Topological automorphisms

Surprisingly, we obtain 13,035 solutions for 20 amino acids (without specifying multiplicities and using the topological automorphism group). A glance at one of the structures generated reveals what has happened ... 

Algorithm 13 Construction of the Geometrically Distinct Topological Automorphisms (Adapted from Ref. 7)... 

Use of geometrical attributes table (GAT) and geometrically distinct topological automorphism (GDTA). 

Remarks on the Group of Automorphism of a Free Topological Tree... 

It follows that an automorphism of an abstract simplicial complex induces a continuous automorphism of its geometric realization, and that the geometric realizations of two isomorphic abstract simplicial complexes are homeo-morphic, with homeomorphisms induced by the isomorphism maps. On the other hand, the existence of a homeomorphism between Z i and A2 does not imply the isomorphism of A and. 2. If a topological space allows a simplicial structure, then it allows infinitely many nonisomorphic simplicial structures. 

An orbit is the set of all atoms which are transformed from one into another by the actions of all automorphisms of a molecular graph. TTie set of all different orbits of a molecular graph G forms a partition of G. If P represents an automorphism then atom i is symmetric (topologically equivalent) with its image p/, and atonis i and p, belong to the same orbit. This symmetry relationship in the molecular graph may or may not be true for the three-dimensional molecular structure. 

Initially, all stereocenters and their type (true or para) are identified. Each of the n stereocenters is then assumed to be a true stereocenter different from every other stereocenter. Since a stereocenter can only have one of two configurations (e.g., R or S), a compound with n different stereocenters will lead to a set of 2" different parity combinations, i.e, stereoisomers, the configuration of each of which can be expressed as a stereoparity (binary) vector of n elements. If the compound actually possesses n different stereocenters, stereoisomer generation is complete. If, on the other hand, the presence of topologically equivalent true stereocenters and/or para stereocenters has been identified, the number of stereoisomers will be less than 2". In operational terms, that means that redundancies exist in the set of 2" stereoparity vectors. The objective in this approach is to identify and remove those redundancies. The automorphism group of such a compound is the key to revealing equivalence among the stereoparity vectors. 

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