The Jones polynomial is an invariant in knot theory. It is known that the Jones polynomial of an alternating link is related to the Tutte polynomial in graph theory. Here, it is shown that the new algorithm [11] of computing the Tutte polynomial can be applied to computing the Jones polynomial of an arbitrary link. Although a problem of computing the Jones polynomial is #P-hard, by using the planarity it can be calculated for some large links, say a link whose signed plane graph is a 10 × 10 grid graph and which has 180 crossings. A new result for the case where a knot is represented as a braid is also given.