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{{short description|Two interlinked loops with five structural crossings}}
{{Infobox knot theory
{{Infobox knot theory
| name= Whitehead link
| name= Whitehead link
Line 12: Line 13:
| linking number= 0
| linking number= 0
| stick number=
| stick number=
| unknotting number= 2
| unknotting number= 1
| conway_notation=
| conway_notation= [212]
| ab_notation= 5{{sup sub|2|1}}
| ab_notation= 5{{sup sub|2|1}}
| dowker notation=
| dowker notation=
| thistlethwaite= L5a1
| thistlethwaite= L5a1
| other=
| other=
| alternating= alternating
| alternating= alternating
| amphichiral=
| amphichiral=
| fibered=
| fibered=
| slice=
| slice=
| tricolorable=
| tricolorable=
| last link= L4a1
| last link= L4a1
| next link= L6a1
| 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.
[[File:Whitehead-link.svg|thumb|100px|Simple depiction]]
⚫
[[File:Torshammare Ödeshög (Montelius 1906 s309).jpg|thumb|175px|Old [[Mjölnir|Thor's hammer]] archaeological artefact]]


⚫
==Structure==
⚫
In [[knot theory]], the '''Whitehead link''', discovered by [[J.H.C. Whitehead]], is one of the most basic [[link (knot theory)|links]].
{{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 [[Face-transitive#Related terms|isotopic]] to the [[unlink]], but it is [[link homotopic]] to the unlink.
⚫
J.H.C. Whitehead spent much of the 1930s looking for a proof of the [[Poincaré conjecture]]. In 1934, the Whitehead link was used as part of his construction of the now-named [[Whitehead manifold]], which refuted his previous purported proof of the conjecture.


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}}
⚫
==Structure==
The link is created with two projections of the [[unknot]]: one circular loop and one figure eight-shaped loop (i.e., a loop with a [[Reidemeister move|Reidemeister Type I move]] applied) intertwined such that they are inseparable and neither loses its form. Excluding the instance where the figure eight thread intersects itself, the Whitehead link has four crossings. Because each underhand crossing has a paired upperhand crossing, its [[linking number]] is 0. It is not [[Face-transitive#Related terms|isotopic]] to the [[unlink]], but it is [[link homotopic]] to the unlink.


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


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


Its [[Jones polynomial]] is
Its [[Jones polynomial]] is


:<math>V(t)=t^{- {3 \over 2}}(-1+t-2t^2+t^3-2t^4+t^5).</math>
:<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]]. (Note that <math>V(1/t)</math> is the Jones polynomial for the mirror image of a link having Jones polynomial <math>V(t)</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==
==See also==
{{commonscat|Whitehead links}}
* [[Solomon's knot]]
* [[Solomon's knot]]
* [[Weeks manifold]]
* [[Weeks manifold]]
* [[Whitehead double]]
* [[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==
==External links==
*{{Knot Atlas|L5a1|L5a1 knot-theoretic link}}
*{{Knot Atlas|L5a1|L5a1 knot-theoretic link}}
*{{mathworld|urlname=WhiteheadLink|title=Whitehead link}}
*{{mathworld|urlname=WhiteheadLink|title=Whitehead link|mode=cs2}}


{{Knot theory|state=collapsed}}
{{Knot theory|state=collapsed}}
Line 59: Line 117:
[[Category:Algebraic topology]]
[[Category:Algebraic topology]]
[[Category:Geometric topology]]
[[Category:Geometric topology]]
[[Category:Hyperbolic knots and links]]
[[Category:Prime knots and links]]

Latest revision as of 18:56, 26 December 2021

Whitehead link
Braid length5
Braid no.3
Crossing no.5
Hyperbolic volume3.663862377
Linking no.0
Unknotting no.1
Conway notation[212]
A–B notation52
1
ThistlethwaiteL5a1
Last / NextL4a1 / L6a1
Other
alternating

In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links. It can be drawn as an alternating link with five crossings, from the overlay of a circle and a figure-eight shaped loop.

Structure

[edit]
Alternating link diagram
Alternative diagram, symmetric by 3d rotation around a vertical line in the plane of the drawing[1]

A common way of describing this knot is formed by overlaying a figure-eight shaped loop with another circular loop surrounding the crossing of the figure-eight. The above-below relation between these two unknots 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 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.[2] 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.[1]

In braid theory notation, the link is written

Its Jones polynomial is

This polynomial and are the two factors of the Jones polynomial of the L10a140 link. Notably, is the Jones polynomial for the mirror image of a link having Jones polynomial .

Volume

[edit]

The hyperbolic volume of the complement of the Whitehead link is 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 (−2, 3, 8).[3]

Dehn filling on one component of the Whitehead link can produce the sibling manifold of the complement of the 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

[edit]
Old 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.[4]

See also

[edit]
Wikimedia Commons has media related to Whitehead links.

References

[edit]
  1. ^ a b Skopenkov, A. (2020), "Fig. 22: Isotopy of the Whitehead link", A user's guide to basic knot and link theory, p. 17, arXiv:2001.01472v1
  2. ^ Cundy, H. Martyn; Rollett, A.P. (1961), Mathematical models (2nd ed.), Oxford: Clarendon Press, p. 59, MR 0124167
  3. ^ Agol, Ian (2010), "The minimal volume orientable hyperbolic 2-cusped 3-manifolds", Proceedings of the American Mathematical Society, 138 (10): 3723–3732, arXiv:0804.0043, doi:10.1090/S0002-9939-10-10364-5, MR 2661571
  4. ^ Gordon, C. McA. (1999), "3-dimensional topology up to 1960" (PDF), in James, I. M. (ed.), History of Topology, Amsterdam: North-Holland, pp. 449–489, doi:10.1016/B978-044482375-5/50016-X, MR 1674921; see p. 480
[edit]
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