In knot theory, the trefoil knot is the simplest nontrivial knot. It can be obtained by joining the loose ends of an overhand knot. It can be described as a (2,3)-torus knot[1], and is the closure of the 2-stranded braid σ1³. It is also the intersection of the unit 3-sphere S3 in C² with the complex plane curve (a cuspidal cubic) of zeroes of the complex polynomial z2 + w3.
Properties
The right and left-handed trefoils are the unique prime knots which have 3-crossing diagrams. They are chiral knots, meaning that the right-handed trefoil is the mirror image of the left-hand trefoil, but they are not themselves isotopic.
The simplest proof that the trefoil is not the unknot is that trefoil is tricolorable while the unknot is not tricolorable.
The trefoil is an alternating knot. However, it is not a slice knot, meaning that it does not bound a smooth 2-dimensional disk in the 4-dimensional ball; one way to prove this is to note that its signature is not zero. Another proof is that its Alexander polynomial does not satisfy the Fox-Milnor condition.
The trefoil is a fibered knot, meaning that its complement in S3 is a fiber bundle over the circle S1. In the model of the trefoil as the set of pairs (z,w) of complex numbers such that | z | 2 + | w | 2 = 1 and z2 + w3 = 0, this fiber bundle has the Milnor map φ(z,w) = (z4 + w3) / | z2 + w3 | as its fibration, and a once-punctured torus as its fiber surface. Since the knot complement is Seifert fibred with boundary, it has a horizontal incompressible surface -- this is also the fiber of the Milnor map.
Invariants
Its Alexander polynomial is t2 − t + 1 and its Jones polynomial is t + t3 − t4. Its knot group is isomorphic to B3, the braid group on 3 strands, which has presentation 
or 
See also
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