Slice obstructions from genus bounds in definite 4-manifolds

We discuss an obstruction to a knot being smoothly slice that comes from minimum-genus bounds on smoothly embedded surfaces in definite 4-manifolds. As an example, we provide an alternate proof of the fact that the (2,1)-cable of the figure eight knot is not smoothly slice, as shown by Dai–Kang–Mallick–Park–Stoffregen in 2022. The main technical input of our argument consists of gauge-theoretic obstructions to smooth small-genus surfaces representing certain homology classes in ${\mathbb{C}\mathbb{P}}^2\mathrm{\#}{\mathbb{C}\mathbb{P}}^2$ proved by Bryan in the 1990s.