Examples of Knots - Maple Programming Help

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:

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

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

$z=0$

The Trefoil Knot

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

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

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

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

The Figure-Eight Knot

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

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

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

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

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:

Star Knot

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

 > f := -x^5+y^2;
 ${f}{≔}{-}{{x}}^{{5}}{+}{{y}}^{{2}}$ (1)

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)
 ${f}{≔}\left({-}{{x}}^{{7}}{+}{{y}}^{{3}}\right){}\left({-}{{x}}^{{5}}{+}{{y}}^{{2}}\right)$ (3)
 ${h}{≔}\left({-}{{x}}^{{7}}{+}{{y}}^{{3}}\right){}\left({100}{}{{x}}^{{13}}{-}{{x}}^{{7}}{+}{{y}}^{{3}}\right){}\left({-}{100}{}{{x}}^{{13}}{-}{{x}}^{{7}}{+}{{y}}^{{3}}\right)$ (4)