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We propose a novel mathematical paradigm for the study of genetic variation in sequence alignments. This framework originates from extending the notion of pairwise relations, upon which current analysis is based on, to k-ary dissimilarity. This dissimilarity naturally leads to a generalization of simplicial complexes by endowing simplices with weights, compatible with the boundary operator. We introduce the notion of k-stances and dissimilarity complex, the former encapsulating arithmetic as well as topological structure expressing these k-ary relations. We study basic mathematical properties of dissimilarity complexes and show how this approach captures watershed moments of viral dynamics in the context of SARS-CoV-2 and H1N1 flu genomic data. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the homology of the simplicial complex induced by a given pair of RNA secondary structures, R = (S , T) . Such a pair induces a bi-secondary structure, whose associated loop nerve X is the simplicial complex obtained by loop intersections. We will provide an algebraic proof of the fact that H 1 (X) = 0 . We will provide a combinatorial interpretation for the generators of H 2 (X) in terms of crossing components of the bi-structure and establish that the rank of H 2 (X) equals the total number of such crossing components. Finally, we shall prove that each crossing component naturally encodes a triangulation of a 2-sphere and provide an analysis of the geometric realization of X. [ABSTRACT FROM AUTHOR]