Topology and phase transition for EPYM AdS black hole in thermal potential

As we all know the local topological properties of thermodynamical systems can be expressed by the winding numbers as the defects. The topological number that is the sum of all winding numbers can be used to classify the global topological nature of thermodynamical systems. In this paper, we construct a kind of thermal potential and then put the Einstein-power-Yang-Mills AdS black hole in it. Through the analysis of the geometric characteristics of the thermal potential based on the complex analysis we find the topological number is an invariant that is same as shown in the way of the Duan's $\phi$-mapping topological current [Sci. Sin. 9, 1072 (1979)]. Furthermore, we adopt the Kramer's escape rate method to investigate the intensity of the first-order phase transition.