5.1 Maximally Inflected Quintics With 9 Flexes

    Here, we draw all the rational plane quintics with 9 flexes at the points -8, -5, -2, -1, 1, 2, 5, 8, and infinity. There are 42 such curves in all. However, since the set of ramification points is fixed under the involution that sends s to -s, we may reparameterize any such curve (reversing the direction of parameterization) to obtain another. This either gives the same parameterized curve back (up to fractional liinear transformation in the plane), or it gives a different curve.

The images of the original curve and the reparameterization, as unparameterized curves in the plane are identical. An artifact of the method we use for drawing such curves is that the reparameterized curve is the image of the original curve, reflected about some vertical axis. Thus the curves which equal their reparameterization are symmetrical while those which do not are assymetrical.

Altogether, there are 6 symmetrical curves and 36 assymetrical curves with this choice of ramification. (In fact, this decomposition of 6 and 36 persists for any similar symmetric (under s maps to -s) choice of the 9 points of ramification.

Below, we display the 42 curves. We mark the positions of the flexes by red circles. Some curves have only 8 flexes displayed; the 9th flex is on the line at infinity, We indicate the positions of the solitary points by green crosses. Some curves have a yellow line those curves have a pair of complex nodes, and the yellow line is the real line on which the complex conjugate nodes lie. Lastly, the MAPLE files used to draw these pictures are found in the subdirectory 9flexes.maple/.

Consider first the curves with three solitary points. There are 2 symmetric such curves and 5 pairs of assymmetric curves, for a total of 12 curves with 3 solitary points.

The two first curves are symmetric. The one on the right has two flexes at its `nose'.
Next, there are three pairs of assymmetric curves. These 8 curves have three real nodes as well as three solitary points.
The chord diagrams of three are hexagons while five are asterixes.
	Crossings in middle
There are four more curves with three solitary points. These each have a pair of complex nodes.
These last two each have two pairs of flexes whose images are very close.