Tightness and computing distances in the curve complex

Abstract

We give explicit bounds on the intersection number between any curve on a tight multigeodesic and the two ending curves. We use this to construct all tight multigeodesics and so conclude that distances in the curve graph are computable. The algorithm applies to all surfaces. We recover the finiteness result of Masur-Minsky for tight goedesics. The central argument makes no use of the geometric limit arguments seen in the recent work of Masur-Minsky (2000) and of Bowditch (2003), and is enough to deduce a computable version of the acylindricity theorem of Bowditch.