View source for Whitehead link

{{short description|Two interlinked loops with five structural crossings}}
{{Infobox knot theory
| name=              Whitehead link
| practical name=    
| image=             Whiteheadlink.png
| caption=           
| arf invariant=     
| braid length=      5
| braid number=      3
| bridge number=     
| crossing number=   5
| hyperbolic volume= 3.663862377
| linking number=    0
| stick number=      
| unknotting number= 1
| conway_notation=   [212]
| ab_notation=       5{{sup sub|2|1}}
| dowker notation=    
| thistlethwaite=    L5a1
| other=             
| alternating=       alternating
| amphichiral=      
| fibered=          
| slice=            
| tricolorable=     
| last link=         L4a1
| next link=         L6a1
}}
In [[knot theory]], the '''Whitehead link''', named for [[J. H. C. Whitehead]], is one of the most basic [[link (knot theory)|links]]. It can be drawn as an [[alternating link]] with five crossings, from the overlay of a circle and a [[lemniscate|figure-eight shaped]] loop.

==Structure==
{{multiple image
| total_width = 300
| image1      = Whitehead-link.svg
| caption1    = Alternating [[link diagram]]
| image2      = Symmetric Whitehead link.svg
| caption2    = Alternative diagram, symmetric by 3d rotation around a vertical line in the plane of the drawing{{r|skopenkov}}
}}

A common way of describing this knot is formed by overlaying a [[lemniscate|figure-eight shaped]] loop with another circular loop surrounding the crossing of the figure-eight. The above-below relation between these two [[unknot]]s is then set as an [[alternating link]], with the consecutive crossings on each loop alternating between under and over. This drawing has five crossings, one of which is the self-crossing of the figure-eight curve, which does not count towards the [[linking number]]. Because the remaining crossings have equal numbers of under and over crossings on each loop, its linking number is 0. It is not [[Homotopy#Isotopy|isotopic]] to the [[unlink]], but it is [[link homotopic]] to the unlink.

Although this construction of the knot treats its two loops differently from each other, the two loops are topologically symmetric: it is possible to deform the same link into a drawing of the same type in which the loop that was drawn as a figure eight is circular and vice versa.{{r|cr}} Alternatively, there exist realizations of this knot in three dimensions in which the two loops can be taken to each other by a geometric symmetry of the realization.{{r|skopenkov}}

In [[braid theory]] notation, the link is written

:<math>\sigma^2_1\sigma^2_2\sigma^{-1}_1\sigma^{-2}_2.\,</math>

Its [[Alexander polynomial]] is

:<math>\Delta (t)=t^{3/2}-3t^{1/2}+3t^{-1/2}-t^{-3/2},</math>

since <math>\begin{pmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & 1 & -1\end{pmatrix}</math> is a possible [[Seifert surface|Seifert matrix]], or because of its [[Alexander-Conway polynomial|Conway polynomial]], which is

:<math>\nabla(z)=z^3 .</math>

Its [[Jones polynomial]] is

:<math>V(t) = t^{-{3 \over 2}}\left(-1 + t - 2t^2 + t^3 - 2t^4 + t^5\right).</math>

This polynomial and <math>V(1/t)</math> are the two factors of the Jones polynomial of the [[L10a140 link]]. Notably, <math>V(1/t)</math> is the Jones polynomial for the mirror image of a link having Jones polynomial <math>V(t)</math>.

==Volume==
The [[hyperbolic volume]] of the complement of the Whitehead link is {{math|4}} times [[Catalan's constant]], approximately 3.66. The Whitehead link complement is one of two two-cusped hyperbolic manifolds with the minimum possible volume, the other being the complement of the [[pretzel link]] with parameters {{math|(−2, 3, 8)}}.{{r|agol}}

Dehn filling on one component of the Whitehead link can produce the sibling manifold of the complement of the [[figure-eight knot (mathematics)|figure-eight knot]], and Dehn filling on both components can produce the [[Weeks manifold]], respectively one of the minimum-volume hyperbolic manifolds with one cusp and the minimum-volume hyperbolic manifold with no cusps.

==History==
[[File:Torshammare Ödeshög (Montelius 1906 s309).jpg|thumb|upright=0.5|Old [[Mjölnir|Thor's hammer]] archaeological artefact]]
The Whitehead link is named for [[J. H. C. Whitehead]], who spent much of the 1930s looking for a proof of the [[Poincaré conjecture]]. In 1934, he used the link as part of his construction of the now-named [[Whitehead manifold]], which refuted his previous purported proof of the conjecture.<ref>{{citation
 | last = Gordon | first = C. McA. | author-link = Cameron Gordon (mathematician)
 | editor-last = James | editor-first = I. M.
 | contribution = 3-dimensional topology up to 1960
 | contribution-url = https://homepages.warwick.ac.uk/~masgar/Teach/2021_3MFDS/References/1999three_dimensional_topology_up_to_1960.pdf
 | doi = 10.1016/B978-044482375-5/50016-X
 | location = Amsterdam
 | mr = 1674921
 | pages = 449–489
 | publisher = North-Holland
 | title = History of Topology
 | year = 1999}}; see p. 480</ref>

==See also==
{{commonscat|Whitehead links}}
* [[Solomon's knot]]
* [[Weeks manifold]]
* [[Whitehead double]]

==References==
{{reflist|refs=

<ref name=agol>{{citation
 | last = Agol | first = Ian | authorlink = Ian Agol
 | arxiv = 0804.0043
 | doi = 10.1090/S0002-9939-10-10364-5
 | issue = 10
 | journal = [[Proceedings of the American Mathematical Society]]
 | mr = 2661571
 | pages = 3723–3732
 | title = The minimal volume orientable hyperbolic 2-cusped 3-manifolds
 | volume = 138
 | year = 2010}}</ref>

<ref name=cr>{{citation
 | last1 = Cundy | first1 = H. Martyn | author1-link = Martyn Cundy
 | last2 = Rollett | first2 = A.P.
 | edition = 2nd
 | location = Oxford
 | mr = 0124167
 | publisher = Clarendon Press
 | title = Mathematical models
 | title-link = Mathematical Models (Cundy and Rollett)
 | year = 1961
 | page = 59}}</ref>

<ref name=skopenkov>{{citation|first=A.|last=Skopenkov|title=A user's guide to basic knot and link theory|year=2020|arxiv=2001.01472v1|contribution=Fig. 22: Isotopy of the Whitehead link|page=17}}</ref>

}}

==External links==
*{{Knot Atlas|L5a1|L5a1 knot-theoretic link}}
*{{mathworld|urlname=WhiteheadLink|title=Whitehead link|mode=cs2}}

{{Knot theory|state=collapsed}}

[[Category:Algebraic topology]]
[[Category:Geometric topology]]
[[Category:Hyperbolic knots and links]]
[[Category:Prime knots and links]]