Polynomialtime algorithms for the curve graph
Abstract
We describe a polynomialtime algorithm to compute a (tight) geodesic between two curves in the curve graph. As well as enabling us to compute the distance between a pair of curves, this has several applications to mapping classes. For example, we can use these geodesics to compute:

the asymptotic translation length,

the Nielsen–Thurston type, and

the canonical curve system
of a mapping class in polynomial time in its word length.
keywords.
curve graph, geodesics, polynomialtime, mapping class group, asymptotic translation length, Nielsen–Thurston classification, canonical curve system.