Smoothing curves carefully

Abstract: This paper proves an elementary topological fact about closed curves on surfaces, namely that by carefully smoothing an intersection point, one can reduce self-intersection by exactly $1$. This immediately implies a positive answer to a problem first raised by Basmajian in the 1990s: among all closed geodesics of a hyperbolic surface that self-intersect at least $k$ times, does the shortest one self-intersect exactly $k$ times? The answer is also shown to be positive for arbitrary Riemannian metrics.