For a knot $K$, let $\Sigma\_K$ be the double cyclic branched cover of a knot. 


By the classical work of **Casson** and **Gordon**, we know that if $K$ is smoothly slice, then $\Sigma\_K$ bounds a rational homology ball.


Is there any well-known counter-example for the reversed direction?


**EDIT** More general statement is true due to the work of Casson & Gordon, just seen in [ACP18](https://arxiv.org/pdf/1811.01433.pdf). 


For any prime $p$ and positive integer $r$, the $p^r$-fold cyclic branched cover of a knot $ K$, denoted by $\Sigma\_{p^r}(K)$, is a $\mathbb Z\_p$-homology sphere.


**Theorem:** If $K$ is smoothly slice, then $\Sigma\_{p^r}(K)$ bounds $\mathbb Z\_p$-homology ball.



A counter-example for the reversed direction of Casson-Gordon's theorem
Here's a particularly subtle counterexample, from the work of Kirk and Livingston (Topology Vol. 38, No. 3, pp. 663--671, 1999). They show that the pretzel knots $J = P(-3,5,7,2)$ and $K = P(5,-3,7,2)$ are not concordant (even locally flat). These two knots are related by mutation (switch the first two pairs of twists in the standard picture of a pretzel knot). On the other hand, they have the same double branched cover, say $\Sigma$, a certain Brieskorn sphere. (This is a general fact about mutations, but is clear in this setting.)


Then the knot $J \# -K$ is not slice, but its double branched cover $\Sigma \# -\Sigma$ bounds an integer homology ball (in fact $(\Sigma - \text{int}\ B^3) \times I$). 


There are many earlier examples in the literature for pairs of knots with the same double branched covers. Presumably it's not hard to check that for some of these, the knots are not concordant, using eg knot signatures. So one could construct many more examples. Likewise there are pairs of knots with the same n-fold branched covers (for a fixed $n$) and I'm sure you could construct counterexamples for those as well. 


