Topic
Average crossing number
About: Average crossing number is a research topic. Over the lifetime, 22 publications have been published within this topic receiving 919 citations.
Papers
TL;DR: In this article, the authors considered the properties of particular geometrical forms which they define as "ideal" for a knot with a given topology and assembled from a tube of uniform diameter.
Abstract: KNOTS are usually categorized in terms of topological properties that are invariant under changes in a knot's spatial configuration1–4. Here we approach knot identification from a different angle, by considering the properties of particular geometrical forms which we define as 'ideal'. 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. Practically, this is equivalent to determining the shortest piece of tube that can be closed to form the knot. Because the notion of an ideal form is independent of absolute spatial scale, the length-to-diameter ratio of a tube providing an ideal representation is constant, irrespective of the tube's actual dimensions. We report the results of computer simulations which show that these ideal representations of knots have surprisingly simple geometrical properties. In particular, there is a simple linear relationship between the length-to-diameter ratio and the crossing number—the number of intersections in a two-dimensional projection of the knot averaged over all directions. We have also found that the average shape of knotted polymeric chains in thermal equilibrium is closely related to the ideal representation of the corresponding knot type. Our observations provide a link between ideal geometrical objects and the behaviour of seemingly disordered systems, and allow the prediction of properties of knotted polymers such as their electrophoretic mobility5.
517 citations
TL;DR: A family of global geometric measures is constructed for protein structure classification that originate from integral formulas of Vassiliev knot invariants and give rise to a unique classification scheme that can better discriminate between many known protein structures.
Abstract: A family of global geometric measures is constructed for protein structure classification. These measures originate from integral formulas of Vassiliev knot invariants and give rise to a unique classification scheme. Our measures can better discriminate between many known protein structures than the simple measures of the secondary structure content of these protein structures.
54 citations
TL;DR: In this paper, the authors investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature, and show that these energies are charge, minimizable in given isotopy classes, tight and strong.
50 citations
TL;DR: The data of ideal knots are reanalyzed and the average crossing number of the ideal knots (ideal) shows a nonlinear behavior with the essential crossing number C, which is contrary to previous claims.
Abstract: The data of ideal knots [Nature, 384, 142 (1996)] are reanalyzed and the average crossing number of the ideal knots $〈X{〉}_{\mathrm{ideal}}$ shows a nonlinear behavior with the essential crossing number C. Supplemented with our Monte Carlo simulations using the bond fluctuation model on flexible knotted polymers, our analysis indicates that $〈X{〉}_{\mathrm{ideal}}$ varies nonlinearly with both C and the corresponding average crossing number of the flexible knot, which is contrary to previous claims. Our extensive simulation data on the average crossing number of flexible knots suggest that it varies linearly with the square root of C. Furthermore, our data on the average writhe number $〈\mathrm{Wr}〉$ indicate that various knots are classified into holonomous groups, and $〈\mathrm{Wr}〉$ has a quantized linear increment with C in all four knot groups in our study.
35 citations
TL;DR: A universal ratio between diffusion constants of the ring polymer with a given knot K and a linear polymer with the same molecular weight in solution through the Brownian dynamics under hydrodynamic interaction is found to be constant with respect to the number of monomers.
Abstract: We have evaluated a universal ratio between diffusion constants of the ring polymer with a given knot $K$ and a linear polymer with the same molecular weight in solution through the Brownian dynamics under hydrodynamic interaction. The ratio is found to be constant with respect to the number of monomers, $N$, and hence the estimate at some $N$ should be valid practically over a wide range of $N$ for various polymer models. Interestingly, the ratio is determined by the average crossing number $({N}_{\text{AC}})$ of an ideal conformation of knotted curve $K$, i.e., that of the ideal knot. The ${N}_{\text{AC}}$ of ideal knots should therefore be fundamental in the dynamics of knots.
22 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2018 | 1 |
| 2013 | 1 |
| 2012 | 1 |
| 2009 | 1 |
| 2008 | 1 |
| 2007 | 1 |