The Non-Abelian Character of Entanglements

The natural generalization of the problem considered above is the calculation of the partition function for the chain entangled with many obstacles on the plane. At first sight, it seems that the approach presented above allows one to solve the problem easily. Actually, let us replace the function f(z) in Eq. (4) by the 

Therefore, the principal difficulty connected with the application of Eq. (12) is due to the incompleteness of the Gauss invariant. So, the use of the Gauss invariant for adequate classification of topologically different states in many-chain systems is very problematic. Nevertheless, that approach was used repeatedly for consideration of such physically important question as the high-elasticity of polymer networks with topological constraints [15]. Unfortunately, 

The model polymer chain in an array of obstacles (PCAO) (see Fig. 4) combines the geometrical clarity of its image with the possibility to investigate the influence of entanglements on equilibrium and dynamic properties of polymers quite precisely. 

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