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3 1 knot

Trefoil
Blue Trefoil Knot.png
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 0141
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 :

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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Wikipedia

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