Smooth spheres in \({\mathbb{R}}^ 4\) with four critical points are standard.

*(English)*Zbl 0559.57019The following result is proved. Main Theorem: Suppose \(\gamma_ 0\) and \(\gamma_ 1\) are knots in the 3-sphere and a certain band sum of \(\gamma_ 0\) and \(\gamma_ 1\) yields the unknot. Then \(\gamma_ 0\) and \(\gamma_ 1\) are the unknot and the band sum is just the connected sum. This answers questions 1.1, 1.2A and 1.3 of R. Kirby’s problem list [Proc. Symp. Pure Math. 32, 273-312 (1978; Zbl 0394.57002)]. Furthermore, an easy consequence is the statement of the title, solving a problem of Kuiper.

This is therefore an interesting and important result in low-dimensional topology, but equally important is the combinatorial technique introduced in the proof. Roughly, given two planar surfaces embedded in general position in a 3-manifold, their intersection gives rise to two planar graphs (one in each surface). A deep combinatorial analysis of these graphs, and the relationship between them, yields a surprising amount of information about the ambient 3-manifold and the embedded surfaces. In the proof of the Main Theorem, the 3-manifold is the 3-sphere with a regular neighbourhood of \(\gamma_ 0\cup \gamma_ 1\cup\) band removed, and the planar surfaces arise from a separating sphere for \(\gamma_ 0\cup \gamma_ 1\) and a spanning disc for the band sum.

This technique has turned out to have far-reaching applications. It has been used by Culler, Gordon, Luecke and Shalen in their work [Bull. Am. Math. Soc. Numer Ser. 13, 43-45 (1985)] on Dehn surgery on knots; by Gabai in his deep work on foliations and the Property R conjecture; and in several other papers of the author.

This is therefore an interesting and important result in low-dimensional topology, but equally important is the combinatorial technique introduced in the proof. Roughly, given two planar surfaces embedded in general position in a 3-manifold, their intersection gives rise to two planar graphs (one in each surface). A deep combinatorial analysis of these graphs, and the relationship between them, yields a surprising amount of information about the ambient 3-manifold and the embedded surfaces. In the proof of the Main Theorem, the 3-manifold is the 3-sphere with a regular neighbourhood of \(\gamma_ 0\cup \gamma_ 1\cup\) band removed, and the planar surfaces arise from a separating sphere for \(\gamma_ 0\cup \gamma_ 1\) and a spanning disc for the band sum.

This technique has turned out to have far-reaching applications. It has been used by Culler, Gordon, Luecke and Shalen in their work [Bull. Am. Math. Soc. Numer Ser. 13, 43-45 (1985)] on Dehn surgery on knots; by Gabai in his deep work on foliations and the Property R conjecture; and in several other papers of the author.

Reviewer: J.Howie

##### MSC:

57Q45 | Knots and links in high dimensions (PL-topology) (MSC2010) |

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

57R40 | Embeddings in differential topology |

57M15 | Relations of low-dimensional topology with graph theory |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

05C75 | Structural characterization of families of graphs |

##### Keywords:

knotted 2-spheres; band-sum of knots; knots in the 3-sphere; planar surfaces embedded in general position in a 3-manifold; planar graphs##### References:

[1] | [Ho] Hosokawa, F.: On trivial 2-spheres in 4-space. Quart. J. Math. Oxford Ser.19, 249-256 (1968) · Zbl 0172.25701 · doi:10.1093/qmath/19.1.249 |

[2] | [KL] Kearton, C., Lickorish, W.B.R.: Piecewise linear critical levels and collapsing. Trans. Amer. Math. Soc.170, 415-424 (1972) · Zbl 0248.57007 · doi:10.1090/S0002-9947-1972-0310899-2 |

[3] | [Ki] Kirby, R.: Problems in low-dimensional manifold theory. Proc. Symp. Pure Math.32, 273-312 (1978) |

[4] | [Ku] Kuiper, N.: Tight embeddings and maps. In: The Chern Symposium, pp. 97-145. Berlin-Heidelberg-New York: Springer 1980 |

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.