Computing Vassilievʹs invariants

We give an explicit algorithm for computing all of Vassilievʹs knot invariants of order ⩽ n, which has been implemented on a computer. In giving a justification of the algorithm, we obtain a simple proof of the sufficiency of the topological 1-term and 4-term relations. We use an example of Taniyama to show that two singular knot diagrams with the same configuration cannot always be made isotopic by crossing changes alone.