Trefoil knot

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Trefoil knot

In knot theory, the trefoil knot is the simplest nontrivial knot.

Contents

1 Descriptions
2 Properties
3 Invariants
4 See also
5 References

It can be obtained by joining the loose ends of an overhand knot.
It can be described as a (2,3)-torus knot.
It is the closure of the braid σ₁³.
It is 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 $z 2 + w 3$ .

It is the unique prime knot with three crossings.
It is chiral, meaning it is not equivalent to its mirror image.
It is alternating.
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 $φ(z, w) = (z 2 + w 3) / | z 2 + w 3 |$ as its fibration, and a once-punctured torus as its fiber surface.

Its Alexander polynomial is $t 2 - t + 1$ .
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 $\langle \sigma_1,\sigma_2 \mid \sigma_1\sigma_2\sigma_1 = \sigma_2\sigma_1\sigma_2 \rangle.\,$

Rolfsen, Dale (1976). Knots and links. Berkeley: Publish or Perish, Inc. ISBN 0-914098-16-0.
Eric W. Weisstein, Trefoil Knot at MathWorld.

Retrieved from "http://en.wikipedia.org../../../t/r/e/Trefoil_knot.html"

Category: Knot theory

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