In Simple ribbon fusions and genera for links, J. Math. Soc. Japan 68 (2016) 1033-1045, we introduced special types of fusions, so-called simple-ribbon fusions on links, and showed that they never decrease the genera of links. In this paper, we also gave a geometric condition for a simple-ribbon fusion to preserve a link type, and it turns out that a "complicated" simple-ribbon fusion may preserve a link type if the link is split. In this paper, we show that only "clearly trivial" simple-ribbon fusion preserves a link type if the link is non-split. This enables us to give a lower bound for the difference between genera of knots related by a simple-ribbon fusion.
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