// Workers AI · dad joke modeWhat did the split link say? I'm chained to relationships.

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, vol. 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, vol. 175, Springer, p. 32, ISBN 9780387982540.

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// Workers AI · dad joke modeWhat did the split link say? I'm chained to relationships.

References