In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Bachelor's thesis, (Milnor 1954).
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How can you play a Rubik's Cube? Not play with it, but play it like a piano? That question doesn't make a lot of sense at first, but an abstract mathematical field called group theory holds the answer, if you'll bear with me. In math, a group is a particular collection of elements. That might be a set of integers, the face of a Rubik's Cube, or anything, so long as they follow four specific rules, or axioms. Axiom one: all group operations must be closed or restricted to only group elements. So in our square, for any operation you do,q like turn it one way or the other, you'll still wind up with an element of the group. Axiom two: no matter where we put parentheses when we're doing a single group operation, we still get the same result. In other words, if we turn our square right two times, then right once, that's the same as once, then twice, or for numbers, one plus two is the same as two plus one. Axiom three: for every operation, there's an element of our group called the identity. When we apply it to any other element in our group, we still get that element. So for both turning the square and adding integers, our identity here is zero, not very exciting. Axiom four: every group element has an element called its inverse also in the group. When the two are brought together using the group's addition operation, they result in the identity element, zero, so they can be thought of as cancelling each other out. So that's all well and good, but what's the point of any of it? Well, when we get beyond these basic rules, some interesting properties emerge. For example, let's expand our square back into a fullfledged Rubik's Cube. This is still a group that satisfies all of our axioms, though now with considerably more elements and more operations. We can turn each row and column of each face. Each position is called a permutation, and the more elements a group has, the more possible permutations there are. A Rubik's Cube has more than 43 quintillion permutations, so trying to solve it randomly isn't going to work so well. However, using group theory we can analyze the cube and determine a sequence of permutations that will result in a solution. And, in fact, that's exactly what most solvers do, even using a group theory notation indicating turns. And it's not just good for puzzle solving. Group theory is deeply embedded in music, as well. One way to visualize a chord is to write out all twelve musical notes and draw a square within them. We can start on any note, but let's use C since it's at the top. The resulting chord is called a diminished seventh chord. Now this chord is a group whose elements are these four notes. The operation we can perform on it is to shift the bottom note to the top. In music that's called an inversion, and it's the equivalent of addition from earlier. Each inversion changes the sound of the chord, but it never stops being a C diminished seventh. In other words, it satisfies axiom one. Composers use inversions to manipulate a sequence of chords and avoid a blocky, awkward sounding progression. On a musical staff, an inversion looks like this. But we can also overlay it onto our square and get this. So, if you were to cover your entire Rubik's Cube with notes such that every face of the solved cube is a harmonious chord, you could express the solution as a chord progression that gradually moves from discordance to harmony and play the Rubik's Cube, if that's your thing.
Contents
Definition
The link group of an ncomponent link is essentially the set of (n + 1)component links extending this link, up to link homotopy. In other words, each component of the extended link is allowed to move through regular homotopy (homotopy through 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 0, and thus is link homotopic to the unlink, but it is not 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 ncomponent unlink is the free group on n generators, , 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.
The link group of the Hopf link, the simplest nontrivial link – two circles, linked once – is the free abelian group on two generators, Note that the link group of two unlinked circles is the free nonabelian group on two generators, of which the free abelian group on two generators is a 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 defined invariants of a link (functions on the link group) in (Milnor 1954), using the character which have thus come to be called "Milnor's μbar invariants", or simply the "Milnor invariants". For each k, there is an kary function which defines invariants according to which k of the links one selects, in which order.
Milnor's invariants can be related to Massey products on the link complement (the complement of the link); this was suggested in (Stallings 1965), and made precise in (Turaev 1976) and (Porter 1980).
As with Massey products, the Milnor invariants of length k + 1 are defined if all Milnor invariants of length less than or equal to k vanish. The first (2fold) Milnor invariant is simply the linking number (just as the 2fold Massey product is the cup product, which is dual to intersection), while the 3fold 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 3fold invariant of 1 or –1, depending on order, but other 3element 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 . Suppose that . Find any Seifert surfaces for link components correspondingly, such that . Then the Milnor 3fold invariant equals minus the number of intersection points in counting with signs; (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 (Habegger & Lin 1990). Viewed from this point of view, Milnor invariants are finite type invariants, and in fact they (and their products) are the only rational finite type concordance invariants of string links; (Habegger & Masbaum 2000).
The number of linearly independent Milnor invariants of length k+1 is where is the number of basic commutators of length k in the free Lie algebra, namely:
where is the Möbius function, which was shown in (Orr 1989). This grows on the order of
Applications
Link groups can be used to classify Brunnian links.
See also
References
 Cochran, Tim D. (1990), "Derivatives of links: Milnor’s concordance invariants and Massey’s Products", Memoirs of the American Mathematical Society, American Mathematical Society, 427
 Habegger, Nathan; Lin, X.S. (1990), "The classification of links up to homotopy", Journal of the American Mathematical Society, 2, American Mathematical Society, 3 (2): 389–419, JSTOR 1990959, doi:10.2307/1990959
 Habegger, Nathan; Masbaum, Gregor (2000), "The Kontsevich integral and Milnor's invariants", Topology, 39 (6): 1253–1289, doi:10.1016/S00409383(99)000415, preprint.
 Milnor, John (March 1954), "Link Groups", Annals of Mathematics, Annals of Mathematics, 59 (2): 177–195, JSTOR 1969685, doi:10.2307/1969685
 Orr, Kent E. (1989), "Homotopy invariants of links", Inventiones Mathematicae, 95 (2): 379–394, doi:10.1007/BF01393902
 Porter, R. (1980), "Milnor's μinvariants and Massey products", Transactions of the American Mathematical Society, American Mathematical Society, 257 (1): 39–71, JSTOR 1998124, doi:10.2307/1998124
 Stallings, John R. (1965), "Homology and central series of groups", Journal of Algebra, 2 (2): 170–181, doi:10.1016/00218693(65)900177
 Turaev, V. G. (1976), "The Milnor invariants and Massey products", Studies in TopologyII, 66: 189–203