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Genus is superadditive under band connected sum. (English) Zbl 0621.57004

The author proves the result given in the title: Let \(k_ 1\), \(k_ 2\) be two knots in \(S^ 3\) which can be separated by an embedded 2-sphere. Theorem 1. If k is a band connected sum of \(k_ 1\) and \(k_ 2\), then genus k \(\geq\) genus \(k_ 1\) \(+\) genus \(k_ 2\). Equality holds if and only if there exists a Seifert surface for k which is a band connected sum (using the same band) of minimal genus Seifert surfaces for \(k_ 1\) and \(k_ 2\).
Reviewer: L.Neuwirth

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
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