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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.
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Unknot
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The unknot can be defined by the following parametric equations:
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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);
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The Trefoil Knot
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The trefoil knot can be defined by the following parametric equations:
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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);
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The Figure-Eight Knot
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The figure-eight can be defined by the following parametric equations:
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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);
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The Lissajous Knot
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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:
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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);
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Star Knot
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A star knot can be defined by using the following polynomial:
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algcurves:-plot_knot(f,x,y,epsilon=0.7,
radius=0.25,tubepoints=10,axes=none,color="Orange",orientation=[60,0],style=surfacecontour);
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