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Helium? |
Lead? |
Nickel? |
From Human Knot to DNA
1. Introduction
Have you ever played a game called �human knot� before? In the game, a group of about 10 people face each other to form a circle. Each player places both hands in the middle of the circle and each hand grasps another hand randomly. In this way, two or more circles may be formed. However, what we want is a single circle only. After ensuring only one circle is formed, the players need to untangle themselves, without letting go of hands, into an untangled circle. Here comes the problem: is it always possible to form an untangled circle?
In mathematics, there is also a branch called knot theory, which deals with distinguishing different types of knots. When talking about knots in mathematics, we assume that knots are closed, and consist of a single loop only. When there is more than one loop, we call it a link. Mathematicians have been trying to find a systematic way to untangle complicated knots into simpler ones for a century. It is the central problem in knot theory. Knot theorists are also curious whether two knots with different appearances are indeed the same, or we may say, are equivalent. One way to do so is by transforming the knots by stretching or flipping so that eventually the two knots look like the same. However, even if we fail to do so, it doesn�t necessarily mean the knots are different. It may happen that some �special techniques� are required.
II. History of Knot Theory
As early as 1794, Gauss had tried to use some theories about knots to investigate inductance in the theory of electromagnetism, which can be regarded as the beginning of knot theory. However, it did not raise much concern at that time.
In the nineteenth century, some people believed that atoms might have knot-like structures. Hence research and classification of knots progressed again.
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Helium? |
Lead? |
Nickel? |
It should not be difficult for one to be convinced that the following four knots are indeed equivalent. We call this equivalence class the trivial knot.
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Trivial knot |
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Another two simple examples are the trefoil knot and figure-8 knot.
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(right-hand) trefoil knot |
figure-8 knot |
III. Reidemeister moves
In 1932, Reidemeister discovered that any two equivalent knots could be transformed to each other through a series of three types of special movements, which are known to be Reidemeister moves.
TYPE 1 Reidemeister moves
TYPE 2 Reidemeister moves
TYPE 3 Reidemeister moves
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Let's demonstrate the moves by two examples:
1. A link with two components
2. trefoil knot
Readers may feel that the knots in the first step and the last step in each of the two examples above are clearly equivalent by direct observation. The introduction of Reidemeister moves seems to complicate the problem. In fact, rather than showing the equivalence of two knots, the power of Reidemeister moves lies in showing that knots are not equivalent. In the study of mathematics, we often investigate what properties are preserved during certain transformations. We call these properties invariants.
IV. Invariants in Knots
Let's do a simple exercise. Are the two knots in the following figures equivalent? Probably everyone knows they are not. Why? It is because they have different number of components.
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The number of components is an invariant in knots. Our common sense tells us that the number of components of a link will not change under transformations like lengthening and flipping. What we need to do in mathematics is to prove things rigorously. How can we show the number of components will remain unchanged under any transformation? To do so, we can use the Reidemeister theorem. By observation, the number of components is an invariant under the three types of Reidemeister moves. Therefore, any two equivalent knots have the same number of components and the number of components is a knot invariant.
Next, we would like to like whether the following knots are equivalent. You may think probably not. But again, why? It is not easy to explain although we believe it. In this case, both knots have a single component only and so we cannot use the number of components to show the non-equivalence of them. Here, we introduce another knot invariant, which is known as tricolorability.
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At each "crossing" of a knot, we may consider it as consisting of three strands.
We say that a knot is tricolorable if each of the strands can be colored one of three different colors A,B and C, so that at least two colors are used and at each crossing, either all the same color comes together or three different colors come together. As shown in the following figures, both the trefoil knot and the trivial link of three components are tricolorable.
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Readers can verify that tricolorability is preserved under Reidemeister moves. As a result, tricolorability is a knot invariant.
Since we can only color the trivial knot (in its simplest form, i.e. a circle) in one color only, the trivial knot is not tricolorable. Therefore the trefoil knot is not equivalent to the trivial knot.
There are many other knot invariants like crossing number, bridge number, unknotting number and linking number. Interested readers can find relevant information from the websites and books listed in the reference. In the next chapter, we will introduce a very useful knot invariant, the Jones polynomial.
V. Jones Polynomial
In 1984, Vaughan F. R. Jones discovered a relation between von Neumann algebra (a branch in mathematics which primarily deals with quantum mechanics) and knot theory. He introduced a new polynomial invariant, called the Jones Polynomial, which highly motivated the development of knot theory. Consequently, Jones was awarded the Fields medal in the 1990 International Congress of Mathematicians.
Before discussing Jones polynomial, we need to introduce the concept of orientation of a knot or a link first. Basically, it is simply a knot or a link with an orientation assigned to each component. For instance, we can assign two possible orientations to the trivial knot, namely the clockwise and anti-clockwise orientation. However, in this case, the two oriented knots are indeed equivalent.
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Jones assigns a �polynomial� of the form 