View source for Hopf link

{{Short description|Simplest nontrivial knot link}}
{{Infobox knot theory
| name=              
| practical name=    
| image=             Hopf Link.png
| caption=           
| arf invariant=     
| braid length=      2
| braid number=      2
| bridge number=     
| crossing number=   2
| hyperbolic volume= 0
| linking number=    1
| stick number=      6
| unknotting number= 1
| conway_notation=[2]
| ab_notation=       2{{sup sub|2|1}}
| dowker notation=   
| thistlethwaite=    L2a1
| other=             
 | alternating=      alternating
 | class=            torus
 | fibered=          fibered
 | slice=            
 | symmetry=         
 | tricolorable=     
| last link=         L0
| next link=         L4a1
}}
[[Image:Skein-relation-link22-plus-sm.png|thumb|right|[[Skein relation]] for the Hopf link.]]
In [[mathematics|mathematical]] [[knot theory]], the '''Hopf link''' is the simplest nontrivial [[link (knot theory)|link]] with more than one component.<ref>{{citation|title=The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots|first=Colin Conrad|last=Adams|publisher=American Mathematical Society|year=2004|isbn=9780821836781|page=151|url=https://books.google.com/books?id=M-B8XedeL9sC&pg=PA151}}.</ref>  It consists of two [[circle]]s linked together exactly once,<ref name="ks98"/> and is named after [[Heinz Hopf]].<ref name="ps97"/>

==Geometric realization==
A concrete model consists of two [[unit circle]]s in perpendicular planes, each passing through the center of the other.<ref name="ks98">{{citation
 | last1 = Kusner | first1 = Robert B.
 | last2 = Sullivan | first2 = John M. | author2-link = John M. Sullivan (mathematician)
 | contribution = On distortion and thickness of knots
 | doi = 10.1007/978-1-4612-1712-1_7
 | location = New York
 | mr = 1655037
 | pages = 67–78
 | publisher = Springer
 | series = IMA Vol. Math. Appl.
 | title = Topology and geometry in polymer science (Minneapolis, MN, 1996)
 | volume = 103
 | year = 1998}}. See in particular [https://books.google.com/books?id=FIPcAxs29ikC&pg=PA77 p.&nbsp;77].</ref> This model minimizes the [[ropelength]] of the link and until 2002 the Hopf link was the only link whose ropelength was known.<ref>{{citation
 | last1 = Cantarella | first1 = Jason
 | last2 = Kusner | first2 = Robert B.
 | last3 = Sullivan | first3 = John M. | author3-link = John M. Sullivan (mathematician)
 | arxiv = math/0103224
 | doi = 10.1007/s00222-002-0234-y
 | issue = 2
 | journal = Inventiones Mathematicae
 | mr = 1933586
 | pages = 257–286
 | title = On the minimum ropelength of knots and links
 | volume = 150
 | year = 2002| bibcode = 2002InMat.150..257C| s2cid = 730891
 }}.</ref> The [[convex hull]] of these two circles forms a shape called an [[oloid]].<ref>{{citation
 | last1 = Dirnböck | first1 = Hans
 | last2 = Stachel | first2 = Hellmuth
 | issue = 2
 | journal = Journal for Geometry and Graphics
 | mr = 1622664
 | pages = 105–118
 | title = The development of the oloid
 | url = http://www.heldermann-verlag.de/jgg/jgg01_05/jgg0113.pdf
 | volume = 1
 | year = 1997}}.</ref>

==Properties==
Depending on the relative [[Orientation (geometry)|orientation]]s of the two components the [[linking number]] of the Hopf link is ±1.<ref>{{harvtxt|Adams|2004}}, [https://books.google.com/books?id=M-B8XedeL9sC&pg=PA21 p.&nbsp;21].</ref>

The Hopf link is a (2,2)-[[torus link]]<ref>{{citation|title=On Knots|volume=115|series=Annals of Mathematics Studies|first=Louis H.|last=Kauffman|publisher=Princeton University Press|year=1987|isbn=9780691084350|page=373|url=https://books.google.com/books?id=BLvGkIY8YzwC&pg=PA373}}.</ref> with the [[braid word]]<ref>{{harvtxt|Adams|2004}}, Exercise 5.22, [https://books.google.com/books?id=M-B8XedeL9sC&pg=PA133 p.&nbsp;133].</ref>
:<math>\sigma_1^2.\,</math>

The [[knot complement]] of the Hopf link is '''R'''&nbsp;&times;&nbsp;''S''<sup>1</sup>&nbsp;&times;&nbsp;''S''<sup>1</sup>, the [[Cylinder (geometry)|cylinder]] over a [[torus]].<ref>{{citation|title=Quantum Invariants of Knots and 3-manifolds|volume=18|series=De Gruyter studies in mathematics|first=Vladimir G.|last=Turaev|publisher=Walter de Gruyter|year=2010|isbn=9783110221831|page=194|url=https://books.google.com/books?id=w7dActmezxQC&pg=PA194}}.</ref> This space has a [[Geometrization conjecture|locally Euclidean geometry]], so the Hopf link is not a [[hyperbolic link]]. The [[knot group]] of the Hopf link (the [[fundamental group]] of its complement) is '''Z'''<sup>2</sup> (the [[free abelian group]] on two generators), distinguishing it from an unlinked pair of loops which has the [[free group]] on two generators as its group.<ref>{{citation|title=Algebraic Topology|year=2002|first=Allen|last=Hatcher|isbn= 9787302105886|page=24|url=https://books.google.com/books?id=xsIiEhRfwuIC&pg=PA24}}.</ref>

The Hopf-link is not [[tricolorable]]: it is not possible to color the strands of its diagram with three colors, so that at least two of the colors are used and so that every crossing has one or three colors present. Each link has only one strand, and if both strands are given the same color then only one color is used, while if they are given different colors then the crossings will have two colors present.

==Hopf bundle==
The [[Hopf fibration]] is a continuous function from the [[3-sphere]] (a three-dimensional surface in four-dimensional Euclidean space) into the more familiar [[sphere|2-sphere]], with the property that the inverse image of each point on the 2-sphere is a circle. Thus, these images decompose the 3-sphere into a continuous family of circles, and
each two distinct circles form a Hopf link. This was Hopf's motivation for studying the Hopf link: because each two fibers are linked, the Hopf fibration is a nontrivial [[fibration]]. This example began the study of [[homotopy groups of spheres]].<ref>{{citation|title=Basic Algebraic Topology|first=Anant R.|last=Shastri|publisher=CRC Press|year=2013|isbn=9781466562431|url=https://books.google.com/books?id=lYMAAQAAQBAJ&pg=PA368|page=368}}.</ref>

== Biology ==
The Hopf link is also present in some proteins.<ref name=":0">{{citation|last1=Dabrowski-Tumanski|first1=Pawel|last2=Sulkowska|first2=Joanna I.|date=2017-03-28|title=Topological knots and links in proteins|journal=Proceedings of the National Academy of Sciences|language=en|volume=114|issue=13|pages=3415–3420|doi=10.1073/pnas.1615862114|issn=0027-8424|pmid=28280100|pmc=5380043|bibcode=2017PNAS..114.3415D |doi-access=free}}</ref><ref>{{citation|last1=Dabrowski-Tumanski|first1=Pawel|last2=Jarmolinska|first2=Aleksandra I.|last3=Niemyska|first3=Wanda|last4=Rawdon|first4=Eric J.|last5=Millett|first5=Kenneth C.|last6=Sulkowska|first6=Joanna I.|date=2017-01-04|title=LinkProt: a database collecting information about biological links|journal=Nucleic Acids Research|volume=45|issue=D1|pages=D243–D249|doi=10.1093/nar/gkw976|issn=0305-1048|pmc=5210653|pmid=27794552}}</ref> It consists of two covalent loops, formed by pieces of [[Peptide bond|protein backbone]], closed with [[Disulfide|disulfide bonds]]. The Hopf link topology is highly conserved in proteins and adds to their stability.<ref name=":0" />

==History==
[[File:Buzanha wachigai mon.jpg|thumb|[[Shingon-shu Buzan-ha|Buzan-ha]] crest]]
The Hopf link is named after topologist [[Heinz Hopf]], who considered it in 1931 as part of his research on the [[Hopf fibration]].<ref>{{citation
 |last= Hopf
 |first= Heinz
 |author-link= Heinz Hopf
 |title= Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche
 |journal= [[Mathematische Annalen]]
 |publisher= [[Springer-Verlag|Springer]]
 |location= Berlin
 |volume= 104
 |issue= 1
 |pages= 637–665
 |year= 1931
 |url= http://resolver.sub.uni-goettingen.de/purl?GDZPPN002274760
 |doi= 10.1007/BF01457962
 |s2cid= 123533891
 }}.</ref> However, in mathematics, it was known to [[Carl Friedrich Gauss]] before the work of Hopf.<ref name="ps97">{{citation
 | last1 = Prasolov | first1 = V. V.
 | last2 = Sossinsky | first2 = A. B.
 | isbn = 0-8218-0588-6
 | location = Providence, RI
 | mr = 1414898
 | page = 6
 | publisher = American Mathematical Society
 | series = Translations of Mathematical Monographs
 | title = Knots, links, braids and 3-manifolds: An introduction to the new invariants in low-dimensional topology
 | url = https://books.google.com/books?id=znCLtJKnZXQC&pg=PA6
 | volume = 154
 | year = 1997}}.</ref> It has also long been used outside mathematics, for instance as the crest of [[Shingon-shu Buzan-ha|Buzan-ha]], a Japanese Buddhist sect founded in the 16th century.

==See also==
*[[Borromean rings]], a link with three closed loops
*[[Catenane]], a molecule with two linked loops
*[[Solomon's knot]], two loops which are doubly linked

==References==
{{reflist|30em}}

==External links==
{{commons category|Hopf links}}
*{{MathWorld|urlname=HopfLink|title=Hopf Link|mode=cs2}}
*{{Knot Atlas|L2a1|Hopf link}}
*[http://linkprot.cent.uw.edu.pl/ "LinkProt" - the database of known protein links.]

{{Knot theory|state=collapsed}}

[[Category:Prime knots and links]]