Fibered knots and Property 2R, II - Mathematics > Geometric TopologyReport as inadecuate

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Abstract: A knot K in the 3-sphere is said to have Property nR if, whenever K is acomponent of an n-component link L and some integral surgery on L produces theconnected sum of n copies of S^1 x S^2, there is a sequence of handle slides onL that converts L into a 0-framed unlink. The Generalized Property R Conjectureis that all knots have Property nR for all n.The simplest plausible counterexample could be the square knot. Exploitingthe remarkable symmetry of the square knot, we characterize all two-componentlinks that contain it and which surger to S^1 x S^2 # S^1 x S^2. We argue thatat least one such link probably cannot be reduced to the unlink by a series ofhandle-slides, so the square knot probably does not have Property 2R. Thisexample is based on a classic construction of the first author.On the other hand, the square knot may well satisfy a somewhat weakerproperty, which is still useful in 4-manifold theory. For the weaker property,copies of canceling Hopf pairs may be added to the link before the handleslides and then removed after the handle slides. Beyond our specific example,we discuss the mechanics of how addition and later removal of a Hopf pair canbe used to reduce the complexity of a surgery description.

Author: Robert E. Gompf, Martin Scharlemann


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