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.

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.

In this paper, we study the average crossing number of equilateral random walks and polygons. We show that the mean average crossing number ACN of all equilateral random walks of length n is of the form 3 16 •n•ln n+O(n). A similar result holds for equilateral random polygons. These results are confirmed by our numerical studies. Furthermore, our numerical studies indicate that when random polygons of length n are divided into individual knot types, the ACN(K) for each knot type K can be described by a function of the form ACN(K) = a • (n − n 0) • ln(n − n 0) + b • (n − n 0) + c where a, b and c are constants depending on K and n 0 is the minimal number of segments required to form K. The ACN(K) profiles diverge from each other, with more complex knots showing higher ACN(K) than less complex knots. Moreover, the ACN(K) profiles intersect with the ACN profile of all closed walks. These points of intersection define the equilibrium length of K, i.e., the chain length ne(K) at which a statistical ensemble of configurations with given knot type K-upon cutting, equilibration and reclosure to a new knot type K-does not show a tendency to increase or decrease ACN(K). This concept of equilibrium length seems to be universal, and applies also to other length-dependent observables for random knots, such as the mean radius of gyration Rg .

In this paper, we study the topological entanglement of uniform random polygons in a confined space. We derive the formula for the mean squared linking number of such polygons. For a fixed simple closed curve in the confined space, we rigorously show that the linking probability between this curve and a uniform random polygon of n vertices is at least 1 − O(1 √ n). Our numerical study also indicates that the linking probability between two uniform random polygons (in a confined space), of m and n vertices respectively, is bounded below by 1 − O(1 √ mn). In particular, the linking probability between two uniform random polygons, both of n vertices, is bounded below by 1 − O(1 n).

It was proved in [4] that the knotting probability of a Gaussian random polygon goes to 1 as the length of the polygon goes to infinity. In this paper, we prove the same result for the equilateral random polygons in R3. More precisely, if EPn is an equilateral random polygon of n steps, then we have [Formula: see text] provided that n is large enough, where ∊ is some positive constant.