Introduction

The first version of this paper was written as a popular text for high-school students. It was published [1] in the the journal ``Kvant'', the third issue of 1988, but in a shortened form. Then we expanded the article in order to encompass or at least mention some related questions. However we decided to keep the style of [1], in the hope that it would also be appreciated by a professional mathematician. We apologized to a reader, who would find the style irritating, and we mentioned that the material in the first two-thirds of the article (through the section on ``sets of five lines'') was announced in the note []3, while the final third of the article is written in a more traditional style.

The expanded version [2] of [1] was published in the first volume of a Russian journal Algebra i Analiz opening a new section ``Light reading for the professional''. English version of the paper became available in a translation made by N. Koblitz and published by American Mathematical Society in the first volume of Leningrad Mathematical Journal.

Unfortunately, the first volumes, even of the first rate journals, are not distributed as well as they deserve. Leningrad Mathematical Journal is an excellent journal, but we would like to bring our paper to more readers. During 10 years which passed since the time of writing [2] some questioned posed in [2] were solved and we decided to refresh the text and make it available to new readers. Partly we fix here some of defects of [2]: wrong pictures and few terminological inaccuracies of translation.

The article in ``Kvant'' was titled Interlacing of Skew Lines. This title sounds a little strange, doesn't it? The word ``interlacing'' suggests something flexible, not straight lines! To be sure, the title refers not to be the process of interlacing, but rather to the result. But is it possible to weave together skew lines which are situated in some clever way with respect to one another? At first glance this may seem not to be possible. Yet where do we get this impression? In daily life we never come across anything that really resembles a straight line. What bothers us is not that there is not such thing as an infinitely thin object--we are prepared to neglect the thickness--but rather that there is no such thing as an infinitely long object. Even light rays--which are models of linearity--become scattered and dispersed, and cannot be detected at a large distance. In practice one deals only with line segments.

Any set of disjoint line segments can be moved around to any other relative location in such a way that they remain disjoint. This we can see from experience, and it is also not hard to prove. We depict straight lines using line segments, and so it seems to us that straight lines cannot be woven together. But is that really the case?

First of all, let us give a more precise statement of the questions which concern us. The first question is: Can a set of disjoint lines be rearranged? But what do we mean by the term ``rearrange''? Here we shall not be concerned with the angles or distances between the lines. We shall consider the relative position of the lines to be unchanged if we move them in such a way that they never touch. But if one set of lines cannot be obtained from another set by such a movement, then we shall say that the two sets of lines are arranged differently.

The simplest lines for us to visualize are parallel lines. Clearly, any two sets of parallel lines with the same number of lines in each set have the same arrangement. In fact, if we consider the lines of one set to be ``frozen'' in place and then rotate the entire space, we can make them parallel to the lines of the other set; then, moving the lines of the first set one by one in such a way that they remain parallel and do not bump into one another, we can easily make them coincide with the lines of the second set.

We now consider arbitrary sets of lines. Can an arbitrary set of lines be moved (``combed'') into a set of parallel lines? This question has a simple and unexpected answer, which is hard to arrive at by considering concrete sets of lines. If you take a specific set of lines and study it for a while, you can probably find a way to make all of the lines parallel. But this does not give an answer to the question in full generality, because you undoubtedly made use of some specific features of your set of lines. Can one treat all possible sets of lines at once? It turns out that one can, and this is how. Let us take an arbitrary set of disjoint lines. We choose two parallel planes which are not parallel to any of the lines in our set. We fix the points of intersection of the first plane with the lines, fastening the lines at those points. We also fix the intersection of the lines with the second plane, but only as a point on that plane, which we allow to slide along the lines. In other words, we drill small holes in the second plane where it intersects with the lines. We then move the second plane away from the first one in the direction perpendicular to both planes. The lines are pulled through the little holes, and the angles which they form with the planes increase. If we move the second plane to infinity in a finite amount of time, then these angles all reach $90^\circ$, i.e., the lines become parallel to one another. This ``combing'' of our set of lines can be described as follows in a language which is more customary for geometry: we expand the space away from the first plane in a direction perpendicular to it, where the expansion factor increases rapidly to infinity in a finite length of time. Here the straight lines rotate around their points of intersection with the plane, and in the limit they become perpendicular to the plane.

Thus, one cannot have interlaced disjoint lines: all sets of disjoint lines have the same arrangement. But our title refers to skew lines, and so sets of parallel lines are excluded. There is a serious reason for this. Parallel lines are very close to being intersecting lines: one can move one of two parallel lines by an arbitrarily small amount so as to make them intersect. This is not the case for skew lines.

Since we have decided not to allow parallel lines, we must reexamine the question of which sets of lines have the same arrangement and which do not. We shall say that the arrangement of a set of lines remains the same if it is moved in such a way that the lines are always skew, never parallel. In what follows we will often be considering such movements of lines, and so it is useful to have a special word to refer to them. We shall use the word isotopy to denote such a movement of lines. If one set of lines cannot be obtained from another by means of an isotopy, then we say that the two sets have different arrangements. We shall also say that such sets of lines are nonisotopic.

The amount of difficulty in determining whether two sets of lines are isotopic depends most of all on the number of lines in the sets. In general, the more lines, the more clever one must be to find an isotopy which transforms one set into the other. We first treat the simplest case of the isotopy problem.