 In this study, researchers discovered a new type of surface state that connects two types of topological defects, isolated whale phonons, IWPs, and whale nodal walls, WNWs. This surface state is called the fan-shaped surface state and its number of fan leaves corresponds to the churn number of the IWP. Researchers were able to identify these surface states in two different materials, K2MG203 and NB3L2N. These results provide a new method for identifying IWPs and determining their churn numbers in real materials. Additionally, this study reveals a new class of surface states in the topological whale complex composed of IWPs and WNWs. This article was authored by Huahua Fu, Qingbolu, Shuqi Wang, and others.