Trefoil knot

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, and is the closure of the 2-stranded braid σ₁³. It is also the intersection of the unit 3-sphere $S^3$ in C² with the complex plane curve (a cuspidal cubic) of zeroes of the complex polynomial $z^2+w^3$.

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 $S^3$ is a fiber bundle over the circle $S^1$. In the model of the trefoil as the set of pairs $(z,w)$ of complex numbers such that $|z|^2+|w|^2=1$ and $z^2+w^3=0$, this fiber bundle has the Milnor map $phi(z,w)=(z^4+w^3)/|z^2+w^3|$ 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 $t^2-t+1$ and its Jones polynomial is $t+t^3-t^4$. Its knot group is isomorphic to "B"₃, the braid group on 3 strands, which has presentation $langle\; x,y\; mid\; x^2\; =\; y^3\; angle,$ or $langle\; sigma\_1,sigma\_2\; mid\; sigma\_1sigma\_2sigma\_1\; =\; sigma\_2sigma\_1sigma\_2\; angle.,$

ee also

*Figure-eight knot (mathematics)
*Triquetra symbol
*Cinquefoil knot