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 , ).
|||R.H. Crowell, R.H. Fox, "Introduction to knot theory" , Ginn (1963)|
|||J. Milnor, "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968)|
See also Knot theory.
|[a1]||D. Rolfsen, "Knots and links" , Publish or Perish (1976)|
Torus knot. M.Sh. Farber (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Torus_knot&oldid=11550