A knot K bounds a locally flat Z_d-disc representing a class x of divisibility d in a simply-connected 4-manifold N iff its Arf invariant matches a congruence and b_2(N) dominates all Levine-Tristram signature bounds.
On the connected sums of the $(2,1)$-cable of the figure eight knot
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abstract
We show that the 3-fold (resp. 6-fold) connected sum of the $(2,1)$-cable of the figure-eight knot cannot bound a smooth null-homologous disk in a punctured $S^2 \times S^2$ (resp. in a punctured $#_2 S^2 \times S^2$. This result is obtained using a real version of the $10/8$-inequality established by Konno, Miyazawa, and Taniguchi.
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Simply slicing knots
A knot K bounds a locally flat Z_d-disc representing a class x of divisibility d in a simply-connected 4-manifold N iff its Arf invariant matches a congruence and b_2(N) dominates all Levine-Tristram signature bounds.