Deciding whether a curve comes from above or below

[siddhartha-gadgil]

The problems

• Consider two curves C and D with indices i and j so that the vertices of the curves at these indices coincide.
• Throughout, before and after are to be viewed cyclically.
• Let c0 and c1 be the edges of C before and after the coinciding vertex and let d0 and d1 be similar.
• Assume that both the curves are reduced, and both are not positive powers of the same curve (in the latter case we replace one by its inverse for computing intersection numbers).
• We have to decide whether the curve D comes in from above C at (or before) the given intersection. Note that we are not ruling out c0 == d0.
• Thus we have to define a boolean function fromAbove(C, D, i, j) of the curves and indices.
• We will define this recursively.

The function fromAbove

1. If c0 == d0 then we replace indices by i - 1 and j - 1 (cyclically) and compute. The hypothesis guarantees that this is not an infinite loop.
2. if d0 = inverse(c1), then we compute fromAbove(C, D, i, j) = !fromAbove(C, inverse(D), i, -j). This immediately gives the above case, so we do simplify.As the curves are reduced, we can only get this case if we have a single intersection point, and once we flip this way we will not have to flip again for the same intersection.
3. Finally, if neither of the above happen then rotate counterclockwise from inverse(c1) to c0 and return true if and only if we encounter d0 along the way.

[siddhartha-gadgil]

The above is wrong and unnecesarily complicated. In the case (2) we just take as non-crossing.

[siddhartha-gadgil]

Since this is wrong, it is closed. Please see http://math.iisc.ac.in/~gadgil/catg2020/notes/intersections/ for corrected sketch.