Loop Homology of Bi-secondary Structures II

TL;DR

This paper explores the topological properties of bi-secondary structures by analyzing the loop nerve, revealing a relationship between crossing components in arc diagrams and the second homology group.

Contribution

It introduces the concept of crossing components in arc diagrams and proves their correspondence to the rank of the second homology group of the loop nerve.

Findings

Number of crossing components equals the rank of H_2(R).

Identification of combinatorial structures in arc diagrams.

Deeper understanding of the topological features of bi-secondary structures.

Abstract

In this paper we further describe the features of the topological space $K(R)$ obtained from the loop nerve of $R$, for $R=(S,T)$ a bi-secondary structure. We will first identify certain distinct combinatorial structures in the arc diagram of $R$ which we will call crossing components. The main theorem of this paper shows that the total number of these crossing components equals the rank of $H_2(R)$, the second homology group of the loop nerve.