plot knot - Maple Help

algcurves

 plot_knot
 make a tubeplot for a singularity knot

 Calling Sequence plot_knot(f, x, y, opt)

Parameters

 f - algebraic curve with a singularity at the point 0 x, y - variables opt - (optional) a sequence of options

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.

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{algcurves}\right):$
 > $\mathrm{printlevel}≔2:$
 > $\mathrm{plot_knot}\left({y}^{2}-{x}^{3},x,y\right)$
 ${\mathrm{Number of branches:}}{,}{1}$
 > $f≔\left({y}^{3}-{x}^{7}\right)\left({y}^{2}-2{x}^{5}\right)$
 ${f}{≔}\left({-}{{x}}^{{7}}{+}{{y}}^{{3}}\right){}\left({-}{2}{}{{x}}^{{5}}{+}{{y}}^{{2}}\right)$ (1)
 > $\mathrm{plot_knot}\left(f,x,y,\mathrm{\epsilon }=0.8,\mathrm{radius}=0.03,\mathrm{color}=\left[\mathrm{blue},\mathrm{red}\right]\right)$
 ${\mathrm{Number of branches:}}{,}{2}$
 > $\mathrm{plot_knot}\left(f+{y}^{3},x,y,\mathrm{\epsilon }=0.8\right)$
 ${\mathrm{Number of branches:}}{,}{3}$
 > $g≔\left({y}^{3}-{x}^{7}\right)\left({y}^{2}-2{x}^{5}\right)\left({y}^{2}+2{x}^{5}\right)$
 ${g}{≔}\left({-}{{x}}^{{7}}{+}{{y}}^{{3}}\right){}\left({-}{2}{}{{x}}^{{5}}{+}{{y}}^{{2}}\right){}\left({2}{}{{x}}^{{5}}{+}{{y}}^{{2}}\right)$ (2)
 > $\mathrm{plot_knot}\left(g,y,x,\mathrm{\epsilon }=0.8,\mathrm{radius}=0.03,\mathrm{color}=\left[\mathrm{blue},\mathrm{red},\mathrm{pink}\right]\right)$
 ${\mathrm{Number of branches:}}{,}{3}$
 > $h≔\left({y}^{3}-{x}^{7}\right)\left({y}^{3}-{x}^{7}+100{x}^{13}\right)\left({y}^{3}-{x}^{7}-100{x}^{13}\right)$
 ${h}{≔}\left({-}{{x}}^{{7}}{+}{{y}}^{{3}}\right){}\left({100}{}{{x}}^{{13}}{-}{{x}}^{{7}}{+}{{y}}^{{3}}\right){}\left({-}{100}{}{{x}}^{{13}}{-}{{x}}^{{7}}{+}{{y}}^{{3}}\right)$ (3)
 > $\mathrm{plot_knot}\left(h,x,y,\mathrm{\epsilon }=0.8,\mathrm{radius}=0.03,\mathrm{numpoints}=250,\mathrm{color}=\left[\mathrm{blue},\mathrm{red},\mathrm{green}\right]\right)$
 ${\mathrm{Number of branches:}}{,}{3}$

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.

 > $\mathrm{plot_knot}\left(f+{y}^{3},y,x,\mathrm{\epsilon }=0.8\right)$
 ${\mathrm{Number of branches:}}{,}{3}$

For more examples, including ones demonstrating the use of additional plot options, see examples/knots.