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=sint
y=cost
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=sint+2sin2t
y=cost+2sin2t
z=sin3t
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+cos2tcos3t
y=2+cos2tsin3t
z=sin4t
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=cosnxt+φx
y=cosnyt+φy
z=cosnzt+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
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
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
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+y3100x13−x7+y3−100x13−x7+y3
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
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