The region index and the unknotting number of a knot

For any knot in the 3-sphere S3, there exists a diagram such that we have the unknot if we change all crossings on the boundary of some region of the diagram. The minimal number of the crossing changes over all such diagrams is called the region index of a knot. Clearly, the unknotting number is less than or equal to the region index. In this paper, we show that there exists a knot which has a gap between the unknotting number m and the region index for any positive integer m (m≥2) by using the Goeritz invariant. We also show that there exists a knot which has the unknotting number and the region index that are equal to n for any positive integer n (n≥2) by using the Rasmussen invariant.