Fast algorithms for computing Jones polynomials of certain links

We give a fast algorithm for computing Jones polynomials of 2-bridge links. Given the Tait graph with n edges of a 2-bridge diagram, this algorithm runs with O(n) additions and multiplications in polynomials of degree O(n), namely in O(n2logn) time. We also propose an algorithm that, given the Tait graph with n edges of a closed 3-braid diagram, computes the Jones polynomial of the closed 3-braid link in O(n2logn) time.