# A counter-example for the reversed direction of Casson-Gordon's theorem

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.

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.

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.

• Thanks a lot for your explanatory response, Professor Ruberman. Nov 10 '18 at 16:56

Probably, I found some more explicit examples:

Akbulut and Larson recently showed in AL18 that the family of Brieskorn spheres $$\Sigma(2,4n+1,12n+5)$$ and $$\Sigma(3,3n+1,12n+5)$$ bound rational homology balls.

These Brieskorn spheres are respectively $$2$$- and $$3$$-fold cyclic branched cover of torus knots $$T_{4n+1,12n+5}$$ and $$T_{3n+1,12n+5}$$ in $$S^3$$. But these knots are not smoothly slice due to for example Milnor slice genus proven by Kronheimer and Mrowka in KM93.