Examples of Knots - Maple Programming 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:

$x=\mathrm{sin}\left(t\right)$

$y=\mathrm{cos}\left(t\right)$

$z=0$

 > 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:

$x=\mathrm{sin}\left(t\right)+2\mathrm{sin}\left(2t\right)$

$y=\mathrm{cos}\left(t\right)+2\mathrm{sin}\left(2t\right)$

$z=\mathrm{sin}\left(3t\right)$

 > 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:

$x=\left(2+\mathrm{cos}\left(2t\right)\right)\mathrm{cos}\left(3t\right)$

$y=\left(2+\mathrm{cos}\left(2t\right)\right)\mathrm{sin}\left(3t\right)$

$z=\mathrm{sin}\left(4t\right)$

 > 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:

$x=\mathrm{cos}\left({n}_{x}t+{\mathrm{\phi }}_{x}\right)$

$y=\mathrm{cos}\left({n}_{y}t+{\mathrm{\phi }}_{y}\right)$

$z=\mathrm{cos}\left({n}_{z}t+{\mathrm{\phi }}_{z}\right)$

Where ${n}_{x}$, ${n}_{y}$, and ${n}_{z}$ are integers and the phase shifts ${\mathrm{phi}}_{x}$, ${\mathrm{phi}}_{y}$, and ${\mathrm{phi}}_{z}$ are any real numbers.

The 8 21 knot (${n}_{x}=3$, ${n}_{y}=4$, and ${n}_{z}=7$) 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;
 ${f}{≔}{-}{{x}}^{{5}}{+}{{y}}^{{2}}$ (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;
 ${g}{≔}\left({-}{{x}}^{{7}}{+}{{y}}^{{3}}\right){}\left({-}{{x}}^{{5}}{+}{{y}}^{{2}}\right){+}{{y}}^{{3}}$ (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);
 ${f}{≔}\left({-}{{x}}^{{7}}{+}{{y}}^{{3}}\right){}\left({-}{{x}}^{{5}}{+}{{y}}^{{2}}\right)$ (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);
 ${h}{≔}\left({-}{{x}}^{{7}}{+}{{y}}^{{3}}\right){}\left({100}{}{{x}}^{{13}}{-}{{x}}^{{7}}{+}{{y}}^{{3}}\right){}\left({-}{100}{}{{x}}^{{13}}{-}{{x}}^{{7}}{+}{{y}}^{{3}}\right)$ (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]);
 >
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