# Nielsen-Thurston decomposition from the product of Dehn twists

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.

• 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 . – HJRW Mar 21 '18 at 17:24
• @HJRW Would you please write your comment as an answer so that I can accept it? – Cusp Mar 21 '18 at 18:26