Knots - Maple Help
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Examples of Knots

The following example worksheet shows various examples of knots visualized using the plots:-tubeplot and algcurves:-plot_knot commands.

Unknot

The unknot can be defined by the following parametric equations:

x=sin⁡t

y=cos⁡t

z=0

> 

plots:-tubeplot([cos(t),sin(t),0,t=0..2*Pi],
   radius=0.2,axes=none,color="Blue",orientation=[60,60],scaling=constrained,style=surfacecontour);

The Trefoil Knot

The trefoil knot can be defined by the following parametric equations:

x=sin⁡t+2⁢sin⁡2⁢t

y=cos⁡t+2⁢sin⁡2⁢t

z=sin⁡3⁢t

> 

plots:-tubeplot([sin(t)+2*sin(2*t),cos(t)-2*cos(2*t),-sin(3*t),t= 0..2*Pi],
   radius=0.2,axes=none,color="Green",orientation=[90,0],style=surface);

The Figure-Eight Knot

The figure-eight can be defined by the following parametric equations:

x=2+cos⁡2⁢t⁢cos⁡3⁢t

y=2+cos⁡2⁢t⁢sin⁡3⁢t

z=sin⁡4⁢t

> 

plots:-tubeplot([(2+cos(2*t))*cos(3*t),(2+cos(2*t))*sin(3*t),sin(4*t),t=0..2*Pi],
   numpoints=100,radius=0.1,axes=none,color="Red",orientation=[75,30,0],style=surface);

The Lissajous Knot

The Lissajous knot can be defined by the following parametric equations:

x=cos⁡nx⁢t+φx

y=cos⁡ny⁢t+φy

z=cos⁡nz⁢t+φz

Where nx, ny, and nz are integers and the phase shifts phix, phiy, and phiz are any real numbers.

The 8 21 knot (nx=3, ny=4, and nz=7) appears as follows:

> 

plots:-tubeplot([cos(3*t+Pi/2),cos(4*t+Pi/2),cos(7*t),t=0..2*Pi],
   radius=0.05,axes=none,color="Brown",orientation=[90,0,0],style=surface);

Star Knot

A star knot can be defined by using the following polynomial:

> 

f := -x^5+y^2;

f≔−x5+y2

(1)
> 

algcurves:-plot_knot(f,x,y,epsilon=0.7,
   radius=0.25,tubepoints=10,axes=none,color="Orange",orientation=[60,0],style=surfacecontour);

Two different projections of the same polynomial

By switching x and y, different visualizations can be generated:

> 

g:=(y^3-x^7)*(y^2-x^5)+y^3;

g≔−x7+y3⁢−x5+y2+y3

(2)
> 

plots:-display(<
algcurves:-plot_knot(g,y,x,epsilon=0.8,radius=0.1,axes=none,color="CornflowerBlue",orientation=[75,30,0])|
algcurves:-plot_knot(g,x,y,epsilon=0.8,radius=0.1,axes=none,color="OrangeRed",orientation=[75,0,0])>);

More examples

> 

f:=(y^3-x^7)*(y^2-x^5);

f≔−x7+y3⁢−x5+y2

(3)
> 

algcurves:-plot_knot(f,x,y,
  epsilon=0.8,radius=0.1,axes=none,orientation=[35,0,0]);

> 

h:=(y^3-x^7)*(y^3-x^7+100*x^13)*(y^3-x^7-100*x^13);

h≔−x7+y3⁢100⁢x13−x7+y3⁢−100⁢x13−x7+y3

(4)
> 

algcurves:-plot_knot(h,x,y,
   epsilon=0.8,numpoints=400,radius=0.03,axes=none,color=["Blue","Red","Green"],orientation=[60,0,0]);

Return to Index for Example Worksheets

See Also

algcurves:-plot_knot, examples/algcurves, plots:-tubeplot