Knot concordance and homology cobordism

TL;DR

This paper investigates whether knots are concordant based on their zero-framed surgeries being homology cobordant, revealing that key invariants like tau and s do not serve as invariants in this context, with negative results in both smooth and topological categories.

Contribution

It demonstrates that zero-framed surgeries being homology cobordant does not imply knot concordance, and shows tau and s invariants are not invariants of the homology cobordism class of surgeries.

Findings

Tau and s invariants differ for many satellite knots.

Zero-framed surgeries on knots can be homology cobordant without the knots being concordant.

Negative results for both topological and smooth categories regarding rational versions of the question.

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.