Computing with curves on surfaces

Dylan Thurston, Barnard College, Columbia University

Abstract: In order to better understand the geometric intersection number of curves, we introduce a new tool, the \emph{smoothing lemma}. This lets us write i(X, A), where X is a simple curve and A is any curve, as a maximum of various i(X, A'), where the A' are also simple and independent of X.
We can use this for several purposes, including a new derivation for the behaviour of Dehn-Thurston coordinates under an elementary change in the pair of pants decomposition (first described by Penner). This gives, in particular, an efficient algorithm for the word problem in the mapping class group of a closed surface.