Polynomial-time algorithms for the curve graph

Polynomial-time algorithms for the curve graph

Mark C. Bell111Department of Mathematics, University of Illinois: mcbell@illinois.edu    Richard C. H. Webb222DPMMS, Centre for Mathematical Sciences, University of Cambridge: rchw2@cam.ac.uk
Abstract

We describe a polynomial-time 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, polynomial-time, mapping class group, asymptotic translation length, Nielsen–Thurston classification, canonical curve system.

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