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Smooth spheres in \({\mathbb{R}}^ 4\) with four critical points are standard. (English) Zbl 0559.57019
The 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.
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
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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
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