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

Descriptions

• 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 σ₁.
• It is the intersection of the unit 3-sphere ${\displaystyle S^{3}}$ in C with the complex plane curve (a cuspidal cubic) of zeroes of the complex polynomial ${\displaystyle z^{2}+w^{3}}$.

Properties

• 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 ${\displaystyle S^{3}}$ is a fiber bundle over the circle ${\displaystyle S^{1}}$. In the model of the trefoil as the set of pairs ${\displaystyle (z,w)}$ of complex numbers such that ${\displaystyle |z|^{2}+|w|^{2}=1}$ and ${\displaystyle z^{2}+w^{3}=0}$, this fiber bundle has the Milnor map ${\displaystyle \phi (z,w)=(z^{2}+w^{3})/|z^{2}+w^{3}|}$ as its fibration, and a once-punctured torus as its fiber surface.

Invariants

• Its Alexander polynomial is ${\displaystyle t^{2}-t+1}$.
• Its Jones polynomial is ${\displaystyle t+t^{3}-t^{4}}$.
• Its knot group is isomorphic to B₃, the braid group on 3 strands, which has presentation ${\displaystyle \langle x,y\mid x^{2}=y^{3}\rangle \,}$ or ${\displaystyle \langle \sigma _{1},\sigma _{2}\mid \sigma _{1}\sigma _{2}\sigma _{1}=\sigma _{2}\sigma _{1}\sigma _{2}\rangle .\,}$

See also

• Figure-eight knot (mathematics)
• Triquetra symbol

References

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

External link

• Laser Surface Scan of a Compact Trefoil Knot

• Knot theory