Trefoil knot

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This article is about the topological concept. For the protein fold, see trefoil knot fold.

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 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 $z 2 + w 3$ .

[edit] Properties

A trefoil knot is the unique prime knot with three crossings. It is chiral, meaning it is not isotopic to its mirror image. To distinguish between the two isotopy classes of knots, the terminology "right-handed" and "left-handed" trefoils is used.

The right-handed Trefoil knot

The left-handed Trefoil knot

It 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. It 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 $φ(z, w) = (z 4 + w 3) / | z 2 + w 3 |$ as its fibration, and a once-punctured torus as its fiber surface.

[edit] 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 B₃, the braid group on 3 strands, which has presentation $\langle x,y \mid x^2 = y^3 \rangle\,$ or $\langle \sigma_1,\sigma_2 \mid \sigma_1\sigma_2\sigma_1 = \sigma_2\sigma_1\sigma_2 \rangle.\,$