nLab
link

Contents

Idea

A link is a generalisation of a knot where one is allowed more than one component. Many knot invariants extend to link invariants and for many such invariants, one needs to know this extension to compute it even for a knot. Thus the study of links and knots is inextricably intertwined.

Basics

It is possible to generalise this to more varied sources and targets.

Links can be studied in a number of ways depending on the notion of equivalence that is used. Coming from knot theory, one considers equivalence up to isotopy; that is, two links are equivalent if there is a homotopy between them which is an embedding for all times. A weaker notion was consider by Milnor wherein the components of the link are allowed to pass through themselves, but not through other components. That is, when restricted to each component it must be an immersion for all times, and the images of the components must always be disjoint.

Examples

Any knot is a link, and any disjoint union of unknots (called an unlink) is a link. We may call these ‘trivial’ (hopefully this name isn't standard for something different), in the sense of what you would know about before you study links.

The Hopf link is the simplest non-trivial (in the sense above) link, consisting of two components linked once.

It is possible to link together nn circles in such a way that removing any one makes the others fall apart. For n=2n = 2, we have the Hopf link above; for n=3n = 3, we have the Borromean link, or Borromean Rings.

The Whitehead link is an example of a link that shows the difference between the two notions of equivalence. If the links are only allowed to move by isotopies, then the two components are linked. However, if a link is allowed to pass through itself, then they can be unlinked.

A Brunnian link is a link which is not an unlink but which has the property that the removal of any of its components results in an unlink. Technically, this includes the Hopf link and any knot (thanks to this MO question for settling that issue). The Borromean rings above are an example of a Brunnian link with three components.

References

  • Milnor, J. (1954). Link groups. Ann. of Math. (2), 59, 177–195. MR

Revised on October 21, 2010 00:40:22 by Toby Bartels (98.16.136.70)