Non-trivial topology

To emphasize the broad region of applicability of the system described in this section, we would like to stress the following fact. Recently, in Refs. [48,49] during investigation of 3D-quantum field theory with Chem-Simon s action a strong connection was established between expectation values of Wilson lines with non-trivial topology and partition function determining the polynomial invariant of the knot or link. 

There is no lack of beautifully simple MIMs. The quintessential example is SchilFs all-hydrocarbon [2]catenane [167], the simplest non-trivial topology composed of the simplest atoms and bonds (Fig. 24). Other MIMs are simplified by highly symmetrical structures that make them easy on the brain , such as the... 

After the very early achievements due to the real pioneers of chemical topology who forty years ago first acceeded to that intriguing molecule that is a catenane (by statistical threading approach Wasserman, 1960 or by a directed approach Liittringhaus and Schill, 1964), this peculiar field of research concerned with molecules whose prime feature rests without doubt on beauty and non trivial spacial arrangement, has in the beginning of the 80 s literaly exploded with the apparition of various templated approaches. 

This theorem is the main reason why we are able to use topology in dimension two to derive non-trivial combinatorial results. 

After a temperature decrease, the formation of the globular structure is thermodynamically favorable. Supposing that the final state can be described in virial expansion we introduce as usual two- and three-body interaction constants B = b < 0 and C = const > 0. However, in addition to the volume interactions we should also take into account non-local topological constraints having a repulsive character. In this connection, we express our main conjecture the topological constraints lead to special non-trivial fractal properties of line representing the chain trajectory in the globule. Let us describe the structure. 

The topological analysis of p(r, X) then proceeds through the search for and identification of its critical points. In the neighbourhood of a critical point, the field p(r, X) is expanded by Taylor s theorem, the first non-trivial terms being those quadratic in the variables r. The collection of the nine second derivatives of p(r, X) constitute the so-called Hessian matrix A of p(r, X) at the critical point. 

It is generally very difficult to restore the two-dimensional structure (the picture ) of any non-trivial chemical compound having only the topology of... 

The non-trivial character and the structural peculiarities of compounds with topological bonds stimulates the elaboration of equally non-trivial approaches to the solution of synthetic problems in this field. In this respect it seems... 

The synthesis of the models shown above acquired additional meaning from the absolute novelty of the stereochemistry problems which became available to experimental studies. While it is premature to make definite suggestions about the peculiarities of the chemistry of compounds having topological bonds, the uniqueness of their structure seems to guarantee that a number of non-trivial phenomena, such as unusual chelating properties, regulated catalytic activity, etc., will be discovered in this field. ... 

Obviously, the Hamiltonian (7) commutes with any product of Pa which is equal to to the product of azab operators around a set of closed loops, these products are fixed by the constraint. However, for a topologically non-trivial system there appear a number of other integrals of motion for a system with K openings a product of azb operators along contour, 7 that begins at one opening and ends at another (or at the outer boundary, see Fig. 6)... 

The first non-trivial comparison theorem relating the 2 topologies states that the strong topology is not too strong ... 

A natural extension of the work described in Sect. 2.4 was to prepare a Cu(l)-complexed [2]catenane with macrocycles incorporating the Zn and Au porphyrins. [2]catenanes are topologically non-trivial molecules (non-planar molecular graph) in which two rings are interlocked but not linked [23,24]. Therefore, differentiating the rings with Zn- and Au-porphyrins enables us to study photoinduced electron transfer in mechanical bond systems and also to have information on the conformation of the system. 

Transition metals have been used as assembling and templating species in a variety of processes [1]. The three-dimensional template effect of one or two copper(I) centres has extensively been used to construct topologically non-trivial molecules like catenanes and knots [2]. 

In chemistry, self-assembly is defined as a phenomenon by which molecules order themselves into a particular arrangement without external intervention. The process occurs under the guidance of numerous weak non-covalent bonding interactions that represent the core of supramolecular chemistry. It is not surprising that chemists engaged in the synthesis of topologically non-trivial structures are interested in utilizing self-assembly to direct the templation of chemical components into intermediates prone to cyclization, followed by the formation of covalent bonds. 

Now we should decompose C and next S into pieces that can be put together to form slices that are topologically trivial and thus subject to the standard non-Abelian Stokes theorem. Such decomposition is shown in Fig. 9. Explicitly, this reads as... 

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