Simple-ribbon fusions on non-split links.

*(English)*Zbl 1369.57011Let \(\ell\) 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.

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 (GenĂ¨ve)

##### MSC:

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

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\textit{K. Kishimoto} et al., J. Knot Theory Ramifications 26, No. 3, Article ID 1741005, 15 p. (2017; Zbl 1369.57011)

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##### 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 |

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