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 $\left(x\,y\right)=\left(0\,0\right)$. 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.

$\mathrm{with}\left(\mathrm{algcurves}\right)\:$

$\mathrm{printlevel}≔2\:$

$\mathrm{plot\_knot}\left(y^2-x^3\,x\,y\right)$

$\mathrm{Number\; of\; branches:},1$

$f≔\left(y^3-x^7\right)\left(y^2-2x^5\right)$

$f≔\left(-x^7+y^3\right)\left(-2x^5+y^2\right)$

$\mathrm{plot\_knot}\left(f\,x\,y\,\mathrm{\epsilon }=0.8\,\mathrm{radius}=0.03\,\mathrm{color}=\left[\mathrm{blue}\,\mathrm{red}\right]\right)$

$\mathrm{Number\; of\; branches:},2$

$\mathrm{plot\_knot}\left(f+y^3\,x\,y\,\mathrm{\epsilon }=0.8\right)$

$\mathrm{Number\; of\; branches:},3$

$g≔\left(y^3-x^7\right)\left(y^2-2x^5\right)\left(y^2+2x^5\right)$

$g≔\left(-x^7+y^3\right)\left(-2x^5+y^2\right)\left(2x^5+y^2\right)$

$\mathrm{plot\_knot}\left(g\,y\,x\,\mathrm{\epsilon }=0.8\,\mathrm{radius}=0.03\,\mathrm{color}=\left[\mathrm{blue}\,\mathrm{red}\,\mathrm{pink}\right]\right)$

$\mathrm{Number\; of\; branches:},3$

$h≔\left(y^3-x^7\right)\left(y^3-x^7+100x^{13}\right)\left(y^3-x^7-100x^{13}\right)$

$h≔\left(-x^7+y^3\right)\left(100x^{13}-x^7+y^3\right)\left(-100x^{13}-x^7+y^3\right)$

$\mathrm{plot\_knot}\left(h\,x\,y\,\mathrm{\epsilon }=0.8\,\mathrm{radius}=0.03\,\mathrm{numpoints}=250\,\mathrm{color}=\left[\mathrm{blue}\,\mathrm{red}\,\mathrm{green}\right]\right)$

$\mathrm{Number\; of\; branches:},3$

$\mathrm{plot\_knot}\left(f+y^3\,y\,x\,\mathrm{\epsilon }=0.8\right)$

$\mathrm{Number\; of\; branches:},3$