View source for Torus knot

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''of type  $  ( p, q) $''

A curve in  $  \mathbf R  ^ {3} $
that in cylindrical coordinates  $  r, z, \theta $
is given by the equations

$$ 
r  =  2 + \cos  t,\ \ 
z  =  \sin  t,\ \ 
\theta  =  {
\frac{pt }{q}
 } ,
$$

where  $  t \in [ 0, 2 \pi q] $. 
Here  $  p $
and  $  q $
are coprime natural numbers. The torus knot lies on the surface of the unknotted torus  $  ( r - 2)  ^ {2} + z  ^ {2} = 1 $, 
intersecting the meridians of the torus at  $  p $
points and the parallels at  $  q $
points. The torus knots of types  $  ( p, 1) $
and  $  ( 1, q) $
are trivial. The simplest non-trivial torus knot is the trefoil (Fig. a), which is of type  $  ( 2, 3) $. 
The group of the torus knot of type  $  ( p, q) $
has a [[Presentation|presentation]]  $  < a, b $: 
$  a  ^ {p} = b  ^ {q} > $, 
and the Alexander polynomial is given by

$$ 
( t  ^ {pq} - 1)
( t - 1)
( t  ^ {p} - 1)  ^ {-} 1
( t  ^ {q} - 1)  ^ {-} 1 .
$$

All torus knots are Neuwirth knots (cf. [[Neuwirth knot|Neuwirth knot]]). The genus of a torus knot is  $  ( p - 1) ( q - 1)/2 $.

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

$$ 
V  = \ 
\{ {( z _ {1} , z _ {2} ) \in \mathbf C  ^ {2} } : {
z _ {1}  ^ {p} + z _ {2}  ^ {q} = 0 } \}
.
$$

If  $  p $
and  $  q $
are coprime, then the intersection of  $  V $
with a sufficiently small sphere  $  S  ^ {3} \subset  \mathbf C  ^ {2} $
is a knot in  $  S  ^ {3} $
equivalent to the torus knot of type  $  ( p, q) $. 
In the case when  $  p $
and  $  q $
are not coprime, this intersection also lies on an unknotted torus  $  T  ^ {2} \subset  S  ^ {3} $, 
but consists of several components. The link so obtained is called the torus link of type  $  ( p, q) $(
cf. Fig. b, where  $  p = 3 $, 
$  q = 6 $).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t093360a.gif" />

Figure: t093360a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t093360b.gif" />

Figure: t093360b

{{OldImage}}

====Comments====
See also [[Knot theory]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  R.H. Crowell,   R.H. Fox,   "Introduction to knot theory" , Ginn  (1963)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  J. Milnor,   "Singular points of complex hypersurfaces" , Princeton Univ. Press  (1968)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Rolfsen,   "Knots and links" , Publish or Perish  (1976)</TD></TR>
</table>