# Split link

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{{short description|Concept in mathematical knot theory}}
{{For|the connecting link (or split link) of a Roller chain |Master link}}
In the [mathematical](/source/mathematics) field of [knot theory](/source/knot_theory), a '''split link''' is a [link](/source/link_(knot_theory)) 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.&nbsp;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](/source/link_complement) is reducible or irreducible as a [3-manifold](/source/3-manifold).  

A link with an alternating diagram, i.e. an [alternating link](/source/alternating_link), will be non-split if and only if this diagram is connected.  This is a result of the work of [William Menasco](/source/William_Menasco).<ref>{{citation|title=An Introduction to Knot Theory|series=Graduate Texts in Mathematics|volume=175|first=W. B. Raymond|last=Lickorish|authorlink=W. B. R. Lickorish|publisher=Springer|year=1997|isbn=9780387982540|page=32|url=https://books.google.com/books?id=PhHhw_kRvewC&pg=PA32}}.</ref>  A split link has many connected, non-alternating link diagrams.

==References==
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Category:Links (knot theory)

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