The collection of knots in the 3-sphere together with the connected sum operation famously becomes a group when one quotients out by slice knots, i.e. those knots that bound embedded discs in the 4-ball. In contrast to higher dimensions, where the analogous groups are completely understood, the knot concordance group remains mysterious, more than 50 years after Fox and Milnor initiated its study. The classical satellite operation gives a nice and much-studied collection of self-maps of the concordance group; however, it is conjectured that these maps are essentially never homomorphisms. I will discuss new obstructions to satellite-induced homomorphisms, defined in terms of linking numbers of curves and relying on Heegaard Floer homology, coming from recent joint work with Tye Lidman (NC State) and Juanita Pinzón-Caicedo (Notre Dame). This talk will not assume familiarity with concordance, satellite operators, or Heegaard Floer homology.