The multi-region index of a knot

TL;DR

This paper introduces the multi-region index, a new knot invariant based on region crossing changes, and establishes bounds relating it to crossing number and homology generators, using Goeritz matrices.

Contribution

It defines the multi-region index and proves bounds connecting it to crossing number and the homology of the double branched cover, employing Goeritz matrices.

Findings

Multi-region index is bounded above by twice the crossing number.

Minimum generators of the homology are less than the multi-region index.

The paper provides a new invariant with proven bounds using algebraic topology.

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.