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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.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M40 Characterizations of the Euclidean $$3$$-space and the $$3$$-sphere (MSC2010)
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