nLab trefoil knot

Contents

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]]

.

Properties

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

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

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

category : knot theory

Last revised on July 18, 2024 at 18:30:24. See the history of this page for a list of all contributions to it.