In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory. The trefoil knot is named after the three-leaf clover (or trefoil) plant.
Descriptions The trefoil knot can be defined as the curve obtained from the following parametric equations:
x = sin t + 2 sin 2 t y = cos t − 2 cos 2 t z = − sin 3 t {\displaystyle {\begin{aligned}x&=\sin t+2\sin 2t\\y&=\cos t-2\cos 2t\\z&=-\sin 3t\end{aligned}}}
The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus ( r − 2 ) 2 + z 2 = 1 {\displaystyle (r-2)^{2}+z^{2}=1} :
x = ( 2 + cos 3 t ) cos 2 t y = ( 2 + cos 3 t ) sin 2 t z = sin 3 t {\displaystyle {\begin{aligned}x&=(2+\cos 3t)\cos 2t\\y&=(2+\cos 3t)\sin 2t\\z&=\sin 3t\end{aligned}}}
Any continuous deformation of the curve above is also considered a trefoil knot. Specifically, any curve isotopic to a trefoil knot is also considered to be a trefoil. In addition, the mirror image of a trefoil knot is also considered to be a trefoil. In topology and knot theory, the trefoil is usually defined using a knot diagram instead of an explicit parametric equation. In algebraic geometry, the trefoil can also be obtained as the intersection in C2 of the unit 3-sphere S3 with the complex plane curve of zeroes of the complex polynomial z2 + w3 (a cuspidal cubic).
If one end of a tape or belt is turned over three times and then pasted to the other, the edge forms a trefoil knot.
Symmetry The trefoil knot is chiral, in the sense that a trefoil knot can be distinguished from its own mirror image. The two resulting variants are known as the left-handed trefoil and the right-handed trefoil. It is not possible to deform a left-handed trefoil continuously into a right-handed trefoil, or vice versa. (That is, the two trefoils are not ambient isotopic.) Though chiral, the trefoil knot is also invertible, meaning that there is no distinction between a counterclockwise-oriented and a clockwise-oriented trefoil. That is, the chirality of a trefoil depends only on the over and under crossings, not the orientation of the curve. But the knot has rotational symmetry. The axis is about a line perpendicular to the page for the 3-coloured image.
Nontriviality The trefoil knot is nontrivial, meaning that it is not possible to "untie" a trefoil knot in three dimensions without cutting it. Mathematically, this means that a trefoil knot is not isotopic to the unknot. In particular, there is no sequence of Reidemeister moves that will untie a trefoil. Proving this requires the construction of a knot invariant that distinguishes the trefoil from the unknot. The simplest such invariant is tricolorability: the trefoil is tricolorable, but the unknot is not. In addition, virtually every major knot polynomial distinguishes the trefoil from an unknot, as do most other strong knot invariants.
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