Mathematical modelling of biology processes based on the table of prime links in the solid torus up to 4 crossings

Abstract

The motivation to study links in lens spaces can be justified in recent applications to theoretical physics and biology. In this paper, we give a short outline of practical significance and implementation proposals of such links, the study of which begins with the study of links in the solid torus, since a punctured disk diagram of a link in the lens space can be considered as a punctured disk diagram in the solid torus provided with the additional slide move. However, links in the solid torus find their applications themselves as well. In this paper, we propose a method to represent knotted proteins as links in the solid torus. Such a method is based on the existence of correspondence between knotoids and knots in the solid torus using a double branched cover. To this end, the table of links in solid torus is necessary. Therefore, we classify all prime links in the solid torus up to 4 crossings. One of possible future applications of the constructed table is an analysis of the database LinkProt that collects information about protein chains and complexes that form links. Also, our table can be used to construct table of prime links in lens spaces.