
4. Determining concordance

1. Genus $g$ traces and concordance

Problem 4.1.

[Lisa Piccirillo] For a knot $K$ in $S^3$, let $X^g(K)$ denote the 4-ball union $\Sigma_g \times D^2$, where $\Sigma_g$ is a genus $g$ surface with one boundary component, attached to $B^4$ along a 0-framed neighborhood of $K$. Are there nonconcordant knots $K$ and $J$ such that $\partial X^g(K)$ and $\partial X^g(J)$ are integrally homology cobordant?
1. Remark. [Lisa Piccirillo] Note that homeomorphism of $\partial X^g(K)$ and $\partial X^g(J)$ is enough to imply that the knots $K$ and $J$ are isotopic.
• Branched covers and concordance

Problem 4.2.

Suppose $K$ and $J$ are knots in $S^3$ such that for every prime power $q$, the $q$th cyclic branched covers $\Sigma_q(K)$ and $\Sigma_q(J)$ are rationally homology cobordant. Must $K$ and $J$ be concordant?
• Surgeries and concordance

Problem 4.3.

Suppose $K$ and $J$ are knots in $S^3$ such that for all $p/q \in \mathbb{Q}$, the Dehn surgeries $S^3_{p/q}(K)$ and $S^3_{p/q}(J)$ are integer homology cobordant. Must $K$ and $J$ be concordant?
• 0-surgeries and concordance

Problem 4.4.

[Adam Levine] Suppose $K$ and $J$ are knots in $S^3$ such that there is an integer homology cobordism $W$ between $S^3_0(K)$ and $S^3_0(J)$ in which the positively oriented meridians $\mu_K$ and $\mu_J$ are freely homotopic. Must $K$ and $J$ be concordant?
1. Remark. Under the assumption that $\mu_K$ and $\mu_J$ are actually concordant in $W$, it follows that $K$ and $J$ are concordant (in the topological category) or exotically concordant (in the smooth category.)
• Remark. [Arunima Ray] This question for links was addressed by Cha-Powell [arXiv:1309.5051], who showed that the answer was ’no’.

Cite this as: AimPL: Smooth concordance classes of topologically slice knots, available at http://aimpl.org/concordsliceknot.