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Unlinking via simultaneous crossing changes. (English) Zbl 0785.57003
The following problem is considered: given two distinct crossings of a knot or link projection, under what conditions can one obtain the unlink by changing both crossings simultaneously? More precisely, consider two “crossing disks” \(D_ 1\) and \(D_ 2\) for a link \(L\), i.e. disks intersecting \(L\) in exactly two points, and twist the link \(t_ 1\) [resp. \(t_ 2\)] times as it passes through \(D_ 1\) [resp. \(D_ 2\)] (adding \(2t_ 1+2t_ 2\) new crossings to the projection of the link). Denote the new link by \(L(t_ 1,t_ 2)\). The main result of the paper states that if the disks are essential and do not bound an annulus disjoint from \(L\), there exists a pair \((s_ 1,s_ 2)\) of integers with the following property: whenever \(L(t_ 1,t_ 2)\) is the unlink, either \(t_ 1=s_ 1\) or \(t_ 2=s_ 2\). This is a consequence of a more general result giving an analogous statement for pairs \((t_ 1,t_ 2)\) reducing the minimal Thurston norm among all spanning surfaces of the link disjoint from the boundary of the two disks.

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M40 Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
Full Text: DOI
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