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, -class and
-class. Lines of the same class can
be transposed by a isotopy, while the lines of different classes
cannot. Denote the lines of the
-class by
and the lines
of the
-class by
. The orderings which cannot be transformed to
each other by isotopies can be enumerated by 4-letter words made
of letters
and
. Here are all 6 of these words:
,
,
,
,
,
. 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.