# Torus knot

*of type *

A curve in that in cylindrical coordinates is given by the equations

where . Here and are coprime natural numbers. The torus knot lies on the surface of the unknotted torus , intersecting the meridians of the torus at points and the parallels at points. The torus knots of types and are trivial. The simplest non-trivial torus knot is the trefoil (Fig. a), which is of type . The group of the torus knot of type has a presentation : , and the Alexander polynomial is given by

All torus knots are Neuwirth knots (cf. Neuwirth knot). The genus of a torus knot is .

A second construction of a torus knot uses the singularity at the origin of the algebraic hypersurface

If and are coprime, then the intersection of with a sufficiently small sphere is a knot in equivalent to the torus knot of type . In the case when and are not coprime, this intersection also lies on an unknotted torus , but consists of several components. The link so obtained is called the torus link of type (cf. Fig. b, where , ).

Figure: t093360a

Figure: t093360b

#### References

[1] | R.H. Crowell, R.H. Fox, "Introduction to knot theory" , Ginn (1963) |

[2] | J. Milnor, "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968) |

#### Comments

See also Knot theory.

#### References

[a1] | D. Rolfsen, "Knots and links" , Publish or Perish (1976) |

**How to Cite This Entry:**

Torus knot. M.Sh. Farber (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Torus_knot&oldid=11550