Let {Kn} be the family of knots obtained by twisting a knot K along an unknot c. When the winding number of K about c is non-zero, we show the limit of g(Kn)/g4(Kn) is 1 if and only if the winding and wrapping numbers of K about c are equal. When equal, this leads to a description of minimal genus Seifert surfaces of Kn for |n|≫0 and eventually to a characterization of when c is a braid axis for K. We then use this characterization to show that satellite L-space knots are braided satellites*. This is joint work with Kimihiko Motegi that builds upon joint work with Scott Taylor. (* Modulo a conjecture whose solution by Hanselman-Rasmussen-Watson has been announced.)