Link group
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).
The link group of an n-component 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 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.
The link group of the n-component 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.
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