Abstract
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.
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Acknowledgements
We would like to thank Robert Penner for his input and helpful remarks on the manuscript. We also gratefully acknowledge the comments from Fenix Huang. Many thanks to Thomas Li and Ricky Chen for discussions.
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Bura, A.C., He, Q. & Reidys, C.M. Loop homology of bi-secondary structures II. J Algebr Comb 56, 785–798 (2022). https://doi.org/10.1007/s10801-022-01132-3
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DOI: https://doi.org/10.1007/s10801-022-01132-3