Taut smoothings of arcs and curves

Abstract

We study the geometric and combinatorial effect of smoothing an intersection point in a collection of arcs or curves on a surface. We prove that all taut arcs with fixed endpoints and all taut 1-manifolds with at least two non-disjoint components on an orientable surface with negative Euler characteristic admit a taut smoothing, and also that all taut arcs with free endpoints admit a smoothing that is either taut or becomes taut after removing at most one intersection. We deduce that for every Riemannian metric on a surface, the shortest properly immersed arcs with at least $k$ self-intersections have exactly $k$ self-intersections when the endpoints of the arc are fixed, and at most $k+1$ self-intersections otherwise, and that the arc length spectrum is "coarsely ordered" by self-intersection number. Along the way, we obtain partial analogous results in the case of curves.