  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.
