reidemeister 3.

Thus, the linking number Lk(A,B) is an invariant of 2 component oriented links. Even better, it’s symmetric Lk(A,B)=Lk(B,A). So we can calculate it by summing the signs of the crossings where B crosses over A.

We can easily calculate the linking number for the oriented Hopf link pictured above, Lk(A,B)=-1.

What happens if we try to calculate the self-linking number of a knot Lk(K,K). Unfortunately it is no longer invariant under Reidemeister 1, since the argument we had used to prove invariance required that we were calculating linking number Lk(A,B) between different components A and B. You can see that the arguments for Reidemeister 2 and 3 did not require that the components were different so that the self-linking number, which we shall call the writhe Wr(K)=Lk(K,K), is invariant under Reidemeister 2 and 3. Furthermore, it does not depend on the orientation, since switching the orientation will not change the sign of a crossing (the orientation switches on both strands, so the sign is preserved).

In order to remedy the problem of non-invariance of the writhe under Reidemeister 1, we introduce a new property of tangles, “framing”. If orientation can be thought of as arrows going parallel to the tangle, then framing can be thought of as arrows going perpendicular to the tangle. If we extend the tangle along these arrows we obtain a “ribbon”, that is a tangle whose components are 2-dimensional surfaces. Now the self-linking number makes sense, as the linking number of the two edges of the ribbon.

We can project framed tangles in such a way that the ribbon is flattened in the projection. Then we need only draw the ribbon using a string as before and we can extend that string perpendicularly in the plane of projection. The resulting framing is called the “blackboard framing”. Such diagrams represent equivalent tangles if and only if they are connected by a sequence of Reidemeister 2 & 3 moves and the framed Reidemeister 1 move.

Framed Reidemeister 1

Framed Reidemeister 1

Notice that no matter what orientation is chosen, both sides have negative crossings. Since the sign of the crossing cannot change, the writhe is invariant under framed Reidemeister 1. Thus, the writhe, Wr(K), is an invariant of framed knots.

We have introduced two interesting new invariants, the linking number Lk(A,B) and the writhe Wr(K) but in order to do so we had to add more structure to tangles, orientation and framing. That these structures are natural as well as closely related is hinted at by our study of invariants. The linking number is sensitive to orientation but not framing and the writhe is sensitive to framing but not orientation. We will have more to say about these features of tangles in the future.

Reidemeister’s theorem

July 1, 2009

Despite living in 3-space our minds can only really grasp 2 dimensions since our eyes project the 3-dimensional world onto our  2-dimensional retinae. Nevertheless, we have a limited perception of 3 dimensions that comes from “layering” the different views of our retinae.

We perform a similar operation on tangles, projecting the inherently 3-dimensional objects onto a surface. Look at the example from the last post. It was necessary to draw it 2-dimensionally since computer displays are 2-dimensional. Nevertheless, we obtain a perception of the 3rd dimension by drawing “crossings” where one strand crosses over another. Try to locate the crossings in the example.

Example of a tangle

Example of a tangle

These projections are the most common way of representing tangles. They are called “tangle diagrams”. When we project a tangle diagram we take care to allow the only singularities (places where the projection doesn’t look nice) to be “transverse double points” which we represent as crossings. We don’t allow any of the following singularities: cusps, tangencies, or triple points.

Cusp

Cusp

Tangency

Tangency

Triple point

Triple point

We can guarantee that there are no such singularities by slightly tilting our projection if there are.

Now, how can we know if two tangle diagrams represent the same tangle? The answer is Reidemeister’s theorem: two tangle diagrams represent the same tangle if and only if they are connected by a sequence of Reidemeister moves. The pictures below demonstrate the 3 Reidemeister moves.

Reidemeister 1

Reidemeister 1

Reidemeister 2

Reidemeister 2

Reidemeister 3

Reidemeister 3

Looking at these pictures, it should be intuitively clear that performing Reidemeister moves does not change the tangle which a tangle diagram represents. The first Reidemeister move consists of adding or removing a “kink”. The second Reidemeister move consists of sliding strands past each other. The third Reidemeister move consists of moving a strand past a crossing. Look at the third move again and try to understand it physically: grab the middle strand and pull it through the crossing until it’s on the other side.

The difficult part of the theorem is proving the “only if” part, that is proving that the 3 Reidemeister moves suffice in order to transform one tangle diagram into any other tangle diagram which represents the same tangle. Notice, however that in the course of performing each of the Reidemeister moves we run afoul of our disallowed singularities. Perform a Reidemeister 1 on a physical tangle and as you get from one side of the equation to the other there will be a point in time where your projection is a cusp. Similarly, performing Reidemeister 2 will yield a tangency and Reidemeister 3 will yield a triple point.

One more important point is that Reidemeister moves are local. This means that if we have a large tangle we can perform Reidemeister moves on small pieces of the tangle. Let’s do an example to clarify.

Example of Reidemeister's theorem

Example of Reidemeister's theorem

We perform a single Reidemeister move locally in each equality. Try to identify where they occur.

Reidemeister’s theorem gives us the perfect tool for showing that two tangle diagrams represent the same tangle. Just perform Reidemeister moves to get from one diagram to the other. How can we show that two tangle diagrams represent different tangles? We may try to connect them via Reidemeister moves and fail, but that doesn’t show anything. Perhaps if we were smarter we could find the right sequence of Reidemeister moves. There’s an infinite number of such sequences so there’s no hope in testing them all. The answer to this puzzle is to look for invariants. But that’s the subject for another post!

My, what a tangled web we weave

July 1, 2009

Hi, my name is Eitan. I’m a student of mathematics. Welcome to my personal blog. I intend to post mathematical exposition, but since this is a personal blog I will also post political thoughts or anything else that comes to mind. Let’s start with some math.

What’s a tangle? Tangles are very important objects in topology. Physically, they are just a number of strings in our usual 3-dimensional space whose endpoints (if they have endpoints) are attached to some boundary surface. We allow the strings to move around freely in 3-space so long as the endpoints remain fixed and we cannot pass strings through each other (or over endpoints). We also allow the strings to stretch and compress as though they’re made of rubber. Here’s an example of a tangle:

Example of a tangle

Example of a tangle

Try to count how many strings there are.

Tangles are really generalizations of knots and links. A knot is a tangle made of only 1 closed string, meaning the string has no endpoints, it’s just a circle. A link is a tangle made of any number of closed strings. Notice that all knots are links and that all links are tangles. Another way of thinking about tangles is that they are “local” pictures of knots, that is, if we zoom in on a knot and look at a small neighborhood, that neighborhood will contain a tangle.

Let’s look at examples of knots and links.

Trefoil Knot

Trefoil Knot

Hopf Link

Hopf Link

By tracing along with your finger, verify that the trefoil knot has only 1 string while the Hopf link has 2 strings.

Recall that I said the endpoints of the strings in a tangle must be attached to some boundary surface. In the example the boundary surface comes in two pieces, a bottom and a top. This is one convention for tangles, the “monoidal category” convention. Another convention, the “planar algebra” convention, is that the surface has only one piece. Really, there’s no important difference between thinking in either convention. It’s only a question of convenience for a given application.

I think that’s enough for now. Stay tuned to hear about Reidemeister’s theorem.