Symmetry, Integrability and Geometry: Methods and Applications Minkowski Polynomials and Mutations ⋆

Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f, and use this to establish many basic facts about mutations. Mutations can be understood combinatorially in terms of Minkowski rearrangements of slices of P, or in terms of piecewise-linear transformations acting on the dual polytope P^* (much like cluster transformations). Mutations map Fano polytopes to Fano polytopes, preserve the Ehrhart series of the dual polytope, and preserve the period of f. Finally we use our results to show that Minkowski polynomials, which are a family of Laurent polynomials that give mirror partners to many three-dimensional Fano manifolds, are connected by a sequence of mutations if and only if they have the same period.


Introduction
Given a Laurent polynomial f ∈ C x ±1 1 , . . . , x ±1 n , one can form the period of f π f (t) = 1 2πi n |x 1 |=···=|xn|=1 The period of f gives a solution to a GKZ hypergeometric differential system associated to the Newton polytope of f (see [9, § 3]). Periods of Laurent polynomials and the associated differential systems are interesting from the point of view of mirror symmetry, because certain Laurent polynomials arise as mirror partners to n-dimensional Fano manifolds [3,4,5,8,11,17]. In its most basic form (which will suffice for what follows) the statement that a Laurent polynomial f is a mirror partner for a Fano manifold X means that the Taylor expansion of the period of f π f (t) = k≥0 c k t k coincides with a certain generating function for Gromov-Witten invariants of X called the quantum period of X [9, § 4]. The Taylor coefficient c k here is the coefficient of the unit monomial in the Laurent polynomial f k . We refer to the sequence (c k ) k≥0 as the period sequence for f . We expect that if the Laurent polynomial f is a mirror partner to a Fano manifold X, then there is a geometric relationship between f and X as follows (cf. [24]). Let N f be the lattice generated by the exponents of monomials of f . Consider the Newton polytope Newt(f ) of f , and assume that the origin lies in its strict interior. Let X f denote the toric variety defined by the spanning fan of Newt(f ) in N f ⊗ Q; in general X f will be singular. We expect that X f admits a smoothing with general fiber X. Note that our assumption that Newt(f ) contains the origin is not restrictive: if the origin is outside Newt(f ) then the period of f is constant, and hence cannot be the quantum period of a Fano manifold; if the origin is contained in a proper face of Newt(f ) then we can reduce to a lower-dimensional situation. Note also that the lattice N f may be a proper sublattice of Z n , see Example 8 and [6]. The picture described here implies that one might expect many Laurent polynomial mirrors for a given Fano manifold, as a smooth Fano manifold can degenerate to many different singular toric varieties.
The motivating case for this paper is that of three-dimensional Minkowski polynomials. These are a family of Laurent polynomials in three variables, defined in Section 4 below, which provide mirror partners to many of the three-dimensional Fano manifolds. The correspondence between three-dimensional Minkowski polynomials and Fano manifolds is not one-to-one, in part because many Minkowski polynomials give rise to the same period sequence. There are several thousand Minkowski polynomials f : (C × ) 3 → C, up to change of co-ordinates on (C × ) 3 , but between them these give only 165 distinct period sequences 1 . In what follows we give a conceptual explanation for this phenomenon. We define birational transformations, called mutations, that preserve periods and show that any two Minkowski polynomials with the same period are related by a sequence of mutations. Our birational transformations are higher-dimensional generalisations of the mutations defined by Galkin-Usnich [14]. Combining our results in Section 5 with a theorem of Ilten [18] shows that whenever f and g are Minkowski polynomials with the same period sequence, the toric varieties X f and X g occur as fibers of a flat family over a curve. This is consistent with our conjectural picture, which implies that whenever Laurent polynomials f and g are mirror partners for the same Fano manifold X, the toric varieties X f and X g are deformation equivalent.
The paper is organised as follows. We define mutations algebraically in Section 2 and combinatorially in Section 3. Algebraic mutations operate on Laurent polynomials, whereas combinatorial mutations operate on polytopes. An algebraic mutation of a Laurent polynomial f induces a combinatorial mutation of its Newton polytope Newt(f ); the converse statement is discussed in Remark 3. We establish various basic properties of combinatorial mutations: they send Fano polytopes to Fano polytopes (Proposition 2); there are, up to isomorphism, only finitely many mutations of a given polytope (Proposition 3); and mutation-equivalent polytopes have the same Hilbert series (Proposition 4). We define Minkowski polynomials in Section 4, and in Section 5 show by means of a computer search that all Minkowski polynomials with the same period sequence are connected by a sequence of mutations. Period sequences for Minkowski polynomials are listed in Appendix A, and mutations connecting the Minkowski polynomials with the same period sequence are listed in Appendix B.

Mutations
In this section we define mutations. These are a class of birational transformations ϕ : (C × ) n (C × ) n with the property that if two Laurent polynomials f and g are related by a mutation ϕ, so that g = ϕ f , then the periods of f and g coincide. We begin with two examples.
where A, B, C are Laurent polynomials in x and y. The pullback of f along the birational transformation (C × ) 3 (C × ) 3 given by is We say that the Laurent polynomials f and g are related by the mutation (2).
with k < 0 and l > 0 where each C i , i ∈ {k, k + 1, . . . , l}, is a Laurent polynomial in x and y. Let A be a Laurent polynomial in x and y such that C i is divisible by A −i for i ∈ {k, k + 1, . . . , −1}.
The pullback of f along the birational transformation (C × ) 3 (C × ) 3 given by We say that the Laurent polynomials f and g are related by the mutation (3).
Remark 1. Note that the pullback of a Laurent polynomial along a birational transformation of the form (2) or (3) will not, in general, be a Laurent polynomial: the condition For brevity, we write this isomorphism as x → x M .
If f , g are Laurent polynomials and ϕ is a mutation such that ϕ f = g then we say that f and g are related by the mutation ϕ.
Remark 2. One can also define mutations of Laurent polynomials in n variables, using the obvious generalisations of Example 2, Definition 1, and Definition 2.
The Newton polytope P of f has two pairs of parallel facets, and we place one pair of them at heights 1 and −1 by applying the This transforms f into the Laurent polynomial We now apply the birational transformation (2) with A(x, y) = y + y x + 1 x , followed by the This shows that the Laurent polynomials f and g are related by the mutation ϕ, where ϕ(x, y, z) = xy 2 z + yz + 1 y , xy 2 xy 2 z + yz + 1 , z xy 2 z + yz + 1 xy .

Lemma 1.
If the Laurent polynomials f and g are related by a mutation ϕ, then the periods of f and g coincide.
from (3). Since GL 3 (Z)-equivalence preserves periods, it suffices to show that if g = ϕ f then the periods of f and g coincide. Let Z = (x, y, z) ∈ (C × ) 3 : A(x, y) = 0 and let U = (C × ) 3 \ Z. Note that the restriction ϕ| U : U → (C × ) 3 is a morphism, and that Let so that the period of f is 1 2πi n C 1,1,1 The amoeba of Z is a proper subset of R 3 [15, Chapter 6, Corollary 1.8], so there exists (a, b, c) such that C a,b,c ⊂ U . The cycles C a,b,c and C a ,b ,c are homologous in (C × ) 3 for any non-zero a, b, c, a , b , c . Thus π g (t) = 1 2πi n C 1,1,1 by the change of variable formula.
Now the homology class ϕ(C a,b,c ) ∈ H 3 (C × ) 3 , Z is equal to k C 1,1,1 for some integer k, since H 3 (C × ) 3 , Z is freely generated by C 1,1,1 , and from (4) and the change of variable formula we see that k = 1. It follows that π g (t) = 1 2πi n C 1,1,1 Example 4. Consider the two Laurent polynomials Since f 1 and f 2 have the same period sequence π f 1 (t) = π f 2 (t) = 1 + 28t 2 + 216t 3 + 3516t 4 + 49680t 5 + · · · , and since Newt(f 1 ) = Newt(f 2 ), it is tempting to assume that there is some GL 3 (Z)-equivalence that preserves the Newton polytope and sends f 1 to f 2 . This is not the case. However, there does exist a birational map sending f 1 to f 2 . This is a composition of mutations The maps ϕ, ψ and their inverses are given by ϕ : (x, y, z) → z(xyz + (y + 1) 2 ) y , xyz + (y + 1) 2 xy , y , Set P := Newt(f 1 ) = Newt(f 2 ) and Q := Newt(f ). The polytopes P and Q are reflexive, but since Vol(P ) = 32 and Vol(Q) = 28, P and Q are not isomorphic. However, as predicted by Proposition 4 below, the Ehrhart series Ehr P * (t) and Ehr Q * (t) are equal: in other words, the Hilbert series Hilb

Combinatorial mutations
The mutations of a Laurent polynomial f defined in Section 2 induce transformations of the Newton polytope of f . In this section we give a combinatorial and coordinate-free definition of these transformations, which we call combinatorial mutations, in terms of the Newton polytope alone. We then establish some basic properties of combinatorial mutations. Let us begin by fixing our notation. Let N be an n-dimensional lattice and let P ⊂ N Q := N ⊗ Q be a convex lattice polytope such that 1) P is of maximum dimension, dim P = n; 2) the origin lies in the strict interior of P , 0 ∈ P • ; 3) the vertices V(P ) ⊂ N of P are primitive lattice points.
We call such a polytope Fano. Given any lattice polytope P ⊂ N Q , the dual polyhedron P * ⊂ M Q , where M := Hom(N, Z), is defined by Condition 2 ensures that, when P is Fano, P * is a polytope. When P * is a lattice polytope, we say that P is reflexive. Low-dimensional reflexive polytopes have been classified [20,21]: up to the action of GL n (Z) there are 16 reflexive polytopes in dimension two; 4, 319 in dimension three; and 473, 800, 776 in dimension four. A Fano polytope P ⊂ N Q is called canonical if P • ∩ N = {0}. In two dimensions, the reflexive polytopes and canonical polytopes coincide. In general every reflexive polytope is canonical, although the converse is not true: there are 674, 688 canonical polytopes in dimension three [19].
Definition 3. Let w ∈ M be a primitive lattice vector, and let P ⊂ N Q be a lattice polytope. Set We define the width of P with respect to w to be the positive integer If width w (P ) = l then we refer to w as a width l vector for P . We say that a lattice point v ∈ N (resp. a subset If 0 ∈ P • then h min < 0 and h max > 0, hence w must have width at least two. If P is a reflexive polytope then for any w ∈ V(P * ) there exists a facet F ∈ F(P ) at height −1; this is a well-known characterisation of reflexive polytopes [2]. For each height h ∈ Z, w defines a hyperplane H w,h : By definition, w h min (P ) = H w,h min ∩ P and w hmax (P ) = H w,hmax ∩ P are faces of P , hence V(w h min (P )) ⊆ V(P ) and V(w hmax (P )) ⊆ V(P ). Furthermore, the face w h min (P ) is a facet of P if and only if w = u for some vertex u ∈ V(P * ), where u denotes the unique primitive lattice point on the ray from 0 through u. Similarly, w hmax (P ) is a facet if and only if −w = u for some u ∈ V(P * ).

Definition 4.
Recall that the Minkowski sum of two polytopes Q, R ⊂ N Q , is Henceforth we adopt the convention that Q + ∅ := ∅ for any polytope Q.
. We call such an F a factor for P with respect to w. We define the combinatorial mutation given by width vector w, factor F , and polytopes {G h } to be the convex lattice polytope Notice that one need only consider factors up to translation, since for any v ∈ N such that w(v) = 0 we have mut w (P, v + F ; {G h + hv}) ∼ = mut w (P, F ; {G h }). In particular, if w h min (P ) is zero-dimensional then combinatorial mutations with width vector w leave P unchanged.
Example 5. Consider the situation of Example 2, so that f is the Laurent polynomial with k < 0 and l > 0, A is a Laurent polynomial in x and y such that The Laurent polynomials f and g are related by the algebraic mutation ϕ. This algebraic mutation induces a combinatorial mutation of P = Newt(f ) with w = (0, 0, 1), and mut w (P, F ; {G h }) = Newt(g).
Remark 3. Given a Laurent polynomial f 1 with Newton polytope P , there may exist combinatorial mutations of P that do not arise from any algebraic mutation of f 1 . Given a combinatorial mutation of P , however, there exists a Laurent polynomial f 2 with Newt(f 2 ) = P such that the combinatorial mutation arises from an algebraic mutation of f 2 . See Example 6 below.
Hence P = P .
Proof . Pick any vertex v ∈ V(Q), and set h := w(v). First consider the case when h ≥ 0. Notice that v ∈ V(w h (P ) + hF ), since otherwise v lies in the convex hull of two (not necessarily lattice) points v 1 , v 2 ∈ Q, and as such could not be a vertex. In general the vertices of a Minkowski sum are contained in the sum of the vertices of the summands, hence there exist lattice points Proof . Set Q := mut w (P, F ; {G h }) and Q := mut w (P, F ; {G h }), and suppose that Q = Q . Then (possibly after exchanging Q and Q ) there exists some vertex q ∈ V(Q ) such that q ∈ Q, w(q) < 0. In particular, there exists a supporting hyperplane H u,l of Q separating Q from q; i.e. u(x) ≤ l for all x ∈ Q, and u(q) > l. For In light of Proposition 1, we simply write mut w (P, F ) for a mutation mut w (P, F ; {G h }) of P . Example 6. Consider the Laurent polynomial The corresponding Newton polytope P has vertices {(0, 1, 2), (0, −1, 2), (2, −1, 2), (0, 2, 0), (4, −2, 0), (−2, 0, 0), (0, 0, −1)}. Note that the sublattice generated by the non-zero coefficients has index one. Set w = (0, 0, −1) ∈ M . The height −2 slice w −2 (P ) Minkowski-factorizes into two empty triangles, whereas the height −1 slice w −1 (P ) = conv{(0, 1, 1), (−1, 0, 1), (1, −1, 1), (2, −1, 1)} is indecomposable. Since there are no vertices of P at height −1, Definition 5 allows us to take G −1 = ∅. This gives a combinatorial mutation to This corresponds to the algebraic mutation ϕ : (x, y, z) → (x, y, z/(x + y + 1)) sending f to Set g = f + yz. Then Newt(g) = P but, since g has a non-zero coefficient on the slice w −1 (P ), the combinatorial mutation described above does not arise from any algebraic mutation of g. Proof . Begin by assuming that P is Fano, and set Q := mut w (P, F ). Let v ∈ V(Q), and define h := w(v) to be the height of v with respect to w. If h ≥ 0 then v = v P + hv F for some v P ∈ V(P ), v F ∈ V(F ). But v P is primitive by assumption, hence v is primitive.
If h < 0 then v ∈ V(G h ). Without loss of generality we are free to take G h equal to the smallest polytope such that G h +(−h)F ⊇ H w,h ∩V(P ). Suppose that v is not primitive. Then for Finally, the "if and only if" follows by considering the inverse combinatorial mutation.
Corollary 1. Let P ⊂ N Q be a lattice polytope, and let w be a width two vector. The combinatorial mutation mut w (P, F ) is a canonical polytope if and only if P is a canonical polytope.
Proof . Begin by assuming that P is a canonical polytope, and set Q := mut w (P, F ). Since w is of width two, we need only show that H w,0 ∩ V(P ) = H w,0 ∩ V(Q); it follows that Q is canonical. But H w,0 ∩V(Q) ⊆ H w,0 ∩V(P ) ⊆ H w,0 ∩V(Q), where the second inclusion follows by considering the inverse combinatorial mutation. Once more, the "if and only if" follows by exchanging the roles of P and Q via the inverse combinatorial mutation.
Proposition 3. Let P ⊂ N Q be a lattice polytope, 0 ∈ P • . Up to isomorphism, there are only finitely many combinatorial mutations mut w (P, F ).
Proof . For fixed width vector w there are clearly only finitely many factors F , and so only finitely many combinatorial mutations mut w (P, F ). Assume that 0 ∈ P • and consider the spanning fan ∆ in M Q for the dual polytope P * . Since 0 ∈ (P * ) • , any width vector w lies in a cone σ ∈ ∆. If we insist that dim σ is as small as possible, then σ is uniquely determined. If dim σ = dim P then w h min (P ) is zero-dimensional, hence mut w (P, F ) ∼ = P . So we may insist that σ ∈ ∆ (n−1) is of codimension at least one. By definition of duality, w ∈ ∂(−h min P * ). Consider the corresponding face F := w h min (P ). Let l F ∈ Z >0 be the largest integer such that there exist lattice polytopes A, B ⊂ N Q , A = ∅, with F = l F A + B. Then, for a factor to exist, l ≤ l F . Define l P := max{l F | F is a face of P }. Then w is a primitive vector in ∆ (n−1) ∩ l P P * ∩ M , and the right hand side is a finite set that depends only on P .
Remark 4. When dim P = 2, 0 ∈ P • , we see that mut w (P, F ) is a non-trivial combinatorial mutation only if w ∈ {u | u ∈ V(P * )}. In particular, there are at most |V(P )| choices for w, and at most |∂P ∩ N | distinct non-trivial combinatorial mutations.
Let Q ⊂ M Q be a rational polytope, and let r ∈ Z >0 be the smallest positive integer such that rQ is a lattice polytope. In general there exists a quasi-polynomial, called the Ehrhart quasipolynomial, L Q : Z → Z of degree dim Q such that L Q (m) = |mQ ∩ M |. The corresponding generating function Ehr Q (t) := m≥0 L Q (m)t m is called the Ehrhart series of Q, and can be written as a rational function [25] Ehr with non-negative coefficients. We call (δ 0 , δ 1 , . . . , δ r(n+1)−1 ) the δ-vector of Q. In particular, the δ-vector is palindromic if and only if Q * ⊂ N Q is a lattice polytope [12].
Proposition 4. Let P ⊂ N Q be a lattice polytope, 0 ∈ P • , and let Q := mut w (P, F ) be a combinatorial mutation of P . Then Ehr P * (t) = Ehr Q * (t).
Proof . Since P * is the dual of a lattice polytope, for any point u ∈ M there exists a nonnegative integer k ∈ Z ≥0 such that u ∈ ∂(kP * ). By definition of duality, H u,−k := {u(v) = −k | v ∈ N Q } is a supporting hyperplane for P . We begin by showing that the map u → u − u min w, Since H u,−k is a supporting hyperplane for P , there exists some v P ∈ V(P ) such that u(v P ) = −k. If w(v P ) ≥ 0 then for any which is a contradiction. Hence u(v F ) = u min , and so (u − u min w)(v Q ) = −k. Thus H u−u min w,−k is a supporting hyperplane for Q.
Finally, suppose that u, u ∈ ∂(kP * ) are lattice points such that u−u min w = u −u min w. Since for any v F ∈ V(F ), (u − u min w)(v F ) = u(v F ), we see that u min = u min and so u = u . Hence |∂(kP * )| ≤ |∂(kQ * )|. By considering the inverse combinatorial mutation mut −w (Q, F ) = P , we have equality. The result follows.
Proposition 4 thus implies that there is no sequence of mutations connecting f 1 and f 2 . However π f 1 (t) = π f 2 (t) = 1 + 12t 3 + 900t 6 + 94080t 9 + 11988900t 12 + · · · , which we recognise as the period sequence for P 2 × P 2 . As in Section 1, let N f i be the sublattice of N generated by the exponents of monomials of f i , and let X f i be the toric variety defined by the spanning fan of P i in N f i ⊗Q. The lattice N f 1 is equal to N , and X f 1 is isomorphic to P 2 ×P 2 .
The lattice N f 2 is an index-three sublattice of N , but the restriction of P 2 to N f 2 is isomorphic to P 1 and hence X f 2 is also isomorphic to P 2 × P 2 . Consider now the toric variety X 2 defined by the spanning fan of P 2 in N Q . This is a three-fold cover of X 2 , and is not deformation-equivalent to P 2 × P 2 : X 2 has Hilbert δ-vector δ 2 , whereas P 2 × P 2 has Hilbert δ-vector δ 1 . As this example illustrates, sublattices are invisible to period sequences.
Let Q := mut w (P, F ) be a combinatorial mutation of P , and let r P denote the smallest dilation k of P * such that kP * is integral. When P is a Fano polytope, we call r P the Gorenstein index. The minimum period common to the cyclic coefficients of L P * divides r P ; when the period does not equal r P we have a phenomena known as quasi-period collapse [7,16]. In general r P = r Q but, by Proposition 4, P * and Q * are Ehrhart equivalent. Hence we have examples of quasi-period collapse. At its most extreme, when P is reflexive r P = 1 and so the period of L Q * is one. Combinatorial mutations give a systematic way of producing families of rational polytopes that exhibit this behaviour.

Corollary 2.
A three-dimensional canonical polytope P ⊂ N Q is combinatorial mutation equivalent to a reflexive polytope if and only if P is reflexive.
Proof . By inspecting the classification of canonical polytopes [19] we see that the period of L P * equals r P for each P .
The following example demonstrates that Corollary 2 does not hold in higher dimensions.
Since r P k = 2, we have quasi-period collapse.
The map u → u − u min w on M described in the proof of Proposition 4 is a piecewise-linear transformation analogous to a cluster transformation. Let ∆ F be the normal fan of F in M Q . Notice that the normal fan is well-defined up to translation of F . Furthermore, the cones of ∆ F are not strictly convex: they each contain the subspace Zw. Let σ be a maximum-dimensional cone of ∆ F . Then there exists M σ ∈ GL n (Z) such that, for any u ∈ −σ, the map u → u − u min w is equal to the map u → uM σ .

Three-dimensional Minkowski polynomials
Three-dimensional Minkowski polynomials are a family of Laurent polynomials in three variables with Newton polytopes given by three-dimensional reflexive polytopes. We consider Laurent polynomials f ∈ C[x ±1 , y ±1 , z ±1 ], and write Definition 6. Let Q be a convex lattice polytope in Q 3 .
1. We say that Q is a length-one line segment if there exist distinct points v, w ∈ Z 3 such that the vertices of Q are v, w (in some order) and Q ∩ Z 3 = {v, w}. In this case, we set 2. We say that Q is an A n triangle if there exist distinct points u, v, w ∈ Z 3 such that the vertices of Q are u, v, w (in some order) and Q ∩ Z 3 = {u, v 0 , v 1 , . . . , v n }, where v 0 = v, v n = w, and v 0 , v 1 , . . . , v n are consecutive lattice points on the line segment from v to w.
In this case, we set Observe that, in each case, the Newton polytope of f Q is Q.
Definition 7. A lattice Minkowski decomposition of a lattice polytope Q ⊂ Q m is a decomposition Q = Q 1 + · · · + Q r of Q as a Minkowski sum of lattice polytopes Q i , i ∈ {1, 2, . . . , r}, such that the affine lattice generated by Q ∩ Z m is equal to the sum of the affine lattices generated by Q i ∩ Z m , i ∈ {1, 2, . . . , r}.
Example 11. The Minkowski decomposition is not a lattice Minkowski decomposition, because the affine lattices in the summands generate an index-two affine sublattice of Z 2 .
Definition 8. Let Q be a lattice polytope in Q m . We say that lattice Minkowski decompositions Definition 9. Let Q be a lattice polytope in Q 3 . A lattice Minkowski decomposition Q = Q 1 + · · · + Q r of Q is called admissible if each Q i is either a length-one line segment or an A n triangle. Given an admissible Minkowski decomposition Q = Q 1 + · · · + Q r of Q, we set Observe that the Newton polytope of f Q:Q 1 ,...,Qr is Q. Observe also that if Definition 10. Let P be a three-dimensional reflexive polytope in Q 3 . Let f be a Laurent polynomial in three variables. We say that f is a three-dimensional Minkowski polynomial with Newton polytope P if: 1. Newt(f ) = P , so that 2. a 0 = 0.
3. For each facet Q ∈ F(P ) of P , we have v∈Q∩Z 3 a v x v = f Q:Q 1 ,...,Qr for some admissible lattice Minkowski decomposition Q = Q 1 + · · · + Q r of Q.
Given a three-dimensional reflexive polytope P , there may be: 1) no Minkowski polynomial with Newton polytope P , if some facet of P has no admissible lattice Minkowski decomposition; 2) a unique Minkowski polynomial with Newton polytope P , if there is a unique equivalence class of admissible lattice Minkowski decomposition for each facet of P ; 3) many Minkowski polynomials with Newton polytope P , if there is some facet of P with several inequivalent admissible lattice Minkowski decompositions.
Up to isomorphism, there are 4319 three-dimensional reflexive polytopes [20]; 1294 of them support no Minkowski polynomial, and the remaining 3025 together support 3747 distinct 3 Minkowski polynomials. As facets, P has four A 1 triangles, one A 2 triangle, and a pentagon. Remark 5. Recall from Section 1 that we denote by N f the lattice generated by the exponents of monomials of a Laurent polynomial f . For a three-dimensional reflexive polytope P , the lattice generated by P ∩ Z 3 is equal to Z 3 ; thus whenever f is a three-dimensional Minkowski polynomial we have N f = Z 3 . In dimension four there are precisely 16 reflexive polytopes P such that the lattice generated by P ∩ Z 4 is a proper sublattice of Z 4 [6].
Remark 6. If f is a Minkowski polynomial and g is a Laurent polynomial related to f via a mutation ϕ then in general g will not be a Minkowski polynomial; indeed Newt(g) will in general not be reflexive or even canonical (cf. Corollary 2).

Mutations between Minkowski polynomials
Consider now the set of all three-dimensional Minkowski polynomials, and partition this into smaller sets, which we call buckets, such that f and g are in the same bucket if and only if the first eight terms of the period sequences for f and g agree. There are 165 buckets. Using Proposition 3 and a computer search, we have determined all mutations of elements of each bucket. For all except two buckets any two Laurent polynomials f , g in the bucket are connected by a sequence of mutations that involves only Minkowski polynomials (and hence involves only Laurent polynomials from that bucket). In fact: 1. For 156 of the buckets, any two Laurent polynomials f , g in the bucket are connected by a sequence of width two mutations that involves only Minkowski polynomials in that bucket.
2. For 7 of the buckets, any two Laurent polynomials f , g in the bucket are connected by a sequence of width two and width three mutations that involves only Minkowski polynomials in that bucket.
For the remaining two buckets 4 , any two Laurent polynomials f , g in the bucket are connected by a sequence of mutations, but it is not possible to insist that all of the Laurent polynomials involved are Minkowski polynomials. (It is, however, possible to insist that all of the Laurent polynomials have reflexive Newton polytope.) The sequence of mutations connecting two Minkowski polynomials f and g is far from unique, but representative such mutations are shown in Appendix B. The mutations shown there suffice to connect any two three-dimensional Minkowski polynomials in the same bucket. Proof . Any two Minkowski polynomials in the same bucket are connected by a sequence of mutations.
Proposition 5. If f and g are three-dimensional Minkowski polynomials with the same period sequence, then there exist a flat family π : X → Σ over a possibly-reducible rational curve Σ and two distinct points 0, ∞ ∈ Σ such that π −1 (0) is isomorphic to the toric variety X f and π −1 (∞) is isomorphic to the toric variety X g .
Proof . Combine the results in Appendix B with [18,Theorem 1.3].
In particular this gives a geometric proof that if f and g are three-dimensional Minkowski polynomials with the same period sequence, then X f and X g have the same Hilbert series. One might hope that substantially more is true: for example that given a bucket of Minkowski polynomials {f i : i ∈ I} corresponding to a Fano manifold X, the toric varieties X f i , i ∈ I, all lie in the same component of the Hilbert scheme, and that X lies in this component too. It is possible that this hope is too naïve, as the geometry of Hilbert schemes is in many ways quite pathological, but in any case, as things currently stand, results of this form seem to be out of reach.
Remark 7. One might regard our results as further evidence that the mirror to a Fano manifold X should be some sort of cluster variety X ∨ together with a function (the superpotential) on X ∨ . This is familiar from the work of Auroux [1]. Given a special Lagrangian cycle and a complex structure on a Fano manifold X, mirror symmetry associates to this choice a Laurent polynomial. The Laurent polynomial depends on our choice: there is a wall-and-chamber structure on the space of parameters, and Laurent polynomials from neighbouring chambers are expected to be related by an elementary birational transformation. Reformulating this: the mirror to X should be a pair (X ∨ , W ) where X ∨ is obtained by gluing tori (one for each chamber) along birational transformations (coming from wall-crossing), and the superpotential W is a global function on X ∨ . When X is two-dimensional, the elementary birational transformations that occur here are closely related to those in the theory of cluster algebras [13]. In our three-dimensional situation, we need to allow a still more general notion of cluster algebra.
More precisely: the fundamental objects in the theory of cluster algebras are seeds and their mutations. A seed is a collection of combinatorial and algebraic data: the so-called exchange matrix and a basis for a field k = Q(x 1 , . . . , x n ) of rational functions of n variables, called a cluster. Seeds can be transformed into new seeds through a process called mutation. The Laurent phenomenon theorem of [13] says that any cluster variable is a Laurent polynomial when expressed in terms of any other cluster variable in any other mutation equivalent seed. This produces a large subalgebra in k, called the upper bound, consisting of those elements that are Laurent polynomials when expressed in terms of cluster variables of any seed. The "Laurent phenomenon" arising from mirror symmetry for Fano manifolds (i.e. the presence of a Laurent polynomial that remains a Laurent polynomial under some collection of elementary birational transformations) does not fit into the framework of cluster algebras. In cluster algebras the number of ways to mutate a seed naturally is always less than or equal to the transcendence degree of k, whereas the Laurent polynomials that are mirror dual to a del Pezzo surface S have χ(S) = ρ(S) + 2 ways to mutate them [10,14] and the transcendence degree of k here is 2. Furthermore, in contrast to the case of cluster algebras, the transcendence degree of the upper bound in this setting equals just one; in fact, the upper bound is just the ring of polynomials generated by W . In dimension three there are additional complications: in the Fano setting the exchange polynomials need not be binomials (cf. [22]), and we do not see how to define the 2-form that occurs in the cluster theory. In this paper we do not consider the problem of defining seeds in dimension three, nor do we single out the relevant notion of mutation of seeds. These are two important questions for further study.