Entanglement topologically invariant

The word W consisting of a sequence of letters corresponding to different entanglements (introduced in Sect. 2.21) plays a role of full topological invariant for the PCAO-model. It is closely connected with the concept of the primitive path obtained by means of roughing of the microscopic chain trajectory up to the scale of the lattice cell and by exclusion of all loop fragments not entangled with the obstacles (Fig. 5). 

We sume that disengagement by pure reptation is negligible for star molecules with diffidently long arms in an entangled medium. (For a contrary opmion, however, based on computer simulation of star molecule motions, see Ref. 33). Relaxation for an f-arm star in a topologically invariant medium is then equivalent to the relaxation of f tethered chains, where is the tethered chain relaxation time (Eq.66) for individual aruK (Rg.l2). 

The Doi-Edwards theory treats monodisperse linear chain liquids by a model which suppresses fluctuations and assumes a topologically invariant medium. Two parameters are required, the monomeric friction coefficient which characterizes the local dynamics and the primitive path step length a which characterizes the topology of the medium. The step length is related to the entanglement molecular weight of earlier theories, = cRqT/Gn, by Eqs. 1 and 37 ... 

Topological invariants offer in principle a rigorous approach to the statistical mechanics of entangled polymer systems, but this approach suffers from a number of disadvantages. First of all, the invariants are so complex that the use of topological invariants for more than two chains is hardly possible jjjjg restriction would... 

In all network theories based on topological invariants (see Sect. 2), it has been assumed that the retractive force exhibited by polymeric networks comes not only from the entropic forces generated in deformed elastically active chains, but also from topological interactions among entangled molecules. 

The models presented in the previous section are of an elementary nature in the sense that they ignore contributions from intermolecular effects (such as entanglements that are permanently trapped on formation of the network). Among the theories that take account of the contribution of entanglements are (1) the treatment of Beam and Edwards [19] in terms of topological invariants, (2) the slip-link model [20, 21], (3) the constrained-]unction and constrained-chain models [22-27], and (4) the trapped entanglement model [11,28]. The slip-link, constrained-junction, and constrained-chain models can be studied under a common format as can be seen from the discussion by Erman and Mark [7]. For illustrative purposes we present the constrained-junction model in some detail here. We then discuss the trapped entanglement models. 

Keywords Gravitational computer, Topological quantum computing, link invariants, entanglement, complexity, computability. 

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