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 is the unique prime knot with three crossings. It can be described as a (2,3)-torus knot, its braid word being σ₁³. Another (closely related) description is as 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

. The trefoil knot 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

\phi(z,w)=(z^2+w^3)/|z^2+w^3|

as its fibration, and a once-punctured torus as its fiber surface.) The Alexander polynomial of the trefoil is -x²+x-1. The knot group of the trefoil is isomorphic to B₃, the braid group on 3 strands, which has presentation

\langle x,y \mid x^2 = y^3 \rangle

\langle \sigma_1,\sigma_2 \mid \sigma_1\sigma_2\sigma_1 = \sigma_2\sigma_1\sigma_2 \rangle

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Trefoil Knot

See also