Algebraic Topological Indices of Molecular Chirality

Abstract

In this chapter we shall focus attention upon those aspects of the mathematical analysis of molecular graphs which relate to the geometric and topological properties of their spatial symmetries and of their knotting and linking. Interconnections with the theory of knots and links (which can also serve as models for small molecules) will also be presented. In this first section we shall briefly review the fundamental concepts which delineate the interrelationships between the molecular structures, the geometric structures, and the topological structures. In the second section we shall present several of the significant test families of molecular graphs which will be employed in the comparison of the stereotopological indices used to detect chirality. This discussion continues in the third section where some of the elements of the knot theory of molecular graphs are introduced. Chimerical graphs, i.e. graphs with a fixed structure near vertices, their elementary properties, and their structural properties are presented in the fourth section. In the fifth section we present a survey of the algebraic topological methods that have been employed to detect and quantify the chirality of classical knots and links as well as molecular graphs, This discussion concludes with the discussion of the development, by E. Witten, of stereotopological indices arising from interactions with, statistical mechanics and gauge theories from mathematical physics.