View source for Torus knot

''of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093360/t0933602.png" />''

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

<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>

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

<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>

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" />.

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>

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" />).

<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

====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>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Rolfsen,   "Knots and links" , Publish or Perish  (1976)</TD></TR></table>