Kishimoto, Kengo; Shibuya, Tetsuo; Tsukamoto, Tatsuya Simple-ribbon fusions on non-split links. (English) Zbl 1369.57011 J. Knot Theory Ramifications 26, No. 3, Article ID 1741005, 15 p. (2017). Let \(\ell\) be an ordered, oriented link in \(S^3\) and \(\mathcal{O}=O_1\sqcup \dots \sqcup O_m\) be an \(m\)-component trivial link that is split from \(\ell\). An \(m\)-ribbon fusion \(L\) of \(\ell\) is obtained by performing a band sum of \(\ell\) and \(\mathcal{O}\), using \(m\) disjoint bands \(B_1,\dots,B_m\) such that each \(B_i\) joins \(O_i\) with some component of \(\ell\). A natural question is the following: are there conditions on the ribbon fusion which ensure that \(L\) is isotopic to \(\ell\)?Kishimoto-Shibuya-Tsukamoto provided a characterization of such fusions by defining the notion of a trivial simple-ribbon fusion (trivial SR-fusion): In [K. Kishimoto et al., J. Math. Soc. Japan 68, No. 3, 1033–1045 (2016; Zbl 1357.57015)], they showed that \(L\) is isotopic to \(\ell\) if and only if the SR-fusion is trivial. This generalises an older result independently obtained by D. Gabai [Topology 26, 209–210 (1987; Zbl 0621.57004)] and M. Scharlemann [J. Differ. Geom. 29, No. 3, 557–614 (1989; Zbl 0673.57015)] in the case of knots.The paper under review deals with a further study of trivial SR-fusions. Namely, in the case where \(\ell\) is non-split, the authors provide a more precise description of trivial SR-fusions. In their own words, they “show that only ‘clearly trivial’ simple-ribbon fusion preserves a link type if the link is non split”. As a second main result, the authors use their characterization to relate the genera of \(\ell\) and \(L\), when \(\ell\) is non-split. Reviewer: Anthony Conway (Genève) Cited in 1 Document MSC: 57M25 Knots and links in the \(3\)-sphere (MSC2010) Keywords:ribbon links; fusions; genera of knots PDF BibTeX XML Cite \textit{K. Kishimoto} et al., J. Knot Theory Ramifications 26, No. 3, Article ID 1741005, 15 p. (2017; Zbl 1369.57011) Full Text: DOI References: [1] C. Goldberg, On the genera of links, Ph.D. thesis, Princeton University (1970). [2] Kishimoto, K.; Shibuya, T.; Tsukamoto, T., Simple ribbon fusions and genera for links, J. Math. Soc. Japan, 68, 1033-1045, (2016) · Zbl 1357.57015 [3] Kobayashi, K.; Shibuya, T.; Tsukamoto, T., Simple ribbon moves for links, Osaka J. Math., 51, 545-571, (2014) · Zbl 1322.57009 [4] Shibuya, T.; Tsukamoto, T., Simple ribbon moves and primeness of knots, Tokyo J. Math., 51, 147-161, (2013) · Zbl 1282.57012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.