Lecture 1 (28 Sept 2015)

Chapter 1: Introduction

We begin with our everyday experience with physical knots: Take a piece of string and tie some knot in it. Independent of how complicated the knot you create is, you can — in principle — always undo it: Just reverse the steps you did to form it.

That is, all such knots are equivalent in the sense that given any two knots, one can be transformed into the other by knotting, i.e., without cutting or tearing the rope. This is not very interesting, and we therefore consider only knots which consist of some knotted string with the two ends glued together. When considering knots with the ends joined, there exist many inequivalent knots.

Examples 1.1:

unknot-3d

This is the simplest possible knot, called “the unknot”. Intuitively, it is not knotted at all. The next simplest example is called a “trefoil” (because of its shape):

trefoil-3d

If you make a rope model of the trefoil and experiment with it a bit, you might soon be convinced that it can not be transformed into the unknot. But how can you be sure? Could you prove that? Another example of a knot is a “figure of eight knot” as depicted below.

fig8-3d

To create it, take a piece of string, tie a figure eight in it, and join the ends.

There are many (actually, infinitely many) more knots. Here are three more examples:

3knots-3d

These more complicated knots don’t all have common names like “trefoil”, “figure 8”, just bureaucratic ones like  “$10_{117}$” according to a scheme that we will introduce later.

It has been known for a long time that knots look beautiful:

pdf_iconKnot Design

Exercise 1.2: Experiment with three-dimensional knots. 

For example, form a few knots out of string (or wire/cable, which better holds its shape). Make rope models of some of the knots depicted above. Can you transform one of the two knots below into the other? (Problem 3 d))

 

fig8+mirror

You can also use software to visualize and manipulate three-dimensional knots, such as the program knotplot which was used in the lecture (go to “Cat” (catalogue), then click on A,B,C to load some knots). This program has a free trial version which can be downloaded here. Another possibility is to use Wolfram’s “Knot Explorer”:

Knot Explorer from the Wolfram Demonstrations Project by Tom Verhoeff

The Knot Explorer works interactively if you have the corresponding browser plugins installed. Alternatively, you can use the knot explorer (and all other applets from the Wolfram Demonstrations Project) if you download the free CDF Player.

Also watch the video below to appreciate that a knot can look much more complicated than it actually is.

 

 

Although we have not given a mathematically precise definition of a knot yet, these pictures already give us a good intuitive understanding of the concept. In this course, we will study the properties of knots. Before entering into that, we might however well ask why we should do that. As explained in the lecture, knot theory has found applications in the following fields:

The mathematical definition of a knot

To arrive at a mathematical definition of what a knot should be, we have to abstract from the physical knots made of string or wire — in particular, we will forget about the material such knots are made of, and treat them as having zero width. Our definition should capture the idea that a knot is a closed loop in three-dimensional space without self-intersections.

Definition 1.3: A geometrical knot is (the graph of) a map $\varphi$ from the unit interval $[0,1]$ to three-dimensional space ${\mathbb R}^3$ such that

  • $\varphi(x)=\varphi(y)$ implies one of the three possibilities i) $x=y$ (then $\varphi(x)=\varphi(y)$ is trivial), or ii) $x=0$, $y=1$, or iii) $x=1$, $y=0$ (the last two mean that the ends are joined) This means in particular that the loop has no self-intersections.
  • $\varphi$ is smooth. (continuity of $\varphi$ corresponds to the knot being closed without gaps, and the stronger assumption of smoothness is needed to avoid pathologies of “infinitely knotted curves” — see the book of Cromwell for more details)

Although we will be mainly interested in knots, it will turn out to be necessary to study the more general links:

Definition 1.4: A geometrical link $L$ is the disjoint union of finitely many geometrical knots, which are called the components of $L$. A geometrical link of $n$ components is also called a geometrical $n$-link. In particular, a geometrical $1$-link is nothing else but a geometrical knot, i.e. every geometrical knot is a (special) geometrical link.

The adjective “geometrical” that appears in these definitions indicates that these knots/links are fixed, rigid curves in ${\mathbb R}^3$. They are not yet topological knots/links (defined in the next lecture), which will be flexible, “elastic” curves that can be deformed without changing them.

But let us first look at some examples of links (We drop the adjective “geometrical” for the time being). As in the case of knots, there exist trivial links, called “unlinks”. The unlink of two components is

2-unlink

Of course, there also exist unlinks with more than two components.

The first non-trivial 2-link is the so-called Hopf link:

Hopf link

 

In the table below(taken from the knotplot website) you see many more links, with two and three components, and different components in different colors:

links

As with knots, some of these links have common names: $4_1^2$ is known as Solomon’s Knot, and $5_1^2$ as the Whitehead Link. Another interesting one, is $6_2^3$, called “the Borromean Rings”. It has the property that if one removes one of its three components, the remaining two fall apart. Can you imagine that for any integer $n$, there exist $n$-links with this property? (Remove any component, the remaining $n-1$ fall apart). Puzzle Question: Come up with a drawing of such a link with four components. An animation of such a link with six components can be found in knotplot, under “DemoA > Knot theory > Brunnian”.