Five Lines

It can be shown (although it is not so easy as in the case of four lines) that any interlacing of five lines is isotopic to one of the seven interlacings shown in Figure 25. Six of them are nonamphicheiral and completely decomposable; they are given by the following symbols:

$$\langle +5\rangle ,\ \langle -5\rangle ,\ \langle \langle +3\rang... ...gle -2\rangle \rangle ,\ \langle \langle -3\rangle ,\langle +2\rangle \rangle ,$$

$$\langle +\langle 1\rangle ,\langle -2\rangle ,\langle -2\rangle \... ...le ,\ \langle -\langle 1\rangle ,\langle +2\rangle ,\langle +2\rangle \rangle .$$

The seventh is the interlacing in Figure 19. One can prove that the seven interlacings are not isotopic to one another by computing in each case the sum of the linking numbers of the ten triples contained in the interlacing. The results are indicated under the diagrams in Figure 25. This sum is clearly preserved under isotopy, and we see that the sums for the seven interlacings are all different.

Figure 25: