# Geometry and physics of knots

### Details

Serval ID

serval:BIB_0E7B92EC5132

Type

**Article**: article from journal or magazin.

Collection

Publications

Fund

Title

Geometry and physics of knots

Journal

Nature

ISSN

0028-0836

Publication state

Published

Issued date

1996

Peer-reviewed

Oui

Volume

384

Number

6605

Pages

142-145

Language

english

Abstract

KNOTS are usually categorized in terms of topological properties that are invariant under changes in a knot's spatial configuration(1-4). 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 mobility(5).

DOI

Web of science

Create date

24/01/2008 10:25

Last modification date

18/11/2016 11:43