The multi-region index of a knot

Abstract

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.

13 pages, 12 figures. Added Theorems 4.6 and 4.8. Added computations of multi-region index of ten new knots to the tables at the end of the paper as a result of Theorem 4.8