Seven Lines

Interlacings of seven lines have been classified by Borobia and Mazurovski[8]. There are 74 types of interlacings of seven lines and 48 of these types are isotopy join. As in the case of interlacings of 6 lines, it turns out that Drobotukhina's polynomial invariant distinguishes all the 74 types.

A key observation which allowed Borobia and Mazurovskito obtain this classification was a possibility to move by an isotopy each interlacing of seven lines into a very special position. In this position the lines are projected to a plane onto extensions of sides of a convex polygon with seven sides, and the lines can be ordered in such a way that the line with number $i$ passes over all the lines whose numbers are greater than $i$. In other words, the first line lies over all the other lines, the second one passes over all the lines besides the first one, the third line passes over all lines with numbers greater than three, etc.