The Erd} Os-nagy Theorem and Its Ramiications

os-Nagy Theorem and its Rami cations Godfried Toussaint School of Computer Science McGill University Montr eal, Qu ebec, Canada June 29, 1999 Abstract Given a simple polygon in the plane, a ip is dened as follows: consider the convex hull of the polygon. If there are no pockets do not perform a ip. If there are pockets then re ect one pocket across its line of support of the polygon to obtain a new simple polygon. In 1934 Paul Erd} os introduced the problem of repeatedly ipping all the pockets of a simple polygon simultaneously and he conjectured that the polygon would become convex after a nite number of ips. In 1939 B ela Nagy pointed out that ipping several pockets simultaneously may result in a nonsimple polygon. Modifying the problem slightly he then proved that if at each step only one pocket is ipped the polygon will become convex after a nite number of ips. We call this result the Erd}os-Nagy Theorem. Since then this theorem has been rediscovered many times in di erent contexts, apparently, with none of the authors aware of each other's work. One purpose of this paper is to bring to light this \hidden" work. We review the history of this problem, provide a simple elementary proof of a stronger version of the theorem and consider variants, generalizations and applications of interest in computational knot theory, polymer physics and molecular biology. We also improve several results in the literature with the application of the Erd}os-Nagy theorem. We close with a list of open problems. This research was supported by NSERC and FCAR. email: [email protected] A1 z