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In the [[mathematics|mathematical]] field of [[knot theory]], a '''split link''' is a [[link (knot theory)|link]] that has a (topological) 2-sphere in its complement separating one or more link components from the others.<ref>{{citation|title=Knots and Links|first=Peter R.|last=Cromwell|publisher=Cambridge University Press|year=2004|isbn=9780521548311|at=Definition 4.1.1, p. 78|url=https://books.google.com/books?id=djvbTNR2dCwC&pg=PA78}}.</ref> 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]]. |
In the [[mathematics|mathematical]] field of [[knot theory]], a '''split link''' is a [[link (knot theory)|link]] that has a (topological) 2-sphere in its complement separating one or more link components from the others.<ref>{{citation|title=Knots and Links|first=Peter R.|last=Cromwell|publisher=Cambridge University Press|year=2004|isbn=9780521548311|at=Definition 4.1.1, p. 78|url=https://books.google.com/books?id=djvbTNR2dCwC&pg=PA78}}.</ref> 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]]. |
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Revision as of 17:41, 30 June 2020
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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