In his 1968 book on singularities of complex hypersurfaces, Milnor asked a question about the "unknotting number" of knots that arise as the links of singular points of complex plane curves. The question was eventually answered in the affirmative, using gauge theory, by Kronheimer and Mrowka in 1992. A proof requiring only combinatorial techniques was found much later, by Rasmussen, using Khovanov homology. In this talk we will explore a surprising relationship between these two proofs: an interplay between gauge theory and Khovanov homology.