Topological data analysis, including persistent homology, has undergone significant development in recent years. However, one outstanding challenge is to build a coherent statistical inference procedure on persistent diagrams. The paired dependent data structure, as birth and death in persistent diagrams, adds additional complexity to the development . In this paper, we present a new lattice path representation for persistent diagrams. A new exact statistical inference procedure is developed for lattice paths via combinatorial enumerations. The proposed lattice path method is applied to  the topological characterization of the protein structures of  COVID-19 viruse . We demonstrate that there are topological changes during the conformation change of spike proteins that are needed to initiate the infection of host cells.
Topological data analysis, including persistent homology, has undergone significant development in recent years. However, one outstanding challenge is to build a coherent statistical inference procedure on persistent diagrams. The paired dependent data structure, which are the births and deaths in persistent diagrams, adds complexity to statistical inference . In this paper, we present a new lattice path representation for persistent diagrams. A new exact statistical inference procedure is developed for lattice paths via combinatorial enumerations. The proposed lattice path method is applied to study the topological characterization of the protein structures of the COVID-19 virus . We demonstrate that there are topological changes during the conformational change of spike proteins , a necessary step in infecting host cells.
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which are the births and deaths
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virus
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conformational
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