nLab trefoil knot

Idea

The trefoil knot is a famous example of a knot. 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 the traditional knot diagram for the trefoil knot:

Here is an alternative depiction with bridge number 2:

category: svg

Remark

To include one of the above svg pictures on a page, write

[[!include trefoil knot - SVG]]

or

[[!include trefoil knot (2 bridge) - SVG]]

.

Related entries

• unknot

• figure eight knot

Properties

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.)

category : knot theory