View source for Torus knot

{{Distinguish|Torus}}
[[Image:TorusKnot3D.png|thumb|A (3,−7)-[[3D computer graphics|3D]] torus knot.]]
[[Image:Eurelea.png|thumb|[[EureleA]] Award showing a (2,3)-torus knot.]]
[[Image:(2,4)-Torus Link.svg|thumb|(2,8) torus link]]

In [[knot theory]], a '''torus knot''' is a special kind of [[knot (mathematics)|knot]] that lies on the surface of an unknotted [[torus]] in '''R'''<sup>3</sup>. Similarly, a '''torus link''' is a [[link (knot theory)|link]] which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of [[coprime]] [[integer]]s ''p'' and ''q''. A torus link arises if ''p'' and ''q'' are not coprime (in which case the number of components is [[greatest common divisor|gcd]](''p, q'')). A torus knot is [[unknot|trivial]] (equivalent to the unknot) [[if and only if]] either ''p'' or ''q'' is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the [[trefoil knot]].

[[Image:Trefoil knot left.svg|thumb|right|the (2,−3)-torus knot, also known as the left-handed [[trefoil knot]]]]

== Geometrical representation ==

A torus knot can be rendered geometrically in multiple ways which are [[topologically equivalent]] (see Properties below) but geometrically distinct. The convention used in this article and its figures is the following.

The (''p'',''q'')-torus knot winds ''q'' times around a circle in the interior of the torus,
and ''p'' times around its axis of [[rotational symmetry]]. 
{Note, this use of the roles of p and q is contrary to what appears on:  http://mathworld.wolfram.com/TorusKnot.html
It is also inconsistent with the "List" of torus knots below and with the pictures that appear in:  "36 Torus Knots", The Knot Atlas.}
If ''p'' and ''q'' are not relatively prime, then we have a torus link with more than one component.

The direction in which the strands of the knot wrap around the torus is also subject to differing conventions. The most common is to have the strands form a right-handed screw for ''p q > 0''.<ref name="livingston">Livingston, Charles (1993). ''Knot Theory'', {{page needed|date=November 2017}}. Mathematical Association of America. {{ISBN|0-88385-027-3}}</ref><ref name="murasugi">Murasugi, Kunio (1996). ''Knot Theory and its Applications'', {{page needed|date=November 2017}}. Birkhäuser. {{ISBN|3-7643-3817-2}}</ref><ref name="kawauchi">Kawauchi, Akio (1996). ''A Survey of Knot Theory'', {{page needed|date=November 2017}}. Birkhäuser. {{ISBN|3-7643-5124-1}}</ref>

The (''p'',''q'')-torus knot can be given by the [[Parametrization (geometry)|parametrization]]
:<math>\begin{align}
  x &=  r\cos(p\phi) \\
  y &=  r\sin(p\phi) \\
  z &= -\sin(q\phi)
\end{align}</math>

where <math>r = \cos(q\phi)+2</math> and <math>0<\phi<2\pi</math>. This lies on the surface of the torus given by <math>(r-2)^2 + z^2 = 1</math> (in [[cylindrical coordinates]]).

Other parameterizations are also possible, because knots are defined up to continuous deformation. The illustrations for the (2,3)- and (3,8)-torus knots can be obtained by taking <math>r = \cos(q\phi)+4</math>, and in the case of the (2,3)-torus knot by furthermore subtracting respectively <math>3\cos((p-q)\phi)</math> and <math>3\sin((p-q)\phi)</math> from the above parameterizations of ''x'' and ''y''. The latter generalizes smoothly to any coprime ''p,q'' satisfying <math>p<q<2p</math>.

==Properties==
[[Image:TorusKnot-3-8.png|thumb|right|Diagram of a (3,−8)-torus knot.]]

A torus knot is [[unknot|trivial]] [[if and only if|iff]] either ''p'' or ''q'' is equal to 1 or −1.<ref name="murasugi"/><ref name="kawauchi"/>

Each nontrivial torus knot is [[prime knot|prime]]<ref>Norwood, Rick (1982). "Every Two generator Knot is Prime", Proceedings of the American Mathematical Society, Vol. 86, No. 1, p 143 7.</ref> and [[chirality (mathematics)|chiral]]<ref name="murasugi"/>.

The (''p'',''q'') torus knot is equivalent to the (''q'',''p'') torus knot.<ref name="livingston"/><ref name="kawauchi"/> This can be proved by moving the strands on the surface of the torus, which is nicely illustrated [http://sketchesoftopology.wordpress.com/2011/03/28/pqisqp/ here]. The (''p'',−''q'') torus knot is the obverse (mirror image) of the (''p'',''q'') torus knot.<ref name="kawauchi"/> The (−''p'',−''q'') torus knot is equivalent to the (''p'',''q'') torus knot except for the reversed orientation.

[[File:(3, 4) torus knot.svg|thumb|The (3, 4) torus knot on the unwrapped torus surface, and its braid word]]
Any (''p'',''q'')-torus knot can be made from a [[braid theory|closed braid]] with ''p'' strands. The appropriate [[braid word]] is <ref name="lickorish">Lickorish, W. B. R. (1997). ''An Introduction to Knot Theory'', {{page needed|date=November 2017}}. Springer. {{ISBN|0-387-98254-X}}</ref>
:<math>(\sigma_1\sigma_2\cdots\sigma_{p-1})^q.</math>
(This formula assumes the common convention that braid generators are right twists,<ref name="murasugi" /><ref name="lickorish" /><ref name="dehornoy">Dehornoy, P. et al. (2000). ''Why are Braids Orderable?'', {{page needed|date=November 2017}}. {{cite web |url=http://www.math.unicaen.fr/~dehornoy/Books/Why/Dgr.pdf |title=Archived copy |accessdate=2011-11-12 |url-status=dead |archiveurl=https://web.archive.org/web/20120415032136/http://www.math.unicaen.fr/~dehornoy/Books/Why/Dgr.pdf |archivedate=2012-04-15 }}</ref><ref name="birman">Birman, J. S., and [[Tara E. Brendle|Brendle, T. E.]] ''Braids: a Survey''. In: Menasco, W., and Thistlethwaite, M. (Eds.) (2005). ''Handbook of Knot Theory'', {{page needed|date=November 2017}}. Elsevier. {{ISBN|0-444-51452-X}}.</ref> which is not followed by the Wikipedia page on braids.)

The [[Crossing number (knot theory)|crossing number]] of a (''p'',''q'') torus knot with ''p'',''q'' > 0 is given by
:''c'' = min((''p''&minus;1)''q'', (''q''&minus;1)''p'').
The [[knot genus|genus]] of a torus knot with ''p'',''q'' > 0 is
:<math>g = \frac{1}{2}(p-1)(q-1).</math>
The [[Alexander polynomial]] of a torus knot is <ref name="livingston"/><ref name="lickorish"/>
:<math>\frac{(t^{pq}-1)(t-1)}{(t^p-1)(t^q-1)}.</math>
The [[Jones polynomial]] of a (right-handed) torus knot is given by
:<math>t^{(p-1)(q-1)/2}\frac{1-t^{p+1}-t^{q+1}+t^{p+q}}{1-t^2}.</math>

The complement of a torus knot in the [[3-sphere]] is a [[Seifert fiber space|Seifert-fibered manifold]], fibred over the disc with two singular fibres.

Let ''Y'' be the ''p''-fold [[Dunce hat (topology)|dunce cap]] with a disk removed from the interior, ''Z'' be the ''q''-fold dunce cap with a disk removed its interior, and ''X'' be the quotient space obtained by identifying ''Y'' and ''Z'' along their boundary circle. The knot complement of the (''p'', ''q'') -torus knot [[deformation retract]]s to the space ''X''. Therefore, the [[knot group]] of a torus knot has the [[presentation of a group|presentation]]

:<math>\langle x,y \mid x^p = y^q\rangle.</math>

Torus knots are the only knots whose knot groups have nontrivial [[center of a group|center]] (which is infinite cyclic, generated by the element <math>x^p = y^q</math> in the presentation above).

The [[stretch factor]] of the (''p'',''q'') torus knot, as a curve in [[Euclidean space]], is &Omega;(min(''p'',''q'')), so torus knots have unbounded stretch factors. Undergraduate researcher [[John Pardon]] won the 2012 [[Morgan Prize]] for his research proving this result, which solved a problem originally posed by [[Mikhail Leonidovich Gromov|Mikhail Gromov]].<ref>{{citation|title=2012 Morgan Prize|journal=[[Notices of the American Mathematical Society]]|date=April 2012|doi=10.1090/noti825|pages=569–571|volume=59|issue=4|first=Elaine|last=Kehoe}}.</ref><ref>{{citation
 | last = Pardon | first = John
 | doi = 10.4007/annals.2011.174.1.21
 | issue = 1
 | journal = [[Annals of Mathematics]]
 | mr = 2811613
 | pages = 637–646
 | series = Second Series
 | title = On the distortion of knots on embedded surfaces
 | arxiv = 1010.1972| volume = 174
 | year = 2011}}.{{page needed|date=November 2017}}</ref>

== Connection to complex hypersurfaces ==

The (''p'',''q'')&minus;torus knots arise when considering the link of an isolated complex hypersurface singularity. One intersects the complex hypersurface with a hypersphere, centred at the isolated singular point, and with sufficiently small radius so that it does not enclose, nor encounter, any other singular points. The intersection gives a submanifold of the hypersphere.

Let ''p'' and ''q'' be coprime integers, greater than or equal to two. Consider the [[holomorphic function]] <math> f: \Complex^2 \to \Complex</math> given by <math>f(w,z) := w^p + z^q.</math> Let <math>V_f \subset \Complex^2</math> be the set of <math>(w,z) \in \Complex^2</math> such that <math>f(w,z) = 0.</math> Given a real number <math>0 < \varepsilon \ll 1, </math> we define the real three-sphere <math>\mathbb{S}^3_{\varepsilon} \subset \R^4 \hookrightarrow \Complex^2</math> as given by <math>|w|^2 + |z|^2 = \varepsilon^2.</math> The function <math>f</math> has an isolated [[critical point (mathematics)|critical point]] at <math>(0,0) \in \Complex^2</math> since <math>\partial f/\partial w = \partial f/ \partial z = 0</math> if and only if <math>w = z = 0.</math> Thus, we consider the structure of <math>V_f</math> close to <math>(0,0) \in \Complex^2.</math> In order to do this, we consider the intersection <math>V_f \cap \mathbb{S}^3_{\varepsilon} \subset \mathbb{S}^3_{\varepsilon}.</math> This intersection is the so-called link of the singularity <math>f(w,z) = w^p + z^q.</math> The link of <math>f(w,z) = w^p + z^q</math>, where ''p'' and ''q'' are coprime, and both greater than or equal to two, is exactly the (''p'',''q'')&minus;torus knot.<ref name="MILNOR">{{Cite book|first=J.|last=Milnor|title=Singular Points of Complex Hypersurfaces|page={{page needed|date=November 2017}}|publisher=Princeton University Press|year=1968|ISBN=0-691-08065-8}}</ref>

==List==
[[File:Torus link (36,3).png|thumb|(36,3) torus link]]  The figure on the right is  torus link (72,4) .

*[[Unknot]], [[Trefoil knot|3<sub>1</sub> knot]] (3,2), [[Cinquefoil knot|5<sub>1</sub> knot]] (5,2), [[7₁ knot|7<sub>1</sub> knot]] (7,2), 8<sub>19</sub> knot (4,3), '''9<sub>1</sub> knot''' (9,2), '''10<sub>124</sub> knot''' (5,3)

{| class="wikitable sortable" style="text-align:center""
! [[Knot tabulation|Table<br/> #]]
! [[Alexander-Briggs notation|A-B]]
! Image
! ''P''
! ''Q''
! [[Crossing number (knot theory)|Cross<br/> #]]
|-
|0
| 0<sub>1</sub>
| [[File:Blue Unknot.png|60px]]
| 
| 
| 0
|-
| 3a1
| 3<sub>1</sub>
| [[File:(3-2) torus knot.png|60px]]
| 3
| 2
| 3
|-
| 5a2
| 5<sub>1</sub>
| [[File:A (5,2)-torus knot.png|60px]]
| 5
| 2
| 5
|-
| 7a7
| 7<sub>1</sub>
| [[File:A (7,2)-torus knot.png|60px]]
| 7
| 2
| 7
|-
| 8n3
| 8<sub>19</sub>
| [[File:A (4,3)-torus knot.png|60px]]
| 4
| 3
| 8
|-
| 9a41
| 9<sub>1</sub>
| [[File:A (9,2)-torus knot.png|60px]]
| 9
| 2
| 9
|-
| 10n21
| 10<sub>124</sub>
| [[File:A (5,3)-torus knot.png|60px]]
| 5
| 3
| 10
|-
| 11a367
| <!--11<sub>Y</sub>-->
| [[File:Simple-knotwork-cross-12crossings.svg|60px]]
| 11
| 2
| 11
|-
| 13a4878
| <!--13<sub>Y</sub>-->
| <!--[[File:A (13,2)-torus knot.png|60px]]-->
| 13
| 2
| 13
|-
| 
| <!--14<sub>Y</sub>-->
| [[File:A (7,3)-torus knot.png|60px]]
| 7
| 3
| 14
|-
| 
| <!--15<sub>Y</sub>-->
| [[File:A (5,4)-torus knot.png|60px]]
| 5
| 4
| 15
|-
| 
| <!--15<sub>Y</sub>-->
| <!--[[File:A (15,2)-torus knot.png|60px]]-->
| 15
| 2
| 15
|-
| 
| <!--16<sub>Y</sub>-->
| [[File:A (8,3)-torus knot.png|60px]]
| 8
| 3
| 16
|-
| 
| <!--17<sub>Y</sub>-->
| <!--[[File:A (17,2)-torus knot.png|60px]]-->
| 17
| 2
| 17
|-
| 
| <!--19<sub>Y</sub>-->
| <!--[[File:A (19,5)-torus knot.png|60px]]-->
| 19
| 2
| 19
|-
| 
| <!--20<sub>Y</sub>-->
| <!--[[File:A (10,3)-torus knot.png|60px]]-->
| 10
| 3
| 20
|-
| 
| <!--21<sub>Y</sub>-->
| [[File:A (7,4)-torus knot.png|60px]]
| 7
| 4
| 21
|-
| 
| <!--21<sub>Y</sub>-->
| <!--[[File:A (21,2)-torus knot.png|60px]]-->
| 21
| 2
| 21
|-
| 
| <!--22<sub>Y</sub>-->
| <!--[[File:A (11,3)-torus knot.png|60px]]-->
| 11
| 3
| 22
|-
| 
| <!--23<sub>Y</sub>-->
| <!--[[File:A (23,2)-torus knot.png|60px]]-->
| 23
| 2
| 23
|-
| 
| <!--24<sub>Y</sub>-->
| [[File:A (6,5)-torus knot.png|60px]]
| 6
| 5
| 24
|-
| 
| <!--25<sub>Y</sub>-->
| <!--[[File:A (25,2)-torus knot.png|60px]]-->
| 25
| 2
| 25
|-
| 
| <!--26<sub>Y</sub>-->
| <!--[[File:A (13,3)-torus knot.png|60px]]-->
| 13
| 3
| 26
|-
| 
| <!--27<sub>Y</sub>-->
| [[File:A (9,4)-torus knot.png|60px]]
| 9
| 4
| 27
|-
| 
| <!--27<sub>Y</sub>-->
| <!--[[File:A (27,2)-torus knot.png|60px]]-->
| 27
| 2
| 27
|-
| 
| <!--28<sub>Y</sub>-->
| [[File:A (7,5)-torus knot.png|60px]]
| 7
| 5
| 28
|-
| 
| <!--28<sub>Y</sub>-->
| <!--[[File:A (14,3)-torus knot.png|60px]]-->
| 14
| 3
| 28
|-
| 
| <!--29<sub>Y</sub>-->
| <!--[[File:A (29,2)-torus knot.png|60px]]-->
| 29
| 2
| 29
|-
| 
| <!--31<sub>Y</sub>-->
| <!--[[File:A (31,2)-torus knot.png|60px]]-->
| 31
| 2
| 31
|-
| 
| <!--32<sub>Y</sub>-->
| [[File:A (8,5)-torus knot.png|60px]]
| 8
| 5
| 32
|-
| 
| <!--32<sub>Y</sub>-->
| <!--[[File:A (16,3)-torus knot.png|60px]]-->
| 16
| 3
| 32
|-
| 
| <!--33<sub>Y</sub>-->
| <!--[[File:A (11,4)-torus knot.png|60px]]-->
| 11
| 4
| 33
|-
| 
| <!--33<sub>Y</sub>-->
| <!--[[File:A (33,2)-torus knot.png|60px]]-->
| 33
| 2
| 33
|-
| 
| <!--34<sub>Y</sub>-->
| <!--[[File:A (17,3)-torus knot.png|60px]]-->
| 17
| 3
| 34
|-
| 
| <!--35<sub>Y</sub>-->
| [[File:A (7,6)-torus knot.png|60px]]
| 7
| 6
| 35
|-
| 
| <!--35<sub>Y</sub>-->
| <!--[[File:A (35,2)-torus knot.png|60px]]-->
| 35
| 2
| 35
|-
| 
| <!--36<sub>Y</sub>-->
| [[File:A (9,5)-torus knot.png|60px]]
| 9
| 5
| 36
|-
| 
| <!--48<sub>Y</sub>-->
| [[File:A (8,7)-torus knot.png|60px]]
| 8
| 7
| 48
|-
| 
| <!--54<sub>Y</sub>-->
| [[File:A (9,7)-torus knot.png|60px]]
| 9
| 7
| 54
|-
| 
| <!--63<sub>Y</sub>-->
| [[File:A (9,8)-torus knot.png|60px]]
| 9
| 8
| 63
|}

==''g''-torus knot==
A '''g-torus knot''' is a closed curve drawn on a [[Genus g surface|g-torus]].  More technically, it is the homeomorphic image of a circle in '''S³''' which can be realized as a subset of a [[genus (mathematics)|genus]] ''g'' [[handlebody]] in '''S³'''.  If a [[link (knot theory)|link]] is a subset of a genus two handlebody, it is a '''double torus link'''.<ref>Rolfsen, Dale. ''Knots and Links'', {{page needed|date=November 2017}}. Publish or Perish, Inc. 1976. {{ISBN|0-914098-16-0}}.</ref>

For genus two, the simplest example of a double torus knot that is not a torus knot is the [[figure-eight knot (mathematics)|figure-eight knot]].<ref>Hill, P. "On double-torus knots (I)", ''Journal of Knot Theory and its Ramifications'', 1999, {{pages needed|date=November 2017}}.</ref><ref>Norwood, Rick. "Curves on Surfaces", ''Topology and its Applications'' 33 (1989) 241-246.</ref>

==See also==
*[[Alternating knot]]
*[[Irrational winding of a torus]]

== References ==

{{reflist}}

==External links==
*{{Knot Atlas|36_Torus_Knots|36 Torus Knots}}
*{{MathWorld|urlname=TorusKnot|title=Torus Knot}}
*[http://www.flexr.org/blog/?p=135 Torus knot renderer in Actionscript]
*[http://www.blackpawn.com/texts/pqtorus/ Fun with the PQ-Torus Knot]

{{Knot theory|state=collapsed}}

[[Category:Algebraic topology]]
[[Category:Fibered knots and links]]
[[Category:Torus knots and links| ]]