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

Zbl 0394.57002
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References:

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