Two Lines

We take any two pairs of skew lines, and try to decide whether they are isotopic. In this case it is perhaps too pretentious to use the word ``problem'', because it is completely obvious that we have an isotopy. Nevertheless, we shall make a detailed examination of the proof.

Using a rotation around a line which is perpendicular to both lines in one of the pairs, we can make the angle between the lines the same in both pairs; in fact, we can make both angles $90^\circ$. We note that the smallest line segment joining the two lines in a pair is the segment of the common perpendicular which is contained between them. We next bring the two lines closer together (or move them farther apart) along this perpendicular, so that the segments have the same length for the two pairs; after that we move one pair so that the segment between the two lines coincides with the segment for the other pair. We use a rotation around this segment to make one of the lines of the first pair coincide with a line of the second pair (this can be done because all of the lines are perpendicular to the segment). In the process the second lines of the pairs also come together. In fact, they both pass through a common point--an endpoint of the perpendicular segment--and are perpendicular to the same plane--the plane determined by the perpendicular and the first lines of the pairs (which now coincide). The proof is complete.

At the end of the proof, after we made the distances between the two lines the same for the two pairs, we moved a pair of lines in a rigid manner--without changing either the distance or the angle between them. The question arises: Suppose that both the distances and angles between the two lines are the same for two pairs of skew lines. Is it always possible to find an isotopy between the two pairs during which the distance and angle remain fixed? The previous argument shows that this question has an affirmative answer if the angle is $90^\circ$. However, if the angle is not $90^\circ$, then it may happen that after the isotopy in the previous paragraph the second lines in the pairs do not coincide. This unlucky case is illustrated in Figure 1. The second lines in the pairs form an angle whose bisector is parallel to the first (skew) lines, and the plane containing the second lines is perpendicular to the plane containing the bisector and the first lines. Thus, there was a good reason why we wanted to make the angles $90^\circ$ in the beginning of the above proof: for any other choice of the angle, the construction would not give the desired result. But this was not simply an artifact of our particular construction; it turns out that any two pairs of skew lines with equal distance and angle which do not coincide after the above construction cannot be made to coincide using any isotopy during which the distance and angle remain fixed. This is connected with a remarkable phenomenon, which we shall encounter often in the sequel. It merits a more detailed discussion.

Figure 1: