Amphicheiral and Nonamphicheiral Sets

Note that a triple of skew lines is never isotopic to its mirror image, while a pair of lines is isotopic to its mirror image. In general, we say that a set of pairwise skew lines is amphicheiral if it is isotopic to its mirror image; otherwise we say it is nonamphicheiral. Thus, a triple is always nonamphicheiral, and a pair is amphicheiral. The following questions arise:

1) Are there other values of $p$ such that any interlacing of $p$ lines is nonamphicheiral?

2) Are there other values of $p$ such that any interlacing of $p$ lines is amphicheiral?

3) For what $p$ does there exist a nonamphicheiral interlacing of $p$ lines?

4) For what $p$ does there exist an amphicheiral interlacing of $p$ lines?

Although this does not take us very far in the direction of an answer to our original question (of describing the set of interlacings of $p$ lines up to isotopy), it is worthwhile to take up these four questions. They are rough and somewhat superficial questions, but at the same time they have a more qualitative character. Because of this roughness and superficiality we can be confident of early success, and the result will undoubtedly be useful in our classification.

We do not yet have at our disposal very many tools for proving that a set is nonamphicheiral. But we do know that every triple is nonamphicheiral, and this is already a lot. After all, any set of more than three lines contains triples. Each triple changes its linking number in the course of a mirror reflection. Thus, if the interlacing is amphicheiral, then it must have the same number of triples with linking number $+1$ as with linking number $-1$. In particular, the total number of triples in the interlacing must be even. This simple argument leads us to the following unexpected result.

Theorem 1. If $p\equiv 3\mod4$, then every interlacing of $p$ lines is nonamphicheiral.

Proof. The number of triples in an interlacing of $p$ lines is equal to $p(p-1)(p-2)/6$, and this is odd if and only if $p\equiv 3 \mod4.$ $\qedsymbol$

Theorem 1 gives an affirmative answer to the first of the four questions above. The second question has a negative answer: for any $p\ge3$ one can construct a nonamphicheiral interlacing of $p$ lines. This also answers question 3). The simplest nonamphicheiral interlacings are shown in Figure 16 for $p=4,5$, and 6. It is easy to continue with this sequence of examples. All of the triples of lines in the interlacings in this sequence have the same linking number, and for this reason we know that the interlacings are nonamphicheiral.

Figure 16:

It remains to answer Question 4). We do not yet know whether or not there are amphicheiral interlacings of $p$ lines when $p\not\equiv 3\mod4$. It is convenient to consider separately the two cases: $p$ even, and $p\equiv1 \mod4$, although in both cases the question turns out to have a positive answer. In Figure 17 (in which $p=4$) we show the simplest example of an amphicheiral interlacing of $p$ lines with $p$ even. For any even number $p$, we take two sets of $p/2$ lines, one behind the other. The lines of the set nearest us are taken from the sequence of nonamphicheiral interlacings constructed above (see Figure 16). The other set of $p/2$ lines is obtained from the first by rotating and then reflecting in a mirror. How do we see that the interlacing in Figure 17 is amphicheiral? We move the set that is nearest us in such a way that the part of its projection which contains all of the intersections (in the projection) passes over and above the projection of the other set (Figure 18). If we then rotate Figure 18 by $90^\circ$ clockwise, we obtain the mirror image of the original interlacing.

Figure 17: An amphicheiral interlacing of 4 lines.

Figure 18: The same interlacing after sliding the nearest two lines to the left and down.

We now turn to the case $p\equiv1 \mod4$, i.e., $p=4k+1$. An amphicheiral interlacing with $k=1$ is shown in Figure 19. Four of the lines form two pairs which are situated as in the amphicheiral interlacing of four lines constructed above. The fifth line is placed so as to separate the two lines in each pair.

Figure 19: An amphicheiral interlacing of 5 lines.

An isotopy between this interlacing and its mirror image can be constructed as follows. We rotate the lines of the pair nearest us around the fifth line by almost $180^\circ$--until the lines of the other pair are in the way (Figure 20).

Figure 20:

We then move the fifth line so that its projection passes to the other side of the intersection (in the projection) of the lines that we moved before (Figure 21).

Figure 21:

It remains simply to look at the resulting interlacing from the opposite side. We do this by rotating it by $180^\circ$ around a vertical line (Figure 22). Now we see that we have the mirror image of the original interlacing.

Figure 22:

Using this example, it is easy to manufacture amphicheiral interlacings of $4k+1$ lines for $k>1$. Each line in Figure 19 except for the fifth is replaced by an interlacing of $k$ lines which either is taken from the sequence in Figure 16 or else is the mirror reflection of an interlacing in that sequence. This must be done in such a way that the interlacings which replace the lines of one of the pairs form an interlacing of the same type. There is no work needed to prove that the final result is an amphicheiral interlacing, since the required isotopy can be obtained in the obvious way from the one in the previous paragraph.