Invisible Knots and Rainbow Rings: Knots Not Determined by Their Determinants

摘要

M¨obius strip experiments are surefire triggers of Aha! experiences. Maybe you don’t remember the first time someone challenged you to color one side blue and the other red, or asked you to guess the result of cutting a M¨obius strip in half, but you surely recall the outcome. As part of a research experience for undergraduates (REU), we discovered that generalizing these experiments results in many more confounding constructions. Rather than simply bisecting the M¨obius strip, try cutting it into n sections. Or, instead of joining the ends of the strip with a single half twist, make two twists, or three, or, in general, m half twists. You have just created examples of paradromic rings, which we’ll denote P(m,n).