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In [[knot theory]], an area of [[mathematics]], the '''link group''' of a [[Link (knot theory)|link]] is an analog of the [[knot group]] of a [[Knot (mathematics)|knot]]. They were described by [[John Milnor]] in his Bachelor's thesis, {{harv|Milnor|1954}}.
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[[File:Whiteheadlink.png|thumb|The [[Whitehead link]] is link homotopic to the [[unlink]], but not [[isotopy (disambiguation)|isotopic]] to the unlink.]]
The link group of an ''n''-component link is essentially the set of (''n''&nbsp;+&nbsp;1)-component links extending this link, up to '''link homotopy.''' {{anchor|link homotopy}} In other words, each component of the extended link is allowed to move through [[regular homotopy]] (homotopy through [[immersion (mathematics)|immersions]]), knotting or unknotting itself, but is not allowed to move through other component. This is a weaker condition than isotopy: for example, the [[Whitehead link]] has [[linking number]]&nbsp;0, and thus is link homotopic to the [[unlink]], but it is not [[isotopy (disambiguation)|isotopic]] to the unlink.
 
It is not the [[fundamental group]] of the [[link complement]], since the components of the link are allowed to move through themselves, though not each other, but thus is a [[quotient group]] of the link complement, since one can start with this, and then by knotting or unknotting components, some of these elements may become equivalent to each other.
 
== Examples ==
The link group of the ''n''-component unlink is the [[free group]] on ''n'' generators, <math>F_n</math>, as the link group of a single link is the [[knot group]] of the [[unknot]], which is the integers, and the link group of an unlinked union is the [[free product]] of the link groups of the components.
 
[[File:Hopf link.svg|thumb|The link group of the [[Hopf link]] is <math>\mathbf{Z}^2.</math>]]
The link group of the [[Hopf link]], the simplest non-trivial link – two circles, linked once – is the [[free abelian group]] on two generators, <math>\mathbf{Z}^2.</math> Note that the link group of two ''unlinked'' circles is the free ''non''abelian group on two generators, of which the free abelian group on two generators is a [[quotient group|quotient]]. In this case the link group is the fundamental group of the link complement, as the link complement deformation retracts onto a torus.
 
The [[Whitehead link]] is link homotopic to the unlink – though it is not isotopic to the unlink – and thus has link group the free group on two generators.
 
==Milnor invariants<!--[[Milnor invariants]] links directly here-->==
Milnor defined invariants of a link (functions on the link group) in {{harv|Milnor|1954}}, using the character <math>\bar{\mu},</math> which have thus come to be called "Milnor's ''μ''-bar invariants", or simply the "Milnor invariants". For each ''k'', there is an ''k''-ary function <math>\bar{\mu},</math> which defines invariants according to which ''k'' of the links one selects, in which order.
 
Milnor's invariants can be related to [[Massey product]]s on the link complement (the complement of the link); this was suggested in {{harv|Stallings|1965}}, and made precise in {{harv|Turaev|1976}} and {{harv|Porter|1980}}.
 
As with Massey products, the Milnor invariants of length ''k''&nbsp;+&nbsp;1 are defined if all Milnor invariants of length less than or equal to ''k'' vanish. The first (2-fold) Milnor invariant is simply the linking number (just as the 2-fold Massey product is the cup product, which is dual to intersection), while the 3-fold Milnor invariant measures whether 3 pairwise unlinked circles are Borromean rings, and if so, in some sense, how many times (i.e., Borromean rings have a Milnor 3-fold invariant of 1 or –1, depending on order, but other 3-element links can have an invariant of 2 or more, just as linking numbers can be greater than 1).
 
Another definition is the following: let's consider a link <math>L=L_1\cup L_2\cup L_3</math>. Suppose that <math>lk(L_i,L_j)=0; i,j=1,2,3; i<j</math>. Find any Seifert surfaces for link components- <math>F_1,F_2,F_3</math> correspondingly, such that <math>F_i\cap L_j=\empty, i\ne j</math>. Then the Milnor 3-fold invariant equals ''minus'' the number of triple points in <math>F_1\cap F_2,\cap F_3</math> counting with signs; {{harv|Cochran|1990}}.
 
Milnor invariants can also be defined if the lower order invariants do not vanish, but then there is an indeterminacy, which depends on the values of the lower order invariants. This indeterminacy can be understood geometrically as the indeterminacy in expressing a link as a closed string link, as discussed below (it can also be seen algebraically as the indeterminacy of Massey products if lower order Massey products do not vanish). 
 
Milnor invariants can be considered as invariants of [[string links]], in which case they are universally defined, and the indeterminacy of the Milnor invariant of a link is precisely due to the multiple ways that a given links can be cut into a string link; this allows the classification of links up to link homotopy, as in {{harv|Habegger|Lin|1990}}. Viewed from this point of view, Milnor invariants are [[finite type invariant]]s, and in fact they (and their products) are the only rational finite type concordance invariants of string links; {{harv|Habegger|Masbaum|2000}}.
 
The number of linearly independent Milnor invariants of length ''k''+1 is <math>mN_k - N_{k+1},</math> where <math>N_k</math> is the number of basic commutators of length ''k'' in the [[free Lie algebra]], namely:
:<math>N_k = \frac{1}{k}\sum_{d|m}\phi(d)\left(m^{k/d}\right),</math>
where <math>\phi</math> is the [[Möbius function]], which was shown in {{harv|Orr|1989}}. This grows on the order of <math>m^{k+1}/k^2.</math>
 
== Applications ==
Link groups can be used to classify [[Brunnian link]]s.
 
== See also ==
* [[Knot group]]
* [[Regular homotopy]]
 
== References ==
* {{citation
|last=Cochran
|first=Tim D.
|title=Derivatives of links: Milnor’s concordance invariants and Massey’s Products
|journal=[[Memoirs of the American Mathematical Society]]
|volume=427
|year=1990
|publisher=American Mathematical Society
}}
* {{citation
|title=The classification of links up to homotopy
|journal=[[Journal of the American Mathematical Society]]
|first1=Nathan
|last1=Habegger
|first2=X.S.
|last2=Lin
|volume=3
|series=2
|year=1990
|pages=389–419
|doi=10.2307/1990959
|jstor=1990959
|issue=2
|publisher=American Mathematical Society
}}
* {{citation
|title=The Kontsevich integral and Milnor's invariants
|journal=Topology
|first1=Nathan
|last1=Habegger
|first2=Gregor
|last2=Masbaum
|volume=39
|year=2000
|pages=1253–1289
|doi=10.1016/S0040-9383(99)00041-5 
|postscript=, [http://people.math.jussieu.fr/~masbaum/K39.ps.gz preprint].
|issue=6
}}
*{{citation
|last=Milnor
|first=John
|author-link=John Milnor
|journal=[[Annals of Mathematics]]
|date=March 1954
|volume=59
|title=Link Groups
|pages=177–195
|jstor=1969685
|doi=10.2307/1969685
|issue=2
|publisher=Annals of Mathematics
}}
* {{citation
|title=Homotopy invariants of links
|last=Orr
|first=Kent E.
|journal=[[Inventiones Mathematicae]]
|volume=95
|pages=379–394
|year=1989
|doi=10.1007/BF01393902
|issue=2
}}
* {{citation
|title=Milnor's ''μ''-invariants and Massey products
|last=Porter
|first=R.
|journal=[[Transactions of the American Mathematical Society]]
|year=1980
|volume=257
|pages=39–71
|doi=10.2307/1998124
|jstor=1998124
|issue=1
|publisher=American Mathematical Society
}}
* {{citation
|title=Homology and central series of groups
|last=Stallings
|first=John R.
|author-link=John R. Stallings
|journal=[[Journal of Algebra]]
|year=1965
|volume=2
|pages=170–181
|doi=10.1016/0021-8693(65)90017-7
|issue=2
}}
* {{citation
|title=The Milnor invariants and Massey products
|last=Turaev
|first=V. G.
|journal=Studies in Topology-II
|year=1976
|volume=66
|pages=189–203
}}
 
{{Knot theory|state=collapsed}}
 
[[Category:Knot invariants]]
[[Category:Links (knot theory)]]

Latest revision as of 22:06, 9 January 2015

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