Abstract. We present a definition of mutations of species with potential that can be applied to the species realizations of any skew-symmetrizable matrix B over cyclic Galois extensions E/F whose base field F has a primitive [E : F ] th root of unity. After providing an example of a globally unfoldable skew-symmetrizable matrix whose species realizations do not admit non-degenerate potentials, we present a construction that associates a species with potential to each tagged triangulation of a surface with marked points and orbifold points of order 2. Then we prove that for any two tagged triangulations related by a flip, the associated species with potential are related by the corresponding mutation (up to a possible change of sign at a cycle), thus showing that these species with potential are non-degenerate. In the absence of orbifold points, the constructions and results specialize to previous work ([22] and [23]) by the second author.The species constructed here for each triangulation τ is a species realization of one of the several matrices that Felikson-Shapiro-Tumarkin have associated to τ in [15], namely, the one that in their setting arises from choosing the number 1 2 for every orbifold point.

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 : O → {1, 4}, we define a chain complex C•(τ, ω) with coefficients in F2 = Z/2Z. Given Σ and ω, we define a colored triangulation of Σω = (Σ, M, O, ω) to be a pair (τ, ξ) consisting of a triangulation of Σ and a 1-cocycle in the cochain complex which 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 [27].We define the flip graph of colored triangulations of Σω as the graph whose vertices are the pairs (τ, x) consisting of a triangulation τ and a cohomology class x ∈ H 1 (C • (τ, ω)), with an edge connecting two such pairs (τ, x) and (σ, z) if and only if there exist 1-cocycles ξ ∈ x and ζ ∈ z such that (τ, ξ) 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 non-degenerate SPs one can associate to any given pair (τ, ω) over cyclic Galois extensions with primitive 4 th roots of unity.The species constructed here are species realizations of the 2 |O| skew-symmetrizable matrices that Felikson-Shapiro-Tumarkin associated in [18] to any given triangulation of Σ. In the prequel [27] to this paper we constructed a species realization of only one of these matrices, but therein we allowed the presence of arbitrarily many punctures.

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We construct an isomorphism between the partially ordered set of tilting modules for the Auslander algebra of K[x]/(x n ) and the interval of rational permutation braids in the braid group on n strands. Hence, there are only finitely many tilting modules.

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