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 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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