We propose a fast algorithm for computing Jones polynomials of
 Montesinos links.
 The Jones polynomial is a useful invariant and Montesinos links are one
 of the fundamental classes in knot theory.
 Given the Tait graph of a Montesinos diagram with $n$ edges, our
 algorithm runs with $\mathcal{O}(n)$ additions and multiplications in
 polynomials of degree $\mathcal{O}(n)$, namely in
 $\mathcal{O}(n^2\log n)$ time.