Weighted Homology Of Bi-Structures Over Certain Discrete Valuation Rings

An RNA bi-structure is a pair of RNA secondary structures that are considered as arc-diagrams. We present a novel weighted homology theory for RNA bi-structures, which was obtained through the intersections of loops. The weighted homology of the intersection complex X features a new boundary operator and is formulated over a discrete valuation ring, R. We establish basic properties of the weighted complex and show how to deform it in order to eliminate any 3-simplices. We connect the simplicial homology, H-i(X), and weighted homology, H-i,H-R(X), in two ways: first, via chain maps, and second, via the relative homology. We compute H-0,H-R(X) by means of a recursive contraction procedure on a weighted spanning tree and H-1,H-R(X) via an inflation map, by which the simplicial homology of the 1-skeleton allows us to determine the weighted homology H-1,H-R(X). The homology module H-2,H-R(X) is naturally obtained from H-2(X) via chain maps. Furthermore, we show that all weighted homology modules H-i,H-R(X) are trivial for i>2. The invariant factors of our structure theorems, as well as the weighted Whitehead moves facilitating the removal of filled tetrahedra, are given a combinatorial interpretation. The weighted homology of bi-structures augments the simplicial counterpart by introducing novel torsion submodules and preserving the free submodules that appear in the simplicial homology.