Preserving Hodge Vectors of Lattice Polytopes

PRESER VING HODGE VECTORS OF LA TTICE POL YTOPES V ADYM KUR YLENK O AND BENJAMIN NILL De dic ate d to the memory of Johanna Steinmeyer Abstract In Ehrhart theory , the lo cal h ∗ -p olynomial is a fundamen tal inv ariant of a lattice p olytop e. Its co ef- ficien t vector, the lo c al h ∗ -ve ctor , is the coefficient vector of the top degree part of the Hodge-Deligne p olynomial of the primitiv e cohomology with compact supp ort of the asso ciated generic hypersurface in the algebraic torus. In the literature on hypergeometric motives, the Ho dge ve ctor of a lattice p olytop e is its lo cal h ∗ -v ector with leading and trailing zero es remo v ed. Of recent sp ecial interest are thin p olytop es whose Ho dge vectors v anish. Giv en lattice p olytop es P 1 , . . . , P k con tained in a k -dimensional subspace U ⊆ R d and a d -dimensional lattice p olytop e P ⊂ R d , w e compute the Ho dge vector of the Cayley p olytop e P 1 ∗ · · · ∗ P k ∗ P , and sho w that it equals the mixed volume of P 1 , . . . , P k times the Ho dge vector of the pro jection of P along U . This allows finding infinitely man y high-dimensional lattice polytop es with the same Hodge vector that are not free joins. The pro of relies on a closed formula for the Ho dge-Deligne p olynomial of generic complete intersections in the torus in terms of the biv ariate/mixed h ∗ -p olynomial. A sp ecial case of our construction is what we call L awr enc e twists : extending the Gale transform by centrally-symmetric pairs of vectors. As applications, w e can produce many new thin polytop es answering a question b y Borger, Kretsc hmer and the second author, and w e pro vide an alternative explanation of the thinness of B k -p olytop es answering a question of Selyan in. 1. Introduction and main resul t Let P b e a d -dimensional lattic e p olytop e , i.e., a d -dimensional p olytop e in R d whose vertices are elements of the lattice Z d . Consider the asso ciated affine hypersurface Z P :    x ∈ ( C ∗ ) d : X m ∈ P ∩ Z d c m x m = 0    , where c m are complex num b ers that are non-zero if m is a vertex of P . W e consider situations only when c m are sufficien tly generic, namely w e require Z P to be P -regular in the sense of Bat yrev, see [ Bat93 , Definition 3.3]. As will be recalled in Section 2 , the Ho dge-Deligne p olynomial of the primitive cohomology of Z P can b e written in the following form E prim ( Z P ; u, v ) = X p + q ≤ d − 1 h ∗ p,q u p v q (1.1) for nonnegative in tegers h ∗ p,q that satisfy Ho dge duality h ∗ p,q = h ∗ q ,p . These n um bers were termed b y Katz and Stapledon [ KS16 , Remark 7.7] as co efficien ts of the h ∗ -diamond of P and hav e purely combinatorial expressions [ DK87 , BB96 , BM03 , KS16 ] in terms of inv arian ts of P . Let us recall the main notion in classical Ehrhart theory . The Ehrhart series , the generating series of the famous Ehrhart p olynomial of a lattice p olytop e, is the following rational function: 1 + X k ≥ 1 | k P ∩ Z d | t k = h ∗ ( P , t ) (1 − t ) d +1 . Its n umerator h ∗ ( P , t ) = 1 + P d i =1 h ∗ i t i is called the h ∗ -p olynomial of P , and its co efficient v ector (1 , h ∗ 1 , . . . , h ∗ d ) its h ∗ -ve ctor . A direct relation to algebraic geometry is given by the fact that for i = 1 , . . . , d the i th co efficient of the h ∗ -p olynomial of P equals the sum ov er the i th diagonal 1 PRESER VING HODGE VECTORS OF LA TTICE POL YTOPES 2 of the h ∗ -diamond (i.e., h ∗ i = P d − i j =0 h ∗ i − 1 ,j ). Hence, the sum o v er all en tries of the h ∗ -diamond plus one equals the normalized volume of P (whic h is defined as d ! times the Euclidean volume of P ). The palindromic top degree co efficien t v ector ( h ∗ d − 1 , 0 , h ∗ d − 2 , 1 , . . . , h ∗ 0 ,d − 1 ) of the h ∗ -diamond is called the ℓ ∗ -ve ctor or lo c al h ∗ -ve ctor ℓ ∗ P of P . This inv arian t was originally prop osed b y Stanley [ Sta92 ] in the context of p olyhedral sub divisions and turned up inde- p enden tly in the com binatorial formulas for stringy Ho dge n um bers by Batyrev and Boriso v [ BB96 , BN08 , NS12 ]. The study of lo cal h ∗ -p olynomials can also b e referred to as lo c al Ehrhart the ory . W e refer to [ BKN23 ] for a comprehensive ov erview of the literature. In this pap er, w e emphasize its app earance in the theory of hypergeometric motives [ RR V22 ]. F rom this view- p oin t the question b ecomes relev an t whether lattice p olytop es of differen t dimensions can hav e the same ℓ ∗ -v ector up to a differen t n um ber of outside zero es. Let us mak e the notation used in this research communit y precise: Definition 1.1. The Ho dge ve ctor of P is defined as the zero vector if ℓ ∗ P = 0, and otherwise as the vector ℓ ∗ P without b eginning and trailing zero es. F or instance, for ℓ ∗ P = (0 , 0 , 1 , 0 , 3 , 0 , 1 , 0 , 0) the Ho dge vector of P equals (1 , 0 , 3 , 0 , 1). In this area, the follo wing question arises naturally [ RR V22 , GV24 ]. Supp ose w e hav e a ( d − 1)-dimensional hypersurface Z P that realizes some hypergeometric motiv e with Ho dge v ector of length l < d . Do es there exist a v ariety V of dimension l − 1 realizing the same motive with the same Ho dge vector? F rom this p ersp ective, it is thus imp ortant to kno w what combinatorial constructions for lattice p olytop es preserve their Ho dge vectors in order to recognize whether such a dimension reduction might already b e p ossible in the toric setting. In classical Ehrhart theory , the lattice pyramid construction provides a direct wa y to get from a d -dimensional lattice p olytop e a ( d + 1)-dimensional lattice p olytop e with the same h ∗ -p olynomial. In fact, Bat yrev prov ed in [ Bat06 ] ev en a conv erse result, namely , fixing the h ∗ -p olynomial forces lattice p olytop es in high dimensions to b e lattice pyramids. In particular, there are up to lattice pyramid constructions only finitely man y isomorphism classes of lattice p olytop es with the same h ∗ -v ector (see also [ Nil08 ]). As it turns out, in lo cal Ehrhart theory the situation is quite different and muc h more op en. Note that because of palindromicity , the dimensions of lattice p olytop es with the same Hodge v ector need to ha v e the same parit y (even/odd). In fact, the Ho dge v ector of an y lattice pyramid just v anishes, see [ BN08 , Lemma 4.5]. One easy wa y though to get a ( d + 2)-dimensional lattice p olytop e with the same Ho dge v ector is to take the free join with a lattice interv al of length 2 (see 2.3 ). This motiv ates the question, whether there are other p ossibilities for preserving the Ho dge vector. In this paper, we present a general construction that allo ws finding infinitely many higher- dimensional lattice p olytop es with the same Ho dge v ector that are not free joins. F or this, let us recall that the Cayley p olytop e P 1 ∗ · · · ∗ P k of lattice p olytop es P 1 , . . . , P k in R d is defined as P 1 ∗ · · · ∗ P k := con v( P 1 × { e 1 } , . . . , P k × { e k } ) ⊂ R d × R k . W e note that dim( P 1 ∗ · · · ∗ P k ) = dim( P 1 + · · · + P k ) + k − 1. Here is our main result: Theorem 1.2. Let P 1 , . . . , P k , P k +1 ⊂ R d b e lattice p olytop es, where the first k polytop es are con tained in a k -dimensional rational subspace U and dim P k +1 = d ≥ 1. Let V denote the mixed volume MV( P 1 , . . . , P k ) of P 1 , . . . , P k . Then the Ho dge v ector of P 1 ∗ · · · ∗ P k +1 equals V times the Ho dge vector of the pro jection of P k +1 along U . The pro of can be found in Section 3 . It follows the original setup of the pap er [ DK87 ] b y Danilo v and Khov anskii and the approach used in [ DRHN19 ]. As a crucial tool, w e presen t in Prop osition 2.7 an explicit formula for the Ho dge-Deligne p olynomial of generic affine complete in tersections in the algebraic torus. PRESER VING HODGE VECTORS OF LA TTICE POL YTOPES 3 Let us describ e how to apply Theorem 1.2 to construct infinitely many non-isomorphic high- dimensional lattice polytop es with the same Ho dge v ector. Let P be a lattice p olytop e of dimension d in R d . Cho ose a nonnegativ e in teger k and any full-dimensional lattice polytop e e P in R d + k that pro jects on to P by pro jecting onto the first d co ordinates. Now, choose an y set of k lattice p olytop es P 1 , . . . , P k in { 0 } × R k with mixed volume 1 (for instance 1 , just take P 1 = · · · = P k as the con v ex h ull of the last k standard basis vectors together with origin). Then b y Theorem 1.2 the Cayley p olytop e P 1 ∗ · · · ∗ P k ∗ e P of dimension d + 2 k has the same Ho dge v ector as P . W e call suc h a construction a gener alize d L awr enc e twist , see Theorem 3.1 . W e refer to the most sp ecial case of the construction of Theorem 1.2 as a L awr enc e twist and describ e it closely in Section 4 . The name comes from its Gale-dual description that in v olv es extending the Gale transform b y cen trally-symmetric pairs of vectors. The pap er is organized as follo ws: Section 2 contains the basics on Ho dge-Deligne p olynomials, biv ariate Ehrhart theory and Ca yley polytop es, as w ell as the closed formula for the Ho dge- Deligne p olynomial for complete intersections in the torus. Section 3 contains the description of generalized Lawrence t wists which implies a pro of of Theorem 3.1 . Section 4 discusses La wrence t wists. Applications, op en questions and relations to other papers will b e presen te d in Section 5 . Ac knowledgmen ts. This work is funded b y the Deutsc he F orsch ungsgemeinschaft (DFG, Ger- man Research F oundation) – 539867500 as part of the research priority program Combinatorial Synergies. The first author thanks Giulia Gugiatti for insigh tful con versations, Asem Ab del- raouf and F ernando Rodriguez Villegas for useful discussions. W e also thank Christian Haase for p ointing us to [ Dur18 ]. 2. Hodge-Deligne pol ynomials and biv aria te Ehrhar t theor y This section recalls the main concepts needed for our results. 2.1. Ho dge-Deligne p olynomials of affine h yp ersurfaces in the torus. The cohomology (with compact supp ort) of an algebraic v ariet y Z carries a mixed Hodge structure. This leads to the definition of its Ho dge-Deligne p olynomial , also known as E -p olynomial : E ( Z ; u, v ) : = X p,q X m ( − 1) m h p,q ( H m c ( Z, Q ) ! u p v q ∈ Z [ u, v ] . Recall that E ( C ; u, v ) = uv and E ( C ∗ ; u, v ) = uv − 1. Ho dge-Deligne polynomials ha ve nice prop erties regarding pro ducts and stratifications: • F or v arieties X and Y w e hav e E ( X × Y ; u, v ) = E ( X ; u, v ) · E ( Y ; u, v ); • if X = ⊔ i X i is a disjoint union of a finite set of lo cally closed subv arieties X i , then E ( X ; u, v ) = X i E ( X i ; u, v ) . See [ DK87 ] for pro ofs and [ DRHN19 ] for more examples. Our particular interest lies in affine hypersurfaces Z P in algebraic tori. Let us follow [ DK87 ] and discuss further basic prop erties of the cohomology of Z P . First of all, we can focus on studying P of full dimension d . Otherwise, P w ould define a hypersurface Z ′ P in a torus of low er dimension d ′ and we could write E ( Z P ; u, v ) = ( uv − 1) d − d ′ E ( Z ′ P ; u, v ) . (2.1) 1 Mixed volume one tuples of lattice p olytop es w ere completely classified b y Estero v and Gusev [ EG15 ]. PRESER VING HODGE VECTORS OF LA TTICE POL YTOPES 4 Since Z P is an affine v ariety of dimension d − 1, b y Grothendieck v anishing theorem and P oincar´ e duality we know that H i c ( Z P , Q ) = 0 , for i = 0 , . . . , d − 2 . Moreo ver, the higher cohomology groups are also well understo o d due to the follo wing prop osi- tion. Prop osition 2.1 ([ DK87 ], Prop osition 3.9) . There exist homomorphisms ϕ i : H i c ( Z P , C ) → H i +2 c (( C ∗ ) d , C ) that are isomorphisms for i > d − 1 and surjective for i = d − 1. Therefore, we kno w the cohomology of Z P except for the middle one H d − 1 c ( Z P , Q ). This motiv ates the follo wing definition. cf. [ Bat93 ]. Definition 2.2. Define the primitive c ohomolo gy of Z P to b e the kernel of ϕ i P H i c ( Z P ) : = k er ϕ i . Since mixed Hodge structures form an ab elian category , the cohomology P H i c ( Z P ) carries a mixed Hodge structure. Multiplying with an o v erall sign, so that its coefficients are alwa ys nonnegativ e, w e define E prim ( Z P ; u, v ) : = X p + q ≤ d − 1 h p,q ( P H d − 1 c ( Z P , Q )) u p v q . Since we know that h p,p ( H n + p (( C ∗ ) n , Q )) =  n p  , w e can easily relate E prim to the Ho dge- Deligne p olynomial of Z P : E prim ( Z P ; u, v ) = ( − 1) d − 1 E ( Z P ; u, v ) + (1 − uv ) d − 1 uv . (2.2) 2.2. Biv ariate Ehrhart theory. Let P ⊂ R d b e a d -dimensional lattice polytop e. It is p ossible to express the p olynomials E ( Z P ; u, v ) and th us E prim ( Z P ; u, v ) purely in combinatorial terms. This has b een done b y in [ BB96 , BM03 ], in particular see the last equation of Prop osition 5.5 in [ BM03 ]. W e refer to [ KS16 , BKN23 ] for the details. In [ KS16 , Def. 7.5] Katz and Stapledon define a biv ariate version of the h ∗ -p olynomial h ∗ ( P ; u, v ) whic h they call mixe d h ∗ -p olynomial of P . As the term ‘mixed’ is t ypically used in con vex geometry for tuples of con vex b o dies and, moreov er, a mixed v ersion of the h ∗ -p olynomial had already b een defined in [ HJKST17 ], we prefer the term bivariate h ∗ -p olynomial . Since we w ould lik e to av oid in this pap er an y unnecessary technicalit y , let us just presen t its elegan t algebro-geometric explanation: h ∗ ( P ; u, v ) = 1 + u v E prim ( Z P ; u, v ) . (2.3) Note that this is a biv ariate polynomial with nonnegativ e in teger co efficien ts. Denoting the co efficien ts of the biv ariate h ∗ -p olynomial as elements of an h ∗ -diamond , see ( 1.1 ), we get h ∗ ( P ; u, v ) = 1 + X p + q ≤ d − 1 h ∗ p,q u p +1 v q +1 . W e remark that from the biv ariate h ∗ -p olynomial w e easily recov er the classical h ∗ -p olynomial: h ∗ ( P ; t, 1) = h ∗ P ( t ). The degree of h ∗ P ( t ) is an imp ortan t measure of the complexit y of a lattice p olytop e and is called the de gr e e deg( P ) of P . Note that the degree is at most d . W e ha ve h ∗ ( P ; u, v ) = h ∗ ( P ; v , u ) and the degree of h ∗ ( P ; u, v ) is at most d + 1. W e define the bivariate lo c al h ∗ -p olynomial or bivariate ℓ ∗ -p olynomial as the top degree part of the biv ariate h ∗ -p olynomial: ℓ ∗ ( P ; u, v ) := X p + q = d − 1 h ∗ p,q u p +1 v q +1 . PRESER VING HODGE VECTORS OF LA TTICE POL YTOPES 5 Again, sp ecializing to the univ ariate case we get the lo c al h ∗ -p olynomial or ℓ ∗ -p olynomial ℓ ∗ P ( t ) := ℓ ∗ ( P ; t, 1) = P d i =1 ℓ ∗ i t i , where ℓ ∗ i = h ∗ i − 1 ,d − i for i = 1 , . . . , d . It is a palindromic p olynomial with resp ect to symmetry along d +1 2 . Its co efficient v ector ( ℓ ∗ 1 , . . . , ℓ ∗ d ) is the lo c al h ∗ -ve ctor or ℓ ∗ -ve ctor of P and equals the middle row of the h ∗ -diamond. W e hav e that ℓ ∗ 1 = ℓ ∗ d equals the n umber of interior lattice p oints of P . In the case of sp ecial interest where ℓ P ( t ) = 0, so the Ho dge v e ctor v anishes, w e call P thin . P is called trivial ly thin if the dimension is at least twice the degree. Note that by palindromicity of the ℓ ∗ -v ector, trivially thin p olytop es are automatically thin. F or more about the prop erties of the local h ∗ -p olynomial and about thin p olytop es, we refer to [ BKN23 ]. W e note that if P is just a lattice p oint, then h ∗ ( P ; u, v ) = 1 and ℓ ∗ ( P ; u, v ) = 0. Rewriting ( 2.2 ) using the biv ariate Ehrhart theory notation w e get: E ( Z P ; u, v ) = 1 uv h ( − 1) d − 1 h ∗ ( P ; u, v ) + ( uv − 1) d i . (2.4) W e will generalize this formula in Prop osition 2.7 b elow. Example 2.3. Let us illustrate the h ∗ -diamond of a 3-dimensional lattice p olytop e P : 0 0 0 h ∗ 2 , 0 h ∗ 1 , 1 h ∗ 0 , 2 h ∗ 1 , 0 h ∗ 0 , 1 h ∗ 0 , 0 These num b ers ha v e the following explicit com binatorial expressions (see [ KS16 , Exam- ple 8.9]), where v ∗ denotes the num b er of vertices of P , e ∗ and f ∗ denote the num b er of lattice p oin ts in the relative in terior of edges resp ectively facets of P , i ∗ denotes the num b er of lattice p oin ts in the interior of P , and (2 i ) ∗ the num b er of lattice p oints in the interior of 2 P : 0 0 0 i ∗ (2 i ) ∗ − 4 i ∗ − f ∗ i ∗ f ∗ f ∗ v ∗ + e ∗ − 4 W e remark that a low er b ound theorem of Katz and Stapledon [ KS16 , p. 184] yields the non-trivial combinatorial identit y i ∗ ≤ (2 i ) ∗ − 4 i ∗ − f ∗ , see also [ BKN23 , Cor. 4.6]. 2.3. Isomorphisms, Ca yley p olytop es, free joins and lattice p yramids. Let us first recall that isomorphisms or unimo dular e quivalenc es of lattice p olytop es P , Q ⊂ R d are given b y affine lattice isomorphisms: w e write P ∼ = Q if and only if there are A ∈ GL d ( Z ) and b ∈ Z d with Q = A · P + b . No w, let us give the definition of Cayley p olytop es which turn up quite naturally when considering complete intersections, cf. [ GKZ94 , BN08 , DRHN19 ]. Definition 2.4. Let P 1 , . . . , P k ⊂ R d b e lattice p olytop es. Then P 1 ∗ · · · ∗ P k := con v ( P 1 × { e 1 } , . . . , P k × { e k } ) ⊂ R d + k is called the Cayley p olytop e of P 1 , . . . , P k . Alternatively , P 1 ∗ · · · ∗ P k ∼ = con v ( P 1 × { 0 } , P 2 × { e 1 } , . . . , P k × { e k − 1 } ) ⊂ R d + k − 1 . (2.5) Note that the dimension of P 1 ∗ · · · ∗ P k equals dim( P 1 + · · · + P k ) + k − 1. Its degree is b ounded b y dim( P 1 + · · · + P k ) ≤ d , see [ BN07 , Prop osition 1.12]. In Ehrhart theory , the following sp ecial cases of Cayley p olytop es pla y an imp ortant role. PRESER VING HODGE VECTORS OF LA TTICE POL YTOPES 6 Definition 2.5. Let P ⊂ R n and Q ⊂ R m b e lattice p olytop es. Then P ◦ Q := con v ( P × { 0 } × { 0 } , { 0 } × Q × { 1 } ) ⊂ R n + m +1 is called the fr e e join of P and Q . W e call the free join of P with a lattice point, a lattic e pyr amid pyr( P ) ov er P . W e remark that dim( P ◦ Q ) = dim( P ) + dim( Q ) + 1. Recall that lattice simplices whose vertices form an affine lattice basis are called unimo dular simplic es . The standard example of a d -dimensional unimodular simplex is conv(0 , e 1 , . . . , e d ), whic h is denoted b y ∆ d . One can also describ e unimo dular simplices as successive lattice pyra- mids ov er a lattice p oint. The biv ariate h ∗ -p olynomial is multiplicativ e with resp ect to free joins. Prop osition 2.6. Let P ⊂ R n and Q ⊂ R m b e lattice p olytop es. Then h ∗ ( P ◦ Q ; u, v ) = h ∗ ( P ; u, v ) h ∗ ( Q ; u, v ) . In particular, ℓ ∗ ( P ◦ Q ; u, v ) = ℓ ∗ ( P ; u, v ) ℓ ∗ ( Q ; u, v ) . W e lea v e the proof to the reader. Is is straightforw ard from the m ultiplicativity of the ℓ ∗ - p olynomial [ NS13 , Remark 4.6(5)] with resp ect to free joins and the multiplicativit y of the toric g -p olynomial with resp ect to pro ducts of p osets. In particular, w e recov er the well-kno wn multiplicativit y of the h ∗ -p olynomial [ HT09 , Lemma 1.3] as well as that of the ℓ ∗ -p olynomial. As an imp ortant case, note that h ∗ (p yr( P ); u, v ) = h ∗ ( P ; u, v ) and ℓ ∗ (p yr( P ); u, v ) = 0 . (2.6) 2.4. A closed form ula for the Ho dge-Deligne p olynomial of an affine complete in- tersection in the torus. Consider a system of k equations in ( C ∗ ) d giv en b y f 1 = f 2 = . . . = f k = 0, where f i are generic Lauren t p olynomials with resp ect to their supp orts. Let P i b e the corresp onding Newton p olytop es of f i . Let Y b e the complete in tersection in ( C ∗ ) d defined as the v anishing lo cus of the ab ov e system. F or ∅  = I ⊆ [ k ] w e define P I as the Cayley polytop e of ( P i ) i ∈ I , and d I as the dimension of the Mink owski sum of ( P i ) i ∈ I . Hence, dim( P I ) = d I + | I |− 1. Moreo ver, w e define P ∅ := { 0 } with dimension d ∅ := 0. Prop osition 2.7. In this situation, the following holds: E ( Y ; u, v ) = 1 ( uv ) k X I ⊆ [ k ] ( − 1) d I −| I | ( uv − 1) d − d I h ∗ ( P I ; u, v ) . Note that with our con ven tion for I = ∅ this formula agrees for k = 1 with ( 2.4 ). Pr o of. W e remark that it is enough to prov e this in the case dim( P 1 + · · · + P k ) = d . Otherwise, let Y ′ b e the complete in tersection corresp onding to the d [ k ] -dimensional subspace aff ( P 1 + · · · + P k ) with resp ect to the lattice aff ( P 1 , . . . , P k ) ∩ Z d . As in ( 2.1 ) w e see that E ( Y ; u, v ) = ( uv − 1) d − d [ k ] E ( Y ′ ; u, v ) . If for E ( Y ′ ; u, v ) the resp ectiv e statement in Prop osition 2.7 holds, then we see that it also holds for E ( Y ; u, v ). So, let us assume that dim( P 1 + · · · + P k ) = d . W e define P := P [ k ] = P 1 ∗ · · · ∗ P k , so dim P = d + k − 1. Here, w e use the isomorphic em b edding of P in R d + k − 1 as in ( 2.5 ). Let Z P ⊆ ( C ∗ ) d + k − 1 b e a generic h yp ersurface with Newton p olytop e P . It can b e defined b y the v anishing of Z P : f 1 + y 2 f 2 + . . . + y k f k = 0 . Instead of Z P let us now consider the h yp ersurface e Z ⊆ ( C ∗ ) d × C k e Z : 1 + y 1 f 1 + y 2 f 2 + . . . + y k f k = 0 . PRESER VING HODGE VECTORS OF LA TTICE POL YTOPES 7 Danilo v and Khov anskii [ DK87 , Section 6] gav e the follo wing simple form ula E ( e Z ; u, v ) = ( uv ) k − 1  ( uv − 1) d − e ( Y ; u, v )  . (2.7) Note that e Z is not a hypersurface in an algebraic torus. How ever, there is a stratification of e Z giv en b y affine hypersurfaces in tori that was describ ed in [ DRHN19 ]. F or ∅  = I ⊆ [ k ] define Z I = e Z ∩ { y j  = 0 : j ∈ I } ∩ { y j = 0 : j / ∈ I } . Eac h of these is no w an affine hypersurface in ( C ∗ ) d + | I | . Let us denote by Q I the corresp onding Newton p olytop es. Note that eac h Q I equals the lattice pyramid ov er the Ca yley p olytop e P I , so dim Q I = d I + | I | . F rom the stratification of e Z and ( 2.7 ) we get ( uv ) k − 1  ( uv − 1) d − E ( Y ; u, v )  = X ∅  = I ⊆ [ k ] E ( Z I ; u, v ) . Plugging in the com binatorial form ula ( 2.4 ) for E ( Z I ; u, v ) and applying ( 2.1 ) to tak e in to accoun t the difference d + | I | − ( d I + | I | ) = d − d I of the dimensions of Q I and the ambien t space R d + | I | w e get: ( uv ) k − 1  ( uv − 1) d − E ( Y ; u, v )  = X ∅  = I ⊆ [ k ] ( uv − 1) d − d I uv h ( − 1) d I + | I |− 1 h ∗ ( Q I ; u, v ) + ( uv − 1) d I + | I | i . By ( 2.6 ) w e know that the biv ariate h ∗ -p olynomial is inv arian t under lattice pyramids, so w e get for E ( Y ; u, v ) the following expression: ( uv − 1) d − 1 ( uv ) k X ∅  = I ⊆ [ k ]  ( − 1) d I + | I |− 1 ( uv − 1) d − d I h ∗ ( P I ; u, v ) + ( uv − 1) d + | I |  . (2.8) It is direct to see that X I ⊆ [ k ] ( uv − 1) | I | = ( uv ) k , so X ∅  = I ⊆ [ k ] ( uv − 1) d + | I | = ( uv − 1) d (( uv ) k − 1) . Plugging this into ( 2.8 ) we get the desired formula for E ( Y ; u, v ). □ Note that for k > d , the left-hand side of Prop osition 2.7 v anishes, and th us it gives us a non trivial com binatorial iden tity for the right-hand side. Example 2.8. Supp ose that k = d , in this case the Hodge-Deligne p olynomial E ( Y ; u, v ) is just the num b er of points of Y , whic h equals the mixed volume of P 1 , . . . , P d b y the Bern- stein–Kho v anskii–Kushnirenko (BKK) theorem [ Ber75 , Kou76 ]. Supp ose also that for each ∅  = I ⊊ [ d ] w e hav e d I − | I | > 0, for example, if all P i are full dimensional, then the Ho dge v ector of the Cayley sum P [ d ] is l ∗ ( P 1 ∗ . . . ∗ P d , t ) = ( M V ( P 1 , . . . , P d ) − 1) · t d − 1 . PRESER VING HODGE VECTORS OF LA TTICE POL YTOPES 8 3. Generalized La wrence twists Here is our main result. Theorem 1.2 is a sp ecial case of it (see situation (1)). Theorem 3.1. Let P 1 , . . . , P k , P k +1 ⊂ R d b e lattice p olytop es suc h that dim( P 1 + · · · + P k ) ≤ k and dim( P 1 + · · · + P k +1 ) = d ≥ 2 . Let U b e an y k -dimensional rational subspace of R d con taining P 1 , . . . , P k if k ≤ d , and set U := R d otherwise. W e denote by V the mixed volume MV ( P 1 , . . . , P k ) of P 1 , . . . , P k if k ≤ d , and set V := 0 otherwise. Then the biv ariate h ∗ -p olynomial of the Ca yley polytop e P 1 ∗ · · · ∗ P k +1 equals V ( uv ) k h ∗ (pro j U ( P k +1 ); u, v ) + X I ⊊ [ k +1]: k +1 ∈ I ( − 1) k −| I | (1 − uv ) d − d I h ∗ ( P I ; u, v ) . (3.1) In particular, we hav e ℓ ∗ ( P [ k +1] ; u, v ) = V ( uv ) k ℓ ∗ (pro j U ( P k +1 ); u, v ) (3.2) in each of the following t wo situations: (1) F or each I ⊊ [ k + 1] with k + 1 ∈ I w e hav e • | I | +( d − d I ) ≤ k or • | I | +( d − d I ) = k + 1 and P I is thin. F or instance, this holds if dim( P k +1 ) = d . (2) k ≥ d , which implies that P 1 ∗ · · · ∗ P k +1 is trivially thin. Pr o of. Let us consider the main case k ≤ d first. Let f 1 , . . . , f k +1 ∈ C [ x ± 1 , . . . , x ± d ] b e generic p olynomials with Newton p olytop es P 1 , . . . , P k +1 . As U is a rational subspace, we can choose an appropriate lattice transformation of Z d , so we can assume that U is simply the subspace generated by the first k standard basis v ectors and f 1 , . . . , f k ∈ C [ x ± 1 , . . . , x ± k ]. W e denote the solution set of f 1 , . . . , f k in ( C ∗ ) k b y b Y . Note that b Y is a finite set. It follo ws from the BKK-theorem that its size equals the mixed volume V := MV( P 1 , . . . , P k ). In particular, applying Prop osition 2.7 to P 1 , . . . , P k and using the notation of Prop osition 2.7 yields V = 1 ( uv ) k X I ⊆ [ k ] ( − 1) d I −| I | ( uv − 1) k − d I h ∗ ( P I ; u, v ) . (3.3) Applying Prop osition 2.7 to P 1 , . . . , P k +1 giv es E ( Y ; u, v ) = 1 ( uv ) k +1 X I ⊆ [ k +1] ( − 1) d I −| I | ( uv − 1) d − d I h ∗ ( P I ; u, v ) . By taking ( 3.3 ) into account, we get that E ( Y ; u, v ) equals 1 ( uv ) k +1   ( uv − 1) d − k ( uv ) k V + X I ⊆ [ k +1]: k +1 ∈ I ( − 1) d −| I | (1 − uv ) d − d I h ∗ ( P I ; u, v )   . (3.4) On the other hand, Y is stratified in { x ′ } × Z x ′ for x ′ ∈ b Y , where Z x ′ is the hypersurface in ( C ∗ ) d − k defined by the p olynomial g x ′ ( z ) := f k +1 ( x ′ , z ) ∈ C [ z ± 1 , . . . , z ± d − k ]. Note that each g is a generic 2 p olynomial with resp ect to its Newton p olytop e which equals the pro jection pro j U ( P k +1 ) of P k +1 along U . W e ma y assume k < d as otherwise (in case (2)), Z x ′ and hence 2 These Laurent p olynomials are Newt( g x ′ ( z ))-regular for each x ′ , provided that the co efficients of f k +1 are c hosen appropriately . Sp ecifically , let f k +1 = P m ∈ P k +1 ∩ Z d c m x m . F or f k +1 to b e P k +1 -regular, the co efficient v ector ( c m ) m ∈ P k +1 m ust lie in the complemen t of a finite collection of hypersurfaces in the co efficient space C | P k +1 ∩ Z d | . Similarly , for eac h x ′ , the requirement that g x ′ ( z ) b e regular with resp ect to its own Newton p olytop e excludes another finite set of hypersurfaces. Consequently , there exists a Zariski op en subset of the coefficient space in whic h f k +1 and all g x ′ ( z ) are simultaneously regular. PRESER VING HODGE VECTORS OF LA TTICE POL YTOPES 9 Y is empty . Note that pro j U ( P k +1 ) is a full-dimensional lattice p olytop e in R d − k . By ( 2.4 ) each Z x ′ has the same Ho dge-Deligne p olynomial and as | b Y | = V w e get E ( Y ; u, v ) = V uv h ( − 1) d − k − 1 h ∗ (pro j U ( P k +1 ); u, v ) + ( uv − 1) d − k i . (3.5) Equating ( 3.4 ) and ( 3.5 ) and isolating h ∗ ( P [ k +1] ; u, v ) w e get our desired form ula after some cancellation. F or the case when k > d , the equations ( 3.3 ) and ( 3.4 ) that one derives from Prop osition 2.7 still hold with V = 0. Th us, from these tw o equations one arrives at h ∗ ( P 1 ∗ . . . ∗ P k +1 ; u, v ) = X I ⊊ [ k +1]: k +1 ∈ I ( − 1) k −| I | (1 − uv ) d − d I h ∗ ( P I ; u, v ) . F or the additional statemen ts w e recall that the Ca yley p olytop es P I ha ve dimensions d I + | I |− 1, pro j U ( P k +1 ) has dimension d − k , and the lo cal h ∗ -p olynomial of an n -dimensional lattice p olytop e equals the top degree part of its biv ariate h ∗ -p olynomial whic h has degree n + 1. So, w e need to consider the degree d + k + 1 part in ( 3.1 ). The first term in ( 3.1 ) has exactly d + k + 1 as maximal degree and V ( uv ) k ℓ ∗ (pro j U ( P k +1 ; u, v ) as the top degree part. Therefore, to arrive at ( 3.2 ) we need that for eac h I the parts of h ∗ ( P I ; u, v ) with degrees d + k + 1 − 2 j for j = 0 , . . . , d − d I v anish. The easiest p ossible case is when the maximal degree of h ∗ ( P I ; u, v ), i.e., d I + | I | , is smaller than 2 d I − d + k + 1, i.e., | I | +( d − d I ) ≤ k . Another notable situation, is when the maximal degree of h ∗ ( P I ; u, v ) is exactly 2 d I − d + k + 1, but the part of h ∗ ( P I ; u, v ) of this degree v anishes, i.e. P I is thin. In the situation (2) note that the degree of the Ca yley p olytop e P [ k +1] is at most d while its dimension is d + k ≥ 2 d , so it is trivially thin. The pro jection pro j U ( P k +1 ) is just a p oint, so its Ho dge vector is also 0. □ Definition 3.2. Whenev er ( 3.2 ) in Theorem 3.1 holds, we call P 1 ∗ · · · ∗ P k +1 a gener alize d L awr enc e twist of pro j U ( P k +1 ). Remark 3.3. W e leav e it as an exercise to chec k that if b P is a generalized Lawrence twist of P , and b b P is a generalized Lawrence t wist of b P , then b b P is a generalized Lawrence t wist of P . Let us also remark that Theorem 1.2 is reminiscen t of the follo wing w ell-known pro jection form ula for the mixed v olume, see [ Sch93 , Theorem 5.3.1] whic h one can also pro ve quite directly using the BKK-theorem. Lemma 3.4. Let P 1 , . . . , P d b e lattice p olytop es in R d . If P 1 , . . . , P k (for 1 ≤ k ≤ d ) are con tained in a k -dimensional rational subspace U of R d , then MV( P 1 , . . . , P d ) = MV( P 1 , . . . , P k ) · MV(pro j U ( P k +1 ) , . . . , pro j U ( P d )) . 4. La wrence twists 4.1. The original motiv ation. Let us explain where the motiv ation for the notion of general- ized Lawrence twists comes from. F or this, let us go bac k to the situation of circuits. Recall that a tuple of integers γ = ( γ 1 , . . . , γ d +2 ) with gcd( γ ) = 1 defines uniquely a lattice polytop e in R d with d + 2 vertices, a cir cuit , with circuit relation γ . One wa y to describ e such a circuit is to pro ject the unimo dular simplex ∆ d +1 along the 1-dimensional subspace R ( γ 1 0 + γ 2 e 1 + · · · + γ d +2 e d +1 ). Moreo ver, w e can asso ciate to it a pair of tuples of rational num b ers ( α 1 , . . . , α K ) and ( β 1 , . . . , β K ) for some p ositive in teger K defined by Q i : γ i < 0 ( T − γ i − 1) Q i : γ i > 0 ( T γ i − 1) = Q K i =1 ( T − e 2 π iα i ) Q K i =1 ( T − e 2 π iβ i ) . PRESER VING HODGE VECTORS OF LA TTICE POL YTOPES 10 Recall that from this data Corti, Golyshev and F edoro v [ CG11 , F ed18 ] ga ve a form ula for the r -th co efficient of the lo cal h ∗ -p olynomial of the circuit (for 1 ≤ r ≤ d ): ℓ ∗ r = # { j | # { α i | α i ≤ β j } − j + m − = r } , where m − = |{ i : γ i < 0 }| . No w note the follo wing. If w e extend the tuple ( γ 1 , . . . , γ d +2 ) b y adjoining to it a tuple of integers of the form ( y 1 , − y 1 , y 2 , − y 2 , . . . , y k , − y k ), then this do es not affect the definition of alphas and betas. In the abov e form ula for ℓ ∗ n it affects only m − , therefore, the local h ∗ - p olynomial of the circuit extended this wa y is the lo cal h ∗ -p olynomial of the initial circuit m ultiplied with t k . In other words, they hav e the same Ho dge vector. As it will turn out, this phenomenon can b e explained by viewing this construction as a sp ecial instance of a generalized La wrence t wist. 4.2. La wrence t wists via Gale dualit y. Let us very quic kly recall the basics of Gale duality . Let A ⊆ Z d b e an integral d -dimensional p oint configuration consisting of n > d + 1 p oints. W e assume that A is sp anning , i.e., an y p oint in Z d is an affine integer com bination of elements of A . Let ¯ A denote the corresp onding homogeneous p oint configuration in Z d +1 . There exists an in teger ( n − d − 1) × n -matrix G such that ¯ A · G T = 0 and the columns of G form a v ector configuration that spans Z n − d − 1 . W e note that the sum ov er the columns of G is zero. W e call G a Gale tr ansform of A . F or instance, in the case of the circuit ab ov e, w e hav e n = d + 2 and G = γ . Conv ersely , every suc h in teger ( n − d − 1) × n -matrix G whose columns span Z n − d − 1 and sum up to zero arises in this w ay . Under these assumptions, we get a w ell-defined correspondence b et ween unimo dular equiv alence classes of p oin t configurations A and unimo dular equiv alence classes of vector configurations G . Example 4.1. Let P b e a d -dimensional lattice p olytop e P ⊂ R d . Let us assume that P is sp anning , i.e., P ∩ Z d is a spanning p oint configuration, and | P ∩ Z d | > d + 1. Then w e call the Gale transform of P ∩ Z d the Gale tr ansform of P . It is a challenging op en problem to express the Hodge vector of P using just its Gale transform in a w ay similar to the form ula of Corti, Golyshev and F edoro v for circuits. Definition 4.2. Let A b e a spanning p oin t configuration satisfying | A | > d + 1. W e denote its Gale transform by G A . Let e A ⊂ Z d +2 k b e the spanning p oint configuration with asso ciated Gale transform G A ⊔ S k , where S k is a centrally symmetric configuration consisting of 2 k non-zero in teger v ectors. Then we call the conv ex h ull of e A a L awr enc e twist of the conv ex hull of A . It is a ( d + 2 k )- dimensional spanning lattice p olytop e. In general, a spanning lattice p olytop e e P is a La wrence t w ist of a spanning lattice p olytop e P if one can choose A and e A as describ ed such that e P = conv( e A ) and P = con v( A ). No w, our choice of terminology for a Lawrence twist should hav e become more clear. It has b een motiv ated by the famous class of L awr enc e p olytop es [ BS90 ]: those p olytop es whose Gale transform is given by a cen trally symmetric configuration of v ectors. A La wrence t wist can also b e seen as a v ariant of a L awr enc e lift , where one adds the negative of an existing vector in the Gale transform [ DLRS10 , Section 5.5]. 4.3. La wrence t wists are generalized Lawrence t wists. Let us giv e the interpretation of La wrence twists in the primal space. It suffices to consider the case k = 1, where we extend the Gale transform by one pair of centrally-symmetric vectors. Prop osition 4.3. Let P ⊂ R d b e a spanning lattice p olytop e, a 1 , . . . , a n a spanning p oint configuration consisting of some of the lattice points of P including the vertices of P , and let n > d + 1. Let P 1 := con v (0 , e d +1 ), and P 2 b e the conv ex h ull of ( a 1 , c 1 ) , . . . , ( a n , c n ) ∈ Z d +1 , where c 1 , . . . , c n ∈ Z , with ( c 1 , . . . , c n )  = (0 , . . . , 0). Then P 1 ∗ P 2 is a Lawrence twist of P with k = 1. Moreov er, any Lawrence twist of P with k = 1 is given in this wa y . PRESER VING HODGE VECTORS OF LA TTICE POL YTOPES 11 Pr o of. Denote the columns of G A b y b 1 , . . . , b n ∈ Z n − d − 1 . Choose an in teger non-zero vector v ∈ Z n − d − 1 . By our spanning assumption, there exist c 1 , . . . , c n ∈ Z such that c 1 b 1 + · · · + c n b n = v . No w, it can b e directly chec ked that the p oin t configuration asso ciated to G A ∪ { v , − v } is given b y the columns of e A =          a 11 . . . a n 1 0 0 a 12 . . . a n 2 0 0 . . . . . . . . . 0 0 a 1 d . . . a nd 0 0 c 1 . . . c n 0 1 0 . . . 0 1 1          . Their conv ex hull is precisely the Cayley polytop e P 1 ∗ P 2 . F rom this, the statemen ts follow. □ W e observe from Theorem 3.1 (1) that this is just a sp ecial case of a generalized La wrence t wist. Hence, Remark 3.3 implies directly a result prov en by the first author in his thesis [ Kur24a ]. Corollary 4.4. Lawrence t wists preserv e Ho dge vectors. 4.4. A direct relation b et ween h ∗ -p olynomials of lattice pro jections and Cayley p oly- top es. Theorem 3.1 (1) giv es an explicit and easy description of the biv ariate h ∗ -p olynomial of a La wrence t wist (with k = 1) just using the biv ariate h ∗ -p olynomials of the pro jection and the p olytop e itself. T urning this around, this could also be seen as an explicit formula for the h ∗ -p olynomial of a pro jection. Corollary 4.5. Let P ⊆ R d b e a d -dimensional lattice p olytop e and let I ⊂ R d b e a lattice in terv al of normalized v olume 1. Then we hav e h ∗ ( P ∗ I ; u, v ) = uv · h ∗ (pro j I P ; u, v ) + h ∗ ( P ; u, v ) . In particular, h ∗ ( P ∗ I , t ) = t · h ∗ (pro j I P , t ) + h ∗ ( P , t ) . This implies deg( P ∗ I ) ≥ deg(pro j I P ) + 1 . Prop osition 4.3 implies the follo wing observ ation. Corollary 4.6. Lawrence t wists with S k increase the degree at least by k . Remark 4.7. In the bac helor thesis of Halit Dur [ Dur18 ] the ab ov e form ula w as discussed in the sp ecial case when pro j I P is a face of P and I is the normal v ector of this face. Moreov er, it w as noted that this wa y one can construct a sequence of p olytop es P 0 , P 1 , P 2 , . . . whose h ∗ - p olynomials b ehav e similarly to the Fib onacci p olynomials, albeit with the non-standard starting v alues h ∗ ( P 0 ) = h ∗ ( P 1 ) = 1. 5. Applica tions 5.1. Infinitely man y non-free-joins lattice p olytop es with same Ho dge v ector. This follo ws from the following result, whose pro of we leav e to the reader. A detailed argumen t using Gale duality can b e found in the thesis of the first author [ Kur24a , Lemma 13]. Lemma 5.1. Let P b e a spanning lattice p olytop e of dimension d . F or eac h k ∈ Z ≥ 1 there are infinitely many non-isomorphic Lawrence t wists of P of dimension d + 2 k that are not free joins. Remark 5.2. Note that for the pro of one really has to w ork on the lev el of p oint configurations and not only on the level of p olytop es. F or instance, if P is a lattice p yramid with ap ex v and one takes in Definition 4.2 as A simply P ∩ Z d , then its Lawrence twist is also a lattice pyramid, so a free join. One needs to take as A the p oint configuration P ∩ Z d with v as a double p oint in order for the La wrence t wist to b e not necessarily a lattice pyramid anymore. PRESER VING HODGE VECTORS OF LA TTICE POL YTOPES 12 5.2. Man y more thin p olytop es. Recall that lattice p olytop es with v anishing Ho dge vectors are called thin. So far, tw o wa ys were kno wn to get high-dimensional thin p olytop es: either by taking the free join with a thin p olytop e or by just b eing trivially thin. W e note that trivially thin p olytop es are abundan t as one can simply tak e any Cayl ey p olytop e of at least d + 1 lattice p olytop es in R d to get one. Hence, the follo wing problem was p osed in [ BKN23 ]. Question 5.3. Suppos e d -dimensional P is spanning and thin. Is it true that if P is not a free join, then P is trivially thin? This holds for d ≤ 3 [ BKN23 ]. With the help of Lawrence t wists we can also easily answer 3 it in dimensions d ≥ 5. Corollary 5.4. The answer to Question 5.3 is negativ e for an y d ≥ 5. In fact, there are infinitely-man y non-isomorphic counter-examples in each dimension d ≥ 5. Pr o of. Consider the lattice p yramid P o ver a tw o-dimensional lattice polygon with an interior lattice point. So, dim( P ) = 3, deg( P ) = 2, and P is thin. Cho ose an y k ∈ Z ≥ 1 . Applying Lemma 5.1 yields infinitely man y non-isomorphic Lawrence twists of P in dimension 3 + 2 k that are not free joins. As b y Corollary 4.6 they hav e degree at least 2 + k , they are not trivially thin. This prov es the statement for d ≥ 5 o dd. The result in an ev en dimension d ≥ 6 follo ws in the same wa y starting from a lattice p yramid o ver a three-dimensional lattice p olytop e with an interior lattice p oin t. □ W e remark that we do not know the answ er to Question 5.3 in dimension d = 4. It is affirmativ e in the case of 4-dimensional thin simplices [ Kur24b ]. 5.3. An example of a family of generalized Lawrence t wists. Of course, taking gener- alized La wrence t wists using Theorem 3.1 allows even more freedom in getting parametrized families of high-dimensional thin p olytop es b y just starting with a thin p olytop e. As an applica- tion, let us consider thin simplices. In [ Kur24b ] a complete classification of all 4-dimensional thin simplices w as giv en. There are 6 sp oradic examples and just one infinite family parametrized b y a discrete parameter N ∈ 2 N . Now w e can show that this family arises exactly from the construction describ ed in Theorem 3.1 . T aking the vertices of the family from [ Kur24b ] and p erforming a simple unimo dular transformation one obtains thin 4-dimensional simplices S ( N )  v 0 v 1 v 2 v 3 v 4  =     N / 2 0 0 0 − 1 0 0 1 − 1 1 0 0 0 0 2 0 0 1 1 1     By taking d = 3, k = 1, and tw o lattice p olytop es P 1 = [0 , N / 2] × { 0 } × { 0 } and P 2 = con v ((0 , 1 , 0) , (0 , − 1 , 0) , ( − 1 , 1 , 2)), w e see that S ( N ) ∼ = P 1 ∗ P 2 . Note that P 2 pro jects along U := R × { 0 } × { 0 } isomorphically onto pro j U ( P 2 ) = conv((1 , 0) , ( − 1 , 0) , (1 , 2)). As the latter p olygon is isomorphic to 2∆ 2 , w e see that P 2 and pro j U ( P 2 ) are b oth thin. Hence, we are in the situation (1) of Theorem 3.1 (where one just has to chec k for I = { 2 } ), and S ( N ) is a generalized La wrence t wist of a thin p olygon. 5.4. An alternative pro of of the thinness of B k -p olytop es. In [ Sel25 ] Selyanin defines in the context of Arnold’s monotonicity theorem the following notion: Definition 5.5. A B k -p olytop e is the Cayley polytop e P 0 ∗ · · · ∗ P k for lattice p olytop es in R n − k , where dim( P 1 + · · · + P k ) < k and dim( P 0 ) = n − k . 3 After this question was resolved in the thesis of the first author, an indep enden t answ er was also given b y Sely anin in the preprint [ Sel25 ]. PRESER VING HODGE VECTORS OF LA TTICE POL YTOPES 13 Sely anin sho w ed in [ Sel25 , Corollary 1.7] that B k -p olytop es are thin. He used the theory of ℓ -Newton num b ers to pro ve this and asked in [ Sel25 , Question 1.10] whether there is a simpler w ay to v erify that B k -p olytop es are thin. And indeed there is. Note that w e are precisely in the situation (1) of Theorem 3.1 (where his first polytop e P 0 corresp onds to our last p olytop e P k +1 ). Hence, B k -p olytop es are generalized Lawrence twists. No w, if k ≤ n − k one just has to apply Bernstein’s criterion [ Ber75 ] to P 1 , . . . , P k with dim( P 1 + · · · + P k ) < k to get for the mixed v olume V = 0, so the ℓ ∗ -p olynomial of P 0 ∗ · · · ∗ P k +1 v anishes by ( 3.2 ). Otherwise, if k > n − k , V = 0 by the con ven tion of Theorem 3.1 , and we get again thinness b y ( 3.2 ). Moreo v er, k ≥ n − k actually implies that P 0 ∗ · · · ∗ P k is even trivially thin (e.g., Theorem 3.1 (2)). 5.5. Infinitely man y nearly thin p olytop es. Coming back to free joins the one construction previously kno wn to get high-dimensional lattice p olytop es with the same Ho dge vector was to tak e the free join with a lattice polytop e with Ho dge v ector (1). This motiv ates the follo wing definition: Definition 5.6. A lattice p olytop e with Ho dge vector (1) is called ne arly thin . Note that nearly thin p olytop es are necessarily odd-dimensional b ecause of the palindromicit y of the lo cal h ∗ -p olynomial. The standard example of a nearly thin lattice p olytop e is the interv al [0 , 2]. More generally , for d odd, 2∆ d is nearly thin. What about other examples? No w, using generalized La wrence t wists the following result is immediate. Corollary 5.7. In each o dd dimension d ≥ 3 there are infinitely man y non-isomorphic nearly thin p olytop es that are not free joins. Note also that as the Lawrence twists construction pro duces Cayley p olytop es, any p ossible Ho dge vector can b e giv en b y a lattice p olytop e of lattice width 1. 5.6. Final questions. Summing up our discussion one ma y ask the following: Question 5.8. (1) Apart from free joins with nearly thin p olytop es and generalized Lawrence twists (with mixed v olume V = 1 families) are there other p ossibilities to get an infinite family of high-dimensional lattice p olytop es with the same Ho dge v ector? (2) Apart from b eing trivially thin, a free join with a thin p olytop e or arising from a gener- alized Lawrence twist are there only finitely man y thin p olytop es in given dimension? The reader is in vited to chec k that in dimension 2 ev ery thin p olytop e is a generalized La wrence t wist except for 2∆ 2 . Finally , let us remark that in the thesis [ Kur24a ] of the first author another construction, called total twist , w as presented motiv ated b y [ GV24 ] that conjecturally preserv es the Hodge v ector. How ev er, in con trast to generalized Lawrence t wists that construction is sp or adic , in the sense that it is not alwa ys applicable and do es not allo w a larger degree of freedom. References [Bat93] Victor V. Batyrev. V ariations of the mixed Ho dge structure of affine hypersurfaces in algebraic tori. Duke Mathematic al Journal , 69(2):349–409, F ebruary 1993. Publisher: Duk e Univ ersity Press. [Bat06] Victor V. Bat yrev. 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