The Insertion Chain Complex: A Topological Approach to the Structure of Word Sets

Abstract: We introduce the Insertion Chain Complex, a higher-dimensional extension of insertion graphs, as a new framework for analyzing finite sets of words. We study its topological and combinatorial properties, in particular its homology groups, as measures of the complexity for word sets. After establishing its theoretical foundations, we investigate the computational and combinatorial aspects of these complexes. Among other results, we classify minimal 1-dimensional cycles and prove that every finitely generated abelian group can be realized as the homology of the insertion complex for some set of words. We also identify conditions that guarantee vanishing homology. These results provide new invariants for characterizing finite sets of words through word-based topological structures and their properties.