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 closed curve embedded in real Euclidean space E. At least, this is a first approximation of what we mean.

In fact, we want the knot to be tame, not wild. The adjective closed used above indicates that the curve has no end point and no starting point. But it could also be interpreted as the knot being a closed subset of E with respect to the usual topology on E. Thus, a knot is the image of a continuous embedding of the circle that is closed. The picture below shows that this is still not enough to guarantee that the knot is tame.

It would suffice to require that the embedding is smooth. A better way of avoiding degeneracy (for our purposes) uses tubular neighborhoods: 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 of the knot. A knot with this property is called tame (as opposed to wild). We will restrict attention to tame knots, and so, when referring to a knot we implicitly assume it is tame.

By the way, a segment of the knot is often referred to as a strand.

If we move the curve in a smooth manner, without ever self-intersecting it along the way, we do not want to think of the result as a different curve from the original one. Such a smooth deformation, during which the curve is the closed image of a continuous embedding of the circle at all time, is called an isotopy. Two knots are said to be isotopic if one can be obtained from the other by such a smooth deformation. Isotopy is an equivalence relation.

In every isotopy class of a knot, there is a representative consisting of a finite number of line segments. Such a knot is called piecewise linear. Moreover, given two piecewise linear knots that are isotopic, one can deform one to the other by means of a homotopy within a simplicial complex inside E in which both knots occur as 1-dimensional subcomplexes. Therefore, another approach to handle the nondegeneracy condition for knots is to consider piecewise linear knots and simplicial complexes only.

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 whether the unkot is a knot is similar to issue 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 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 isotopic. It is the quest for an algorithm that, on input two knots, decides the issue 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 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 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.

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 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.

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 node, 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 node and having three terms each, we find that the number of solutions is a power of 3.

A knot is said to be composed 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 (possibly trivial) L and K is called the connected sum of L and K and denoted L+K. If a knot is not trivial and not composed, then it is called prime.

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

It follows that a knot is trivial if and only if its genus is trivial.