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Given a closed surface of genus $g\geq 2$, we know that the mapping class group $Mod(S)$ is generated by the Dehn twists. My question is

Given an element as a product of Dehn twist, is it possible to write down the corresponding Nielsen-Thurston decomposition?

Explicitly, suppose $\phi\in Mod(S)$ such that $\phi=\prod_{i=1}^nT_{x_i}^{k_i}$ where $x_i$'s are simple closed curves, $T_{x_i}$ is the left Dehn twist about $x_i$ and $k_i\in\mathbb{Z}$. Given this data, is it possible to write down the Nielsen-Thurston decomposition of $\phi.$

PS: Rivin's comment have helped me to find this exact link.

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    $\begingroup$ There isn’t a simple recipe. Bell and Webb recently gave a polynomial-time algorithm to compute the Nielsen—Thurston type of a mapping class: arxiv.org/abs/1609.09392v2 . $\endgroup$
    – HJRW
    Commented Mar 21, 2018 at 17:24
  • $\begingroup$ @HJRW Would you please write your comment as an answer so that I can accept it? $\endgroup$
    – Cusp
    Commented Mar 21, 2018 at 18:26

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There isn’t a simple recipe. Bell and Webb recently gave a polynomial-time algorithm to compute the Nielsen—Thurston type of a mapping class. Their paper also contains a summary of previously known algorithms.

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    $\begingroup$ You can now also explore / play with these via Curver which includes a Python implementation of these algorithms. See curver.readthedocs.io $\endgroup$
    – Mark Bell
    Commented Apr 14, 2018 at 22:27
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Not only is it possible, it is implemented by Mark Bell and Saul Schleimer as Twister.

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  • $\begingroup$ This is true, if you use Twister and Snappy together. However, there is a cleaner way, which avoids hyperbolic three-manifolds. Use Flipper, written as part of Mark Bell's thesis. bitbucket.org/Mark_Bell/flipper $\endgroup$
    – Sam Nead
    Commented Mar 22, 2018 at 21:15
  • $\begingroup$ @SamNead Thanks! I have used it via SnapPy, so have a parochial view :) $\endgroup$
    – Igor Rivin
    Commented Mar 22, 2018 at 23:29

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