MINIMAL KNOTTED POLYGONS ON THE CUBIC LATTICE

Abstract

The polygons on the cubic-lattice have played an important role in simulating various circular molecules, especially the ones with relatively big volumes. There have been a lot of theoretical studies and computer simulations devoted to this subject. The questions are mostly around the knottedness of such a polygon, such as what kind of knots can appear in a polygon of given length, how often it can occur, etc. A very often asked and long standing question is about the minimal length of a knotted polygon. It is well-known that there are knotted polygons on the lattice with 24 steps yet it is unproved up to this date that 24 is the minimal number of steps needed. In this paper, we prove that in order to obtain a knotted polygon on the cubic lattice, at least 24 steps are needed and we can only have trefoils with 24 steps.