The topological filtration of $\gamma$-structures

TL;DR

This paper introduces a new mathematical framework for analyzing RNA pseudoknot structures based on their topological genus, providing generating functions, singularity analysis, and a limit theorem for their distribution.

Contribution

It derives a novel bivariate generating function for $b3$-structures, enabling detailed analysis of their topological properties and implications for RNA folding algorithms.

Findings

Derived a new bivariate generating function for $b3$-structures.

Established a singularity analysis of the generating functions.

Proved a central limit theorem for the distribution of topological genus.

Abstract

In this paper we study $\gamma$-structures filtered by topological genus. $\gamma$-structures are a class of RNA pseudoknot structures that plays a key role in the context of polynomial time folding of RNA pseudoknot structures. A $\gamma$-structure is composed by specific building blocks, that have topological genus less than or equal to $\gamma$, where composition means concatenation and nesting of such blocks. Our main results are the derivation of a new bivariate generating function for $\gamma$-structures via symbolic methods, the singularity analysis of the solutions and a central limit theorem for the distribution of topological genus in $\gamma$-structures of given length. In our derivation specific bivariate polynomials play a central role. Their coefficients count particular motifs of fixed topological genus and they are of relevance in the context of genus recursion and novel folding algorithms.