<p>The study of unconventional phases and elucidation of correspondences between topological invariants and their intriguing properties are pivotal in topological physics. Here, we investigate a complex exceptional ring (CER), composed of a third-order exceptional ring and multiple Weyl exceptional rings, and establish a direct correspondence between Chern numbers and the distinctive behaviors of these structures. We show that band braiding during quasistatic encircling processes correlates with nontrivial Chern numbers, resulting in triple (double) periodic spectra for topologically nontrivial (trivial) middle bands. Moreover, Chern numbers predict mode transfer during dynamical encircling. Experimental schemes for realizing CER in cold atoms are proposed, emphasizing the crucial role of Chern numbers as both measurable quantity and descriptor of exceptional physics in dissipative systems. This discovery broadens topological classifications in non-Hermitian systems, with promising applications in quantum computing and metrology.</p>