Knot concordance and homology cobordism

Abstract

We consider the question: "If the zero-framed surgeries on two oriented knots in the 3-sphere are integral homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on K is Z-homology cobordant to the zero-framed surgery on many of its winding number one satellites P(K). Then we prove that in many cases the tau and s-invariants of K and P(K) differ. Consequently neither tau nor s is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show, that a natural rational version of this question has a negative answer in both the topological and smooth categories, by proving similar results for K and its (p,1)-cables.

arXiv comment15 pages, 8 figures

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• On the Upsilon invariant and satellite knots
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• Using rational homology circles to construct rational homology balls
• Ozsváth-Szabó invariants of contact surgeries
• Classification of torus bundles that bound rational homology circles
• Surgeries on torus knots, rational balls, and cabling
• Fibered knots with the same $0$-surgery and the slice-ribbon conjecture
• Slice knots which bound punctured Klein bottles
• Linear independence of cables in the knot concordance group
• Action of the Mazur pattern up to topological concordance