Species with Potential Arising from Surfaces with Orbifold Points of Order 2, Part II: Arbitrary Weights

Let Σ =(Σ, M,O) be either an unpunctured surface with marked points and order-2 orbifold points or a once-punctured closed surface with order-2 orbifold points. For each pair (τ, ω) consisting of a triangulation τ of Σ and a function ω: O1,4, we define a chain complex C• (τ, ω) with coefficients in F_2=Z/2Z. Given Σ and ω , we define a colored triangulation of Σ _ω =(Σ, M,O,ω) to be a pair (τ, \xi) consisting of a triangulation of Σ and a 1-cocycle in the cochain complex that is dual to C• (τ, ω) ; the combinatorial notion of colored flip of colored triangulations is then defined as a refinement of the notion of flip of triangulations. Our main construction associates to each colored triangulation a species and a potential, and our main result shows that colored triangulations related by a flip have species with potentials (SPs) related by the corresponding SP-mutation as defined in [25]. We define the flip graph of Σ _ω as the graph whose vertices are the pairs (τ, x) consisting of a triangulation τ and a cohomology class x\in H1C(τ, ω)) , with an edge connecting two such pairs, (τ, x) and (Σ, z), if and only if there exist 1-cocycles xi in x and ζ in z such that (τ, xi) and (Σ, ζ) are colored triangulations related by a colored flip; then we prove that this flip graph is always disconnected provided the underlying surface Σ is not contractible. In the absence of punctures, we show that the Jacobian algebras of the SPs constructed are finite-dimensional and that whenever two colored triangulations have the same underlying triangulation, the Jacobian algebras of their associated SPs are isomorphic if and only if the underlying 1-cocycles have the same cohomology class. We also give a full classification of the nondegenerate SPs one can associate to any given pair (τ, ω) over cyclic Galois extensions with certain roots of unity. The species constructed here are species realizations of the 2^|O| skew-symmetrizable matrices that Felikson-Shapiro-Tumarkin associated in [17] to any given triangulation of Σ . In the prequel [25] of this paper we constructed a species realization of only one of these matrices, but therein we allowed the presence of arbitrarily many punctures.