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algcurves

  

plot_knot

  

make a tubeplot for a singularity knot

 

Calling Sequence

Parameters

Options

Description

Examples

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 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.

Examples

withalgcurves:

printlevel2:

plot_knoty2x3,x,y

Number of branches:,1

fy3x7y22x5

f:=x7+y32x5+y2

(1)

plot_knotf,x,y,ε=0.8,radius=0.03,color=blue,red

Number of branches:,2

plot_knotf+y3,x,y,ε=0.8

Number of branches:,3

gy3x7y22x5y2+2x5

g:=x7+y32x5+y22x5+y2

(2)

plot_knotg,y,x,ε=0.8,radius=0.03,color=blue,red,pink

Number of branches:,3

hy3x7y3x7+100x13y3x7100x13

h:=x7+y3100x13x7+y3100x13x7+y3

(3)

plot_knoth,x,y,ε=0.8,radius=0.03,numpoints=250,color=blue,red,green

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.

plot_knotf+y3,y,x,ε=0.8

Number of branches:,3

See Also

plots[tubeplot]

 


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