Physical knots

Knot Generalization of a physical knot, thought of without thickness and in an arbitrary number of dimensions. 

Topological quantum field theory has become a fascinating and fashionable subject in mathematical physics. At present, the main applications of topological field theory are in mathematics (topology of low-dimensional manifolds) rather than in physics. Its application to the issue of classification of knots and links is one of the most interesting. To approach this problem, one usually tries to somehow encode the topology of a knot or link. As was first noted by Witten... 

The plectonemic nature of the DNA double-helix makes it a tractable molecule for experiments in molecular topology. This is a very rich vein for the exploration of the topological properties of matter. In this chapter, we have tried to illuminate some of the techniques by which the single-stranded topology of DNA can be directed in synthetic molecules. Catenanes and knots, periodic braids, and Borro-mean rings are available from simple protocols, and it is to be hoped that the physical properties associated with complex topologies (Moffatt 1990) will become available through the medium of DNA constructions. 

It is based on the idea of electromagnetic knot, introduced in 1990 [27-29] and developed later [30-32], An electromagnetic knot is defined as a standard electromagnetic field with the property that any pair of its magnetic lines, or any pair of its electric lines, is a link with linking number i (which is a measure of the extent to which the force lines curl themselves around one another, i.e., of the helicity of the field). These lines coincide with the level curves of a pair of complex scalar fields , 0. The physical space and the complex plane are compactified to Si and S2, so that the scalars can be... 

M. Atiyah, The Geometry and Physics of Knots, Cambridge Univ. Press, 1990. 

Chaotic Evolution and Strange Attractors D. Ruelle Introduction to Polymer Dynamics P. de Gennes The Geometry and Physics of Knots M. Atiyah Attractors for Semigroups and Evolution Equations ... 

Thus, it should be stressed that the mathematical topological theory investigates, as a rule, the problems of classification of knots and links, the construction of topological invariants, definitions of topological classes, etc. whereas the fundamental physical problem in the theory of topological properties of polymer chains is the determination of the entropy, S = In Z with the fixed topological state of chains. Both these problems are very difficult, but important. 

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