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.

Phase diagrams of knotted and unknotted ring polymers

“…With the cinqefoil knot 5 1 , the number of available combinations of (n, h) is dramatically reduced, and the three-twist 5 2 is so restrictive that the simulations find only two bins with a difference of two contacts between them, and density of states equal within statistical uncertainty. This is not surprising as the 36-bead polymer is the shortest which can contain a three-twist knot [21]. Consequently, there is only a very weak collapse transition, below which conformations with the maximum contacts dominate.…”

“…For lattice polymers, there is a minimum length of ring that allows each knot to be formed on the lattice. Determining this length for different knots, and the conformations that achieve them, is a problem itself requiring numerical methods, but much progress has been made recently [21]. Using this work as a guide, it is possible to simulate, using Markov chain Monte Carlo, lattice polymers with a range of knots, down to the shortest lattice polymers known to be capable of supporting each knot.…”

On the mean and variance of the writhe of random polygons

“…Theoretical and computational studies of minimal lattice knots include [16, 31, 32, 36]. Our group reported numerical estimates for the number of minimal edge conformations of lattice polygons with 9 and less crossings in [59]. For example, an ideal polygon for the trefoil knot has length 24 and there are 1664 such configurations for the right-hand trefoil [16, 59].…”

“…Our group reported numerical estimates for the number of minimal edge conformations of lattice polygons with 9 and less crossings in [59]. For example, an ideal polygon for the trefoil knot has length 24 and there are 1664 such configurations for the right-hand trefoil [16, 59]. We define the ideal mean writhe of K, and denote it by w I ( K ), as the writhe averaged over all minimal edge conformations of the knot.…”

Lattice knots in a slab

“…For example p 4 (0 1 ) = 3 while p n (0 1 ) = 0 if n < 4 or if n is odd, where 0 1 is the unknot in standard knot notation. It is also known that p 24 (3 + 1 ) = 1664 while p n (3 + 1 ) = 0 if n < 24 [22], where 3 + 1 is the trefoil knot type. By symmetry, p n (K + ) = p n (K − ) if K is a chiral knot type.…”