Michel Planat, Marcelo Amaral, Fang Fang, David Chester, Raymond Aschheim, Klee Irwin (2022)
Taking a DNA sequence, a word with letters/bases A, T, G and C, as the relation between the generators of an infinite group, one can discriminate between two important families: (i) the cardinality structure for conjugacy classes of subgroups of is that of a free group on one to four bases, and the DNA word, viewed as a substitution sequence, is aperiodic; (ii) the cardinality structure for conjugacy classes of subgroups of is not that of a free group, the sequence is generally not aperiodic and topological properties of have to be determined differently. The two cases rely on DNA conformations such as A-DNA, B-DNA, Z-DNA, G-quadruplexes, etc. We found a few salient results: Z-DNA, when involved in transcription, replication and regulation in a healthy situation, implies (i). The sequence of telomeric repeats comprising three distinct bases most of the time satisfies (i). For two-base sequences in the free case (i) or non-free case (ii), the topology of may be found in terms of the character variety of and the attached algebraic surfaces. The linking of two unknotted curves—the Hopf link—may occur in the topology of in cases of biological importance, in telomeres, G-quadruplexes, hairpins and junctions, a feature that we already found in the context of models of topological quantum computing. For three- and four-base sequences, other knotting configurations are noticed and a building block of the topology is the four-punctured sphere. Our methods have the potential to discriminate between potential diseases associated to the sequences.