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 , R ( X ) , in two ways: first, via chain maps, and second, via the relative homology. We compute H 0 , R ( X ) by means of a recursive contraction procedure on a weighted spanning tree and H 1 , R ( X ) via an inflation map, by which the simplicial homology of the 1-skeleton allows us to determine the weighted homology H 1 , R ( X ) . The homology module H 2 , R ( X ) is naturally obtained from H 2 ( X ) via chain maps. Furthermore, we show that all weighted homology modules H i , 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.