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We introduce bilayer networks in this paper to study the coupled air-sea systems. Results show that the framework of bilayer networks is powerful for studying the statistical topology structure and dynamics in the fields of ocean and atmosphere. Based on bilayer networks, the inner and cross interactions of the sea surface temperature (SST) field and the height field are displayed, and the main three-dimensional air-sea interaction pattern is identified. The formation of the main pattern can be explained by the 'gearing between the Indian and Pacific Ocean (GIP)' model; therefore, the pattern existence can be confirmed reliably. Furthermore, lead-lag analysis reveals the trigger processes of the 'GIP'. That is, the anomalies of the tropical mid-eastern Pacific Ocean SST (TMEPO-SST) appear first; then, through the Walker circulation, the 850-hPa geopotential height over the Pacific Islands responds to the anomalies of the TMEPO-SST 2 months later; finally, the tropical Indian Ocean SST (TIO-SST) responds to the anomalies of the height 1 month later through the Asian monsoon circulation. Therefore, the impacts of the TMEPO-SST to the TIO-SST show 3 months later through the air-sea interactions between the components of the main three-dimensional air-sea interaction mode. The new framework uncovers already-known as well as other novel features of the air-sea systems and general circulation. The application of complex network theory and methodology to understand the complex interactions between the oceans and the atmosphere is promising. [ABSTRACT FROM AUTHOR]