A family of metrics on contact structures based on edge ideals

The measurement of the similarity of RNA secondary structures, and in general of contact structures, of a fixed length has several specific applications. For instance, it is used in the analysis of the ensemble of suboptimal secondary structures generated by a given algorithm on a given RNA sequence, and in the comparison of the secondary structures predicted by different algorithms on a given RNA molecule. It is also a useful tool in the quantitative study of sequence-structure maps. A way to measure this similarity is by means of metrics. In this paper we introduce a new class of metrics $d_{m}$, $m\geq 3$, on the set of all contact structures of a fixed length, based on their representation by means of edge ideals in a polynomial ring. These metrics can be expressed in terms of Hilbert functions of monomial ideals, which allows the use of several public domain computer algebra systems to compute them. We study some abstract properties of these metrics, and we obtain explicit descriptions of them for $m=3,4$ on arbitrary contact structures and for $m=5,6$ on RNA secondary structures.