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''of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t0933602.png" />''
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$#C+1 = 31 : ~/encyclopedia/old_files/data/T093/T.0903360 Torus knot
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A curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t0933603.png" /> that in cylindrical coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t0933604.png" /> is given by the equations
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{{TEX|auto}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t0933605.png" /></td> </tr></table>
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''of type  $  ( p, q) $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t0933606.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t0933607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t0933608.png" /> are coprime natural numbers. The torus knot lies on the surface of the unknotted torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t0933609.png" />, intersecting the meridians of the torus at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336010.png" /> points and the parallels at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336011.png" /> points. The torus knots of types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336013.png" /> are trivial. The simplest non-trivial torus knot is the trefoil (Fig. a), which is of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336014.png" />. The group of the torus knot of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336015.png" /> has a [[Presentation|presentation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336016.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336017.png" />, and the Alexander polynomial is given by
+
A curve in  $  \mathbf R  ^ {3} $
 +
that in cylindrical coordinates  $  r, z, \theta $
 +
is given by the equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336018.png" /></td> </tr></table>
+
$$
 +
= 2 + \cos  t,\ \
 +
= \sin  t,\ \
 +
\theta  = {
 +
\frac{pt }{q}
 +
} ,
 +
$$
  
All torus knots are Neuwirth knots (cf. [[Neuwirth knot|Neuwirth knot]]). The genus of a torus knot is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336019.png" />.
+
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
 
A second construction of a torus knot uses the singularity at the origin of the algebraic hypersurface
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336020.png" /></td> </tr></table>
+
$$
 +
= \
 +
\{ {( z _ {1} , z _ {2} ) \in \mathbf C  ^ {2} } : {
 +
z _ {1}  ^ {p} + z _ {2}  ^ {q} = 0 } \}
 +
.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336022.png" /> are coprime, then the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336023.png" /> with a sufficiently small sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336024.png" /> is a knot in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336025.png" /> equivalent to the torus knot of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336026.png" />. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336028.png" /> are not coprime, this intersection also lies on an unknotted torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336029.png" />, but consists of several components. The link so obtained is called the torus link of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336031.png" /> (cf. Fig. b, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t09336033.png" />).
+
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" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t093360a.gif" />
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====References====
 
====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></table>
 
<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></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 08:26, 6 June 2020


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 $ < 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). 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 $).

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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torus_knot&oldid=49001
This article was adapted from an original article by M.Sh. Farber (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article