  Using region crossing changes, we define a new invariant called the
multi-region index of a knot. We prove that the multi-region index of a knot is
bounded from above by twice the crossing number of the knot. In addition, we
show that the minimum number of generators of the first homology of the double
branched cover of $S^3$ over the knot is strictly less than the multi-region
index. Our proof of this lower bound uses Goeritz matrices.
