Interlacings of Labeled Lines

Of course, the classification considered in the previous section, as well as the ones above, does not take into account any order of lines. We consider unordered interlacings, in which lines are not numerated or labeled. A classification of ordered interlacings is also possible. Mazurovskiand Pavlov [9] have classified ordered interlacings of up to seven lines.

When the number of lines is less than 4, the result does not differ from the classification in unordered case. Indeed, by an isotopy one can change arbitrarily the order of lines.

In the case of 4 lines the number of isotopy classes of ordered interlacings is 8. In a non-amphicheiral interlacing any two lines can be transposed by an isotopy. Therefore the two non-amphicheiral isotopy classes of unordered interlacings do not split when we take into account an order of lines. So there are exactly two isotopy classes of ordered interlacings of 4 lines. In an amphicheiral interlacing of 4 lines, the lines are divided into two pairs of isotopic lines, $+1$-class and $-1$-class. Lines of the same class can be transposed by a isotopy, while the lines of different classes cannot. Denote the lines of the $+1$-class by $A$ and the lines of the $-1$-class by $B$. The orderings which cannot be transformed to each other by isotopies can be enumerated by 4-letter words made of letters $A$ and $B$. Here are all 6 of these words: $AABB$, $BBAA$, $ABAB$, $BABA$, $BAAB$, $ABBA$. Together with the 2 classes of non-amphicheiral interlacings mentioned above, the corresponding 6 amphicheiral ordered interlacings of 4 lines give totally 8 isotopy classes of ordered interlacings of 4 lines.

Seven isotopy classes of unordered interlacings of 5 lines split into 64 isotopy classes of ordered interlacings of 5 lines.

In the case of 6 lines, the 19 classes discussed above split into 1066. In the case of 7 lines, 74 classes split into 43400.