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.

Topological Information Embodied in Local Juxtaposition Geometry Provides a Statistical Mechanical Basis for Unknotting by Type-2 DNA Topoisomerases

“…5,17 For self-avoiding but otherwise unconstrained loops configured on the simple cubic lattice, n = 24 is the minimum loop size that allows a knot (a trefoil) to be formed. 78 Consistent with the general trend, the exact counts in Table 1 show that, for small loops, the fractions of conformations that are knotted are very small. However, although p K is small for all five preformed juxtaposition geometries considered, the differences in p K among them are striking.…”

Knotted Globular Ring Polymers: How Topology Affects Statistics and Thermodynamics

“…3,10 For a lattice knot the "ideal" configurations are those with the minimal number (l min ) of steps. 47,48 The probability maxima discussed above are located at relatively small values of l 1 , not too far from l min k 1 . This suggests that the configurations taken by the loop with k 1 for l 1 ≈ l 1 could be somehow reminiscent, up to temperature dependent moderate deviations, of the configurations of minimal length, with l 1 ∼ l min k 1 .…”

“…Our analysis also shows that the strength of the topological correction is connected to the length of the knots in their ideal, minimal length form. 3,10,[46][47][48] This connection sheds light on the reason why the correction itself seems to be determined primarily and almost exclusively by the minimal crossing number of the knots.…”

Lattice knots in a slab

“…For example, one may deduce that n 2,31 = 24 from reference [4]. However, simulations show that n 1,31 = 26 [8].…”