Natural classification of knots

The principal objective of the knot theory is to provide a simple way of classifying and ordering all the knot types. Here, we propose a natural classification of knots based on their intrinsic position in the knot space that is defined by the set of knots to which a given knot can be converted by individual intersegmental passages. In addition, we characterize various knots using a set of simple quantum numbers that can be determined upon inspection of minimal crossing diagram of a knot. These numbers include: crossing number; average three-dimensional writhe; number of topological domains; and the average relaxation value.

TWISTING: A QUANTITY DESCRIBING THE TOPOLOGICAL RELATION OF TWO PEPTIDE FRAGMENTS IN PROTEIN STRUCTURES

A method for analyzing the topological relation of any two peptide fragments in a protein structure is given by making analogy of the two fragments with a braid. Knotted relations are analyzed by calculating polynomials for the two fragments separately and for the two as a whole, respectively. Twisting is calculated from the writhe number, which is a measure to the internal relative rotation during formation of the protein structure. Statistical analysis on the internal rotations of all known structures found that the relative rotation like a one-dimensional random walk with decreasing step lengths.

Near Helical Conformations of an Elastic dsDNA Ribbon

We find analytically the conditions to form a nearly helical conformation for a model of double-stranded DNA under external force and torque. For such conformations, the tilting angle can be either oscillatory or decaying with increasing arclength, depending on the relationship among the force, torque and initial tilting angle. Both conformations are possible for a small initial tilting angle while only the oscillatory solution is possible for a large initial tilting angle. Phase diagram that depicts the oscillatory and decaying solutions is calculated. The corresponding linking number, twist, writhe number, relative extension and the locus of the chain are also calculated analytically.