First Week of Knot Theory in MasterMath Course on Quantum Groups and Knot Theory

This lecture (September 12, 2007) provides a first introduction to knot theory. The following notions are treated:

At the end you will find Exercises.

A knot is a simple closed curve embedded in real Euclidean space $E={ℝ}^{3}$. At least, this is a first approximation of what we mean.

To begin with, we want to eliminate the possibility of 'wild' knots which can become infinitely complicated, for example:

We can avoid such degeneracies by only considering knots which can be drawn with a finite number of straight line segments. These knots are called piecewise linear . Equivalently, we can use tubular neighbourhoods: demand that the knot has a tubular neighborhood that is not self-intersecting. This means that there is a positive constant $e$ such that the intersection of the knot with the ball of radius $e$ around each point of the knot is a single segment (or strand ) of the knot.

Any knot which is not wild is called tame , and in this course we will restrict our attention to tame knots. Notice that we can also avoid wild knots by assuming that our knot is a smooth embedding. Although this assumption is usually equivalent to the piecewise linear case, there are some areas of knot theory where it is necessary to keep in mind the distinction.

Next, we need to make rigorous our intuition that if we move the curve in a smooth manner, without ever self-intersecting it along the way, the result should not be a different curve from the original one.

As a first try, we could say that two knots are equivalent if there is an isotopy between them. An isotopy of a knot is a continuous deformation of the embedded circle. Unfortunately this definition means that any knot is equivalent to the unknotted circle by just pulling the string tighter and tighter until the knot disappears!

We can fix this problem by distorting all of the ambient space along with the knot. An ambient isotopy between two knots is a continuous deformation of Euclidean space $E$ so that the first knot is carried to the second. (See the glossary for a more precise description of these ideas.) So we consider two knots to be equivalent if they are ambient isotopic.

The embedding of a circle in $E$ as the boundary of a disk is called the unknot or the trivial knot . In many senses, it is the easiest of all knots. The question of whether the unknot is a knot is similar to the issue of whether zero should be counted as a natural number.

A link is the union of a finite number of disjoint knots in $E$. The comments on knots all apply for links after 'circle' has been replaced by 'disjoint union of a finite number of circles'.

In particular, two links are considered equal if they are ambient isotopic.

Many statements about knots also apply to links. This may be a good reason to extend the study to links. However, the most important rationale for including links is that certain operations that we will consider leave invariant the class of links, but not the class of knots.

The knots making up the link are called the components of the link.

The embedding of a set of $n$ circles in $E$ as the boundaries of a set of disjoint disks is called the trivial link on $n$ components.

The fundamental problem of knot theory is to determine, when given two knots, whether or not they are ambient isotopic. It is the quest for an algorithm that can take an input of two knots and decide whether one can be smoothly transformed into the other. Of course, the same question can be asked for links. A theoretic description of an algorithm determining if two knots are ambient isotopic has been given by Hemion.

A more modest problem of knot theory is to determine, when given a knot, whether or not it is ambient isotopic to the unknot. Haken has given a description of an algorithm for this purpose.

Both algorithm descriptions are too involved to program even for simple cases; see Hass.

In this course we mainly discuss the approach through invariants: functions defined on the set of knots, preferably easy to compute, for the purpose of distinguishing nonisotopic links from each other. The big issue then is to find a complete set of invariants, that is, enough in order to tell any two non-isotopic knots apart. In order to be able to compute with links at all, we work with link diagrams.

A link diagram is a planar graph whose nodes have valency four, supplied with a little extra information. The diagram is interpreted as a projection on the plane of a knot in 3-dimensional space. During the projection crossings may occur: points in the plane lying on the projection of two parts of the curves making up the link. The projection is always chosen so that no more than two parts project onto arcs going through the same point of the plane and the points lying in two projected parts are isolated. Conversely, given a planar graph of valency four, the link can be reconstructed by connecting opposite edges at each node and separating the two segments, except that it is not clear which part passes over and which goes under the crossing. This brings us to the little extra information alluded to: it tells us exactly that. Usually, the crossings are drawn in the plane in such a way that it is immediately clear which segment is going over and which is going under at each node.

Two link diagrams are said to be the same when they are equal up to planar isotopy. Here, a planar isotopy is a smooth deformation of the plane such that, at each stage of the deformation, the link diagram represent the same planar graph as the original link diagram.

Each link diagram represents a unique link. Each link can have many link diagrams representing it.

Of course, a knot diagram is just a link diagram corresponding to a knot.

A knot is said to be composite if, by a single cut with scissors, cutting two adjacent parallel strands of the knot diagram, and subsequently joining the ends at the same sides of the cut, we obtain two disjoint nontrivial knots. A link that is composed this way of two links $L$ and $K$ (possibly trivial) is called the connected sum of $L$ and $K$ and denoted $L+K$. If a knot is not trivial and not composite, then it is called prime .

The connected sum turns the set of links into a commutative monoid, with the trivial knot as its zero.

An orientation of a link is a direction of travelling along each of its components. If such an orientation is given, we speak of an oriented link . A link can be oriented in ${2}^{c}$ ways, where $c$ is the number of its components.

The signed Gauss code of a link is a particular encoding of a link diagram for algorithmic use. This is how it works for an oriented link.

The result is the so-called signed Gauss code.

The signed Gauss code is not unique. Not even after the link is oriented, the crossings are labelled, and the components are ordered. There is still the freedom of choosing the starting crossing for each component. By viewing the sequences as cycles (as in permutation cycles), this ambiguity disappears.

Not every Gauss code lookalike leads to a link diagram. An obvious necessary condition is that every label occurs exactly twice, but there are more.

How can we manipulate a link diagram without changing the isotopy class of the link? Kurt Reidemeister came up with three such changes that can be made to a diagram:

A Reidemeister move of type I, II, or III is a transformation of a link diagram by means of substitution of a subconfiguration of the knot as occurring at the top or bottom of a column in the above picture by the other subconfiguration in the same column, where the ends of the strands are matched to the link as they were before. The reflected versions of these rules also count as Reidemeister rules of the same type.

A Reidemeister move of type 0 is a transformation of a link diagram by means of substitution of a strand without crossings at the outside of the diagram by a strand without crossings at the other side of the diagram. This transformation is symbolized in the two pictures below.

A Reidemeister 0 move can be easily seen as a sequence of Reidemeister II and III moves, porting the strand to the other side by moving it under (or over, but stay with one choice) all other strands and crossings. It can be interpreted by viewing the link diagram as embedded in the 2-dimensional sphere rather than the Euclidean plane, by just closing up the plane with an additional point at infinity. The result of the Reidemeister 0 move corresponds to letting the point at infinity cross the strand that is moved.

Peter Vos (TU Eindhoven) wrote a Java program to help understand the Reidemeister moves. If you have Java installed, it can be (web)started by clicking on the Knotweaver link.

It is easy to see that if a link diagram ${D}_{1}$ is obtained from the link diagram ${D}_{2}$ by a Reidemeister move, then ${D}_{1}$ and ${D}_{2}$ represent the same link. The surprise of the result below is that the converse also holds.

The nontrivial implication is a rather straightforward but tedious proof. We omit it here.

Seifert circles form a step towards the construction of a surface whose boundary is a given link. Start from an oriented link diagram. At each crossing follow each arc coming into the crossing and join it to the adjacent arc leaving the crossing. The resulting strands are disjoint circles, called Seifert circles .

A Seifert surface of a link $L$ is an orientable surface whose boundary is $L$.

Here, a surface is the closure of a 2-dimensional subset of $E$ such that the intersection of a small enough sphere around each of its points with the surface is a disk. It is called orientable if the orientation of a disk around an inner point never changes after following a continuous closed path containing the point.

Jack van Wijk's pages provide a pleasant introduction to Seifert surfaces and many ways to visualize them. For example, the Seifert surface for Stevedore's knot created above can be visualised as follows:

Notice that there is no unique Seifert surface for a link. Even by using the algorithm given in the proof above, called Seifert's algorithm , you will produce different surfaces for different link diagrams of the same link.

A link invariant is a map defined on the set of links that is invariant under ambient isotopy. The origin of this name lies in the problem of defining such maps and computing the image of a link under the map. Often, Reidemeister's theorem is used for this purpose as follows.

Reidemeister's theorem then allows us to conclude that the map is actually defined on links. So, a link invariant is often given as a map on link diagrams. The importance of such maps is that it may help to tell two links apart. Some link invariants are defined on oriented links instead of ordinary links. The number of components is a very simple example of a link invariant. The number of tricolorings of a link diagram is a relatively simple example of a link invariant.

Markov's theorem can be also used to define link invariants, by use of braids instead of link diagrams.

The crossing number of a link diagram is simply the number of crossings appearing in that diagram. The crossing number of a link $L$ is the minimum over all crossing numbers of link diagrams representing $L$.

The number of components of a link is a link invariant. This invariant is not helpful in telling knots apart.

By definition, the crossing number is a link invariant. But there is no explicit method of generating, for a given link, all link diagrams with crossing numbers less than a given number. So in terms of telling two knots apart, the crossing number is of no great use. Still it is an intuitively appealing quantity and is often used in classifications of lists with a limited number of crossings.

The genus of a surface is a well-known invariant. Together with its orientability (a Boolean variable) and the number of boundary components, it characterizes the surface up to homotopy.

A Seifert surface is orientable and has just as many boundary components as it has link components. So it is determined by its genus.

The genus of a surface with boundary is, by definition, the genus of the closed surface obtained by capping off each boundary component with a disk.

The genus of a link is the minimal of all genera of Seifert surfaces for the link.

It follows that a knot is trivial if and only if its genus is zero. It also follows that knots cannot have additive inverses: you cannot add two knots together in such a way that they cancel each other and create the unknot.

Using an orientation of a link diagram we can orient its crossings as follows. If, along the travelling direction of the upper segments, the under segment is oriented right-to-left, the crossing is positively oriented and we will say that the sign of the crossing is 1. Otherwise, it is negatively oriented and the sign of the crossing is $-1$.

The writhe of an oriented link diagram is the sum of the signs over all crossings.

Clearly, the writhe of an oriented link diagram changes under a Reidemeister I move.

The writhe is not an invariant as it changes under Reidemeister I moves. By selecting those crossings that involve two distinct components of a link, we guarantee that the self-crossing of Reidemeister I is never counted. The linking number of an oriented link diagram is defined as half the sum of the crossing signs, taking only crossings involving two distinct components of the link diagram. Notice that this is an integer. As an immediate consequence, we have a link invariant.

This invariant does not distinguish between knots.

An arc of a link diagram is a connected component of the link diagram in the plane; it is bounded by ends where the link is not drawn as it passes under a crossing. A tricoloring of a link diagram is a coloring of all of its arcs with (at most) three colors in such a way that, at each crossing, an odd number of colors appears.

At each crossing three arcs pass. Label the colors by the number 0, 1, 2 modulo 3. The requirement that, at a given crossing, an odd number of colors appears means that the labels of the arcs passing through that crossing add up to 0 modulo 3. Therefore, solving the resulting linear equations, one for each crossing and having three terms each, we find that the number of solutions is a power of 3.