In this paper, we firstly focus on catacondensed even ring systems (shortly CERS) without any linearly connected adjacent triple of finite faces. For such a graph \( G \), we describe a bijection between the set of all perfect matchings (Kekulé structures) of \( G \) and the set of all independent sets of the inner dual of \( G \), which enables us to prove the equality between three polynomials: the sextet polynomial of \( G \), the independence polynomial of the inner dual of \( G \), and the newly introduced link polynomial of \( G \). These equalities imply that the number of perfect matchings of \( G \) equals the number of resonant sets of \( G \) and also the number of independent sets of the inner dual of \( G \). Moreover, we show that the number of edges of the resonance graph of \( G \) coincides with the derivative of the mentioned polynomials evaluated at \( x=1 \). Finally, we provide the generalization of the results to all peripherally 2-colorable graphs.