nLab trefoil knot

The trefoil knot is a famous knot. One of the reasons is that in the list of knots, ordered by crossing number, it is the first ‘real’ knot one meets, being the simplest non-trivial knot. (The first knot listed is usually the ‘unknot’, i.e. the unknotted circle.) The trefoil has crossing number 3.

Here is a traditional view:

Here is a depiction with bridge number 2:

category: svg

The knot group of the trefoil knot (calculated either by the Dehn or Wirtinger presentations) has two very useful presentations:

• $\langle x,y \mid x y x=y x y\rangle$, which is the braid group, Br 3;

• $\langle a,b | a^2= b^3\rangle$, in which the pair of numbers, $(2,3)$, is apparent. These reflect the fact that the trefoil is a $(2,3)$-torus knot. (Of course, it is also a (3,2)-torus knot.)