Bounds for the minimum step number of knots in the simple cubic lattice

Abstract

Knots are found in DNA as well as in proteins, and they have been shown to be good tools for structural analysis of these molecules. An important parameter to consider in the artificial construction of these molecules is the minimum number of monomers needed to make a knot. Here we address this problem by characterizing, both analytically and numerically, the minimum length (also called minimum step number) needed to form a particular knot in the simple cubic lattice. Our analytical work is based on improvement of a method introduced by Diao to enumerate conformations of a given knot type for a fixed length. This method allows us to extend the previously known result on the minimum step number of the trefoil knot 31 (which is 24) to the knots 41 and 51 and show that the minimum step numbers for the 41 and 51 knots are 30 and 34, respectively. Using an independent method based on the BFACF algorithm, we provide a complete list of numerical estimates (upper bounds) of the minimum step numbers for prime knots up to ten crossings, which are improvements over current published numerical results. We enumerate all minimum lattice knots of a given type and partition them into classes defined by BFACF type 0 moves.