The Restriction of Efficient Geodesics to the Non-separating Complex of Curves

Abstract

In the complex of curves of a closed orientable surface of genus $g,$ $\mathcal{C}(S_g),$ a preferred finite set of geodesics between any two vertices, called efficient geodesics introduced by Birman, Margalit, and Menasco in birman_margalit_menasco₂016. The main tool used to establish the existence of efficient geodesics was a dot graph, which recorded the intersection pattern of a reference arc with the simple closed curves associated with a geodesic path. The idea behind the construction was that a geodesic that is not initially efficient contains shapes in its corresponding dot graph. These shapes then correspond to surgeries that reduce the intersection with the reference arc. In this paper, we show that the efficient geodesic algorithm is able to be restricted to the non-separating curve complex; the proof of this will involve analysis of the dot graph and its corresponding surgeries. Moreover, we demonstrate that given any geodesic in the complex of curves we may obtain an efficient geodesic whose vertices, with the possible exception of the endpoints, are all nonseparating curves.