Research in Knot Theory: Unknotting and Region Crossing Change
Sarah Goodhill ’22, Valeria Muñoz Gonzales ’22, Jessica Rattray ’22, Amelia Zeh ’21, and Professor Adam Lowrance
Take a string and tie a knot. If you connect the loose ends together, you have a mathematical knot in 3-space. The tabulation of knots is based on how many crossings they have. The simplest version of a knot is called the unknot, a closed loop with zero crossings. How do you unknot a knot that has no loose ends? We focus on achieving the unknot from a knot diagram through an operation called region crossing change. This operation allows us to change crossings in specific regions for any knot diagram. How do we calculate how a knot is unknotted? We define a new invariant called the multi-region index that measures how complex a knot is with respect to the region crossing change operation. The multi-region index of a knot is the fewest number of crossings in a set of regions that must be changed in order to achieve the unknot. Our main theorem provides a computable lower bound for the multi-region index to help us know when it is minimal. We also show that the multi-region index of a knot is at most twice the crossing number of the knot.