Geometry and physics of knots

Knotted Globular Ring Polymers: How Topology Affects Statistics and Thermodynamics

“…The values M 2 − M 1 and M 2 − M 1 are determined with the simulations described in the previous section, see Table 2 N = 500 off-lattice knotted flexible yet impenetrable tubes of thickness D and length L, the ideal length l id of a given knot is the smallest ratio L/D one can achieve. 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 .…”

“…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.…”

Universal growth law for knot energy of Faddeev type in general dimensions

“…In particular, Moffat (1990) articulates to use the minimum knot energy as a new type of topological invariant for knots and links, further emphasizes that any knot or link may be characterized by an 'energy spectrum', a set of positive real numbers determined solely by its topology, and proposes that the lowest energy provides a possible measure of knot or link complexity. Katritch et al (1996) approach knot identification by considering the properties of specific geometric forms of knots that are defined as ideal so that for a knot with a given topology and assembled from a tube of uniform diameter, the ideal form is the geometrical configuration having the highest ratio of volume to surface area. Equivalently, this amounts to determining the shortest piece of tube that can be closed to form the knot.…”

Predicting Knot or Catenane Type of Site-Specific Recombination Products

“…33 c Ideal geometric configurations of knots or catenanes are the trajectories that allow maximal radial expansion of a virtual tube of uniform diameter centered around the axial trajectory of the knot. 12,13 d Note all figures represent (the axis of) duplex DNA.…”

“…7,11 This has not generalized, although recent experiments indicate that knots/catenanes may migrate linearly with respect to the average crossing number (ACN) of a particular conformation -the ideal configuration c of the knot or catenane. 11,13,14 For gel electrophoresis, one must also construct an appropriate knot ladder as a control to determine the exact DNA knot or catenane, since adjacent bands determine only relative MCN or ACN, not precise values. While this can be done in some cases (e.g.…”