make a tubeplot for a singularity knot
plot_knot(f, x, y, opt)
algebraic curve with a singularity at the point 0
(optional) a sequence of options
epsilon=value -- the radius of the sphere. The default is 1. In some cases a smaller number must be chosen for the picture to be correct.
color=list -- specifying a list of colors results in a plot where each branch gets its own color.
The options for tubeplot can be used as well. In plot_knot these options have the following default values: numpoints=150, radius=0.05, tubepoints=5, scaling=constrained, and style=surface.
Let f be a polynomial in x and y giving an algebraic curve in the plane C^2 with a singularity at the point x,y=0,0. The output of this procedure is called the singularity knot of this singularity. This knot is defined as follows: By identifying C^2 with R^4 the curve can be viewed as a two-dimensional surface over the real numbers. This procedure computes the intersection of this surface with a sphere in R^4 with radius epsilon and center 0. The intersection consists of a number of closed curves over the real numbers. After applying a projection from the sphere (which is three-dimensional over R) to R^3 these curves can be plotted by the tubeplot command in the plots package. Such a plot gives information about the singularity of f at the point 0. See also: E. Brieskorn, H. Knörrer: Ebene Algebraische Kurven, Birkhauser 1981.
The curve given by f need not be irreducible, but f must be square-free otherwise this procedure does not work.
If printlevel > 1 the number of branches will be printed to the screen. Each branch (i.e. place above the point 0) corresponds to one component in the knot.
printlevel ≔ 2:
Number of branches:,1
f ≔ y3−x7⁢y2−2⁢x5
f ≔ −x7+y3⁢−2⁢x5+y2
Number of branches:,2
Number of branches:,3
g ≔ y3−x7⁢y2−2⁢x5⁢y2+2⁢x5
g ≔ −x7+y3⁢−2⁢x5+y2⁢2⁢x5+y2
h ≔ y3−x7⁢y3−x7+100⁢x13⁢y3−x7−100⁢x13
h ≔ −x7+y3⁢100⁢x13−x7+y3⁢−100⁢x13−x7+y3
This is the same knot as above, but it looks different because the projection point is different now that x and y are switched. This is the command to create the plot from the Plotting Guide.
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