The Linking Number

Any semi-oriented pair of lines has a characteristic which takes the value $+1$ or $-1$. It is called the linking number. This number is preserved under isotopies, and so if two semi-oriented pairs of lines have different linking numbers, then they are not isotopic. Here is the definition of the linking number. The most economical way of aligning an oriented line with a second oriented line which is skew to it is to place it alongside a common perpendicular to the two lines and then rotate it by the smallest angle that brings it to the same direction as the second line. Here the line rotates either like the right hand around the thumb, or like the left hand (Figure 5). In the first case the linking number is $-1$, and in the second case it is $+1$.

Figure 5:

To help the reader familiar with algebraic topology make the right connection, we give a second equivalent definition of the linking number of a pair of oriented skew lines. Through one of the lines we draw a plane which intersects the other line. We place our right hand so that our thumb rests on the second line and passes through the plane in the direction determined by the orientation of the line, while rotating in the direction our fingers point. On the plane we obtain an oriented circle which is traced by the tips of our fingers. The orientation of the circle may be the same as the orientation of the first line (Figure 6) or different (Figure 7). In the first case the linking number is $+1$, and in the second case it is $-1$. Figure 8 will enable the reader to see that the two definitions of the linking number are equivalent.

Figure 6:

Figure 7:

Figure 8:

It is clear that changing the orientation of one of the lines of the pair changes the linking number. Hence, if the orientation of the pair is changed to the opposite orientation (i.e., the orientation is reversed on both lines), then the linking number does not change. In other words, the linking number is an invariant of a semi-oriented pair: it depends only on the semi-orientation. If we look at the reflection of our pair of oriented lines in the mirror (Figure 9), the linking number changes.

Figure 9:

We now return to the unfortunate situation we encountered when looking for an isotopy between two pairs of skew lines which preserves the distance and angle between the lines (see Figure 10). At the time we could not

Figure 10:

answer the question of whether the sets are isotopic (Figure 11). Now, however, we see that these pairs (with their canonical semi-orientation) are obtained from one another by a mirror reflection, and so they have different linking numbers. Thus, they cannot be connected by an isotopy which preserves the distance and angle between the lines. But if two pairs have the same distance and angle and also the same linking number, then they can be connected by such an isotopy.

Figure 11:

By the way, it is possible to modify the notion of the angle between two skew lines in such a way as to incorporate the linking number and thereby make it unnecessary to work with the linking number separately. The angle between two lines was defined above so as to be in the interval $(0^\circ,90^\circ)$. define the modified angle between two lines to be the product of the angle in the earlier sense and the linking number, if the latter is defined (i.e., if the angle is not $90^\circ$), and to be the angle in the earlier sense (i.e., $90^\circ$) if the linking number is not defined. The modified angle is in one of the intervals $(-90^\circ,0^\circ)$, $(0^\circ,+90^\circ]$. The sign can be determined from the right hand rule, without saying anything about the linking number.

We have thereby completely analyzed the situation with sets of two skew lines.