Trefoil knot

Trefoil
Common name	Overhand knot
Arf invariant	1
Braid length	3
Braid no.	2
Bridge no.	2
Crosscap no.	1
Crossing no.	3
Genus	1
Hyperbolic volume	0
Stick no.	6
Tunnel no.	1
Unknotting no.	1
Conway notation	[3]
A-B notation	31
Dowker notation	4, 6, 2
Last /Next	01 / 41
Other
alternating, torus, fibered, pretzel, prime, slice, reversible, tricolorable, twist

In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together 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.

The trefoil knot can be defined as the curve obtained from the following parametric equations:

The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus ${\displaystyle (r-2)^{2}+z^{2}=1}$:

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).

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