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: 

 

 

 

>	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: 

 

 

 

>	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: 

 

 

 

>	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: 

 

 

 

Where , , and are integers and the phase shifts , , and are any real numbers. 

The 8 21 knot (, , and ) 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;

 

(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;

 

(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);

 

(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);

 

(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