Split link

In the mathematical field of knot theory, a split link is a link that has a (topological) 2-sphere in its complement separating one or more link components from the others.[1] A split link is said to be splittable, and a link that is not split is called a non-split link or not splittable. Whether a link is split or non-split corresponds to whether the link complement is reducible or irreducible as a 3-manifold.

A link with an alternating diagram, i.e. an alternating link, will be non-split if and only if this diagram is connected. This is a result of the work of William Menasco.[2] A split link has many connected, non-alternating link diagrams.

References

Cromwell, Peter R. (2004), Knots and Links, Cambridge University Press, Definition 4.1.1, p. 78, ISBN 9780521548311.
Lickorish, W. B. Raymond (1997), An Introduction to Knot Theory, Graduate Texts in Mathematics, 175, Springer, p. 32, ISBN 9780387982540.

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[1] Cromwell, Peter R. (2004), Knots and Links, Cambridge University Press, Definition 4.1.1, p. 78, ISBN 9780521548311.

[2] Lickorish, W. B. Raymond (1997), An Introduction to Knot Theory, Graduate Texts in Mathematics, 175, Springer, p. 32, ISBN 9780387982540.