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In this paper we introduce abstract string modules and give an explicitbijection between the submodule lattice of an abstract string module and theperfect matching lattice of the corresponding abstract snake graph. Inparticular, we make explicit the direct correspondence between a submodule of astring module and the perfect matching of the corresponding snake graph. Forevery string module, we define a Coxeter element in a symmetric group, and weestablish a bijection between these lattices and the interval in the weakBruhat order determined by the Coxeter element. Using the correspondencebetween string modules and snake graphs, we give a new concise formulation ofsnake graph calculus.

Lattice bijections for string modules, snake graphs and the weak Bruhat order