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The band sum problem. (English) Zbl 0546.57001

Take \(n+1\) unlinked knots in \(S^ 3\), join them by n disjoint thin rectangular strips (”bands” or ”ribbons”) which meet the knots precisely in their narrow ends. The knot obtained by this process: \[ \cup\quad (knots)\quad\cup \quad\partial (bands)\quad -\quad (bands\quad\cap \quad knots) \] is called a band-sum of the original knots. Theorem: if the band-sum is unknotted, then so was each of the original knots. This answers a question of Lickorish [Problem 1.1 in R. Kirby, Proc. Symp. Pure Math. 32, Part 2, 273-312 (1978; Zbl 0394.57002)]. The proof uses recently published results about locally indicable groups. There is an alternative proof based on unpublished results of Thurston; and a somewhat stronger result in the case \(n=1\) has been proved by M. Scharlemann (to appear in a volume of conference proceedings in the Lect. Notes Math. series).

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20F38 Other groups related to topology or analysis

Citations:

Zbl 0394.57002
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