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In this post, we discuss the relation between the concordance of knots in $S^3$ and the integral homology cobordism.

Following its notation, assume that knots $K_0$ and $K_1$ in $S^3$ are concordant. Then the $3$-manifolds $S^3_{n} (K_0)$ and $S^3_{n} (K_1)$ are integral homology cobordant.

Now we perform one more (say $m$) integral surgery on concordant knots in $S^3_{n} (K_0)$ and $S^3_{n} (K_1)$.

My questions are:

  1. Do we still have an integral homology cobordism between the new resulting $3$-manifolds?
  2. Is there any restriction on $m$?
  3. The integer $n$ can be zero?
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  • $\begingroup$ What do you mean by "some null-homologous knots"? If you choose two random null-homologous knots (say, a trefoil and an unknot, each contained in a ball), then there's no reason to expect a rational homology cobordism. $\endgroup$
    – Marco Golla
    Commented Sep 9, 2020 at 16:14
  • $\begingroup$ Sorry for unclarity. I edited the post. $\endgroup$
    – user160180
    Commented Sep 9, 2020 at 16:27

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