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About: Lissajous knot

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In knot theory, a Lissajous knot is a knot defined by parametric equations of the form where , , and are integers and the phase shifts , , and may be any real numbers. The projection of a Lissajous knot onto any of the three coordinate planes is a Lissajous curve, and many of the properties of these knots are closely related to properties of Lissajous curves.

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  • In knot theory, a Lissajous knot is a knot defined by parametric equations of the form where , , and are integers and the phase shifts , , and may be any real numbers. The projection of a Lissajous knot onto any of the three coordinate planes is a Lissajous curve, and many of the properties of these knots are closely related to properties of Lissajous curves. Replacing the cosine function in the parametrization by a triangle wave transforms every Lissajousknot isotopically into a billiard curve inside a cube, the simplest case of so-called billiard knots.Billiard knots can also be studied in other domains, for instance in a cylinder or in a (flat) solid torus (Lissajous-toric knot). (en)
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  • In knot theory, a Lissajous knot is a knot defined by parametric equations of the form where , , and are integers and the phase shifts , , and may be any real numbers. The projection of a Lissajous knot onto any of the three coordinate planes is a Lissajous curve, and many of the properties of these knots are closely related to properties of Lissajous curves. (en)
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  • Lissajous knot (en)
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