Crystal structures are the richest source of information about materials that allows us to analyse material properties and structure-property relations. Understanding regularities of crystal structures would give us a tool for rational material design, opening new possibilities to invent new drugs, functional materials and materials with required properties. Of especial interest are materials with knotted or interpenetrating covalent bond nets, since the properties of such materials are defined not only by the covalent bond network itself but also the topological nature of the knots that they form. Such materials could yield new molecular machines, new types of biologically active substances, new materials for electronics and computer industry.
Unfortunately, at the moment there are no methods that would enable detection of such materials in crystal structures. One of the main difficulties is the absence of reliable knot invariants, i.e. the mathematical expressions that would guarantee that the knots, links or nets with the same invariant are topologically equivalent. As a consequence there are no reliable algorithms to detect such structures in crystallographic databases.
We thus propose to investigate a possibility to apply the known knot invariants to crystallographic net analysis, taking into account the specifics of the crystal structures (e.g. the finite size of the atoms and the low number of crossings of knots that the molecules form). The main goal would be to create an algorithm for discovery of knots, links and interpenetrating nets in crystal structures and to build a database of such structures.