NNON-SPANNING LATTICE -POLYTOPES M ´ONICA BLANCO AND FRANCISCO SANTOS
Abstract.
We completely classify non-spanning 3-polytopes, by which wemean lattice 3-polytopes whose lattice points do not affinely span the lattice.We show that, except for six small polytopes (all having between five andeight lattice points), every non-spanning 3-polytope P has the following simpledescription: P ∩ Z consists of either (1) two lattice segments lying in paralleland consecutive lattice planes or (2) a lattice segment together with three orfour extra lattice points placed in a very specific manner.From this description we conclude that all the empty tetrahedra in a non-spanning 3-polytope P have the same volume and they form a triangulationof P , and we compute the h ∗ -vectors of all non-spanning 3-polytopes.We also show that all spanning 3-polytopes contain a unimodular tetrahe-dron, except for two particular 3-polytopes with five lattice points. Contents
1. Introduction and statement of results 12. Sublattice index 73. Non-spanning 3-polytopes of width one 84. Four infinite non-spanning families 95. Proof of Theorem 1.3 116. (Almost all) spanning 3-polytopes have a unimodular tetrahedron 177. The h ∗ -vectors of non-spanning 3-polytopes 17References 191. Introduction and statement of results
A lattice d -polytope is a polytope P ⊂ R d with vertices in Z d and with aff( P ) = R d . We call size of P its number of lattice points and width the minimum length ofthe image f ( P ) when f ranges over all affine non-constant functionals f : R d → R with f ( Z d ) ⊆ Z . That is, the minimum lattice distance between parallel latticehyperplanes that enclose P .In our papers [2, 3, 4] we have enumerated all lattice 3-polytopes of size 11 orless and of width greater than one. This classification makes sense thanks to the Mathematics Subject Classification.
Key words and phrases.
Lattice polytope, spanning, classification, 3-dimensional, lattice width.Partially supported by grants MTM2014-54207-P and MTM2017-83750-P (both authors) andBES-2012-058920 (M. Blanco) of the Spanish Ministry of Economy and Competitiveness, andby the Einstein Foundation Berlin (F. Santos). Both authors were also supported by the Na-tional Science Foundation under Grant No. DMS-1440140 while they were in residence at theMathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. a r X i v : . [ m a t h . C O ] O c t M ´ONICA BLANCO AND FRANCISCO SANTOS following result [2, Theorem 3]: for each n ∈ N there are only finitely many lattice -polytopes of width greater than one and with exactly n lattice points . Here and inthe rest of the paper we consider lattice polytopes modulo unimodular equivalence or lattice isomorphism . That is, we consider P and Q isomorphic (and write P ∼ = Q )if there is an affine map f : R d → R d with f ( Z d ) = Z d and f ( P ) = Q .As a by-product of the classification we noticed that most lattice 3-polytopesare “lattice-spanning”, according to the following definition: Definition.
Let P ⊂ R d be a lattice d -polytope. We call sublattice index of P the index, as a sublattice of Z d , of the affine lattice generated by P ∩ Z d . P iscalled lattice-spanning if it has index . We abbreviate sublattice index and lattice-spanning as index and spanning . In this paper we completely classify non-spanning lattice 3-polytopes. Part ofour motivation comes from the recent results of Hofscheier et al. [6, 7] on h ∗ -vectorsof spanning polytopes (see Theorem 7.1). In particular, in Section 7 we computethe h ∗ -vectors of all non-spanning 3-polytopes and show that they still satisfy theinequalities proved by Hofscheier et al. for spanning polytopes, with the exceptionof empty tetrahedra that satisfy them only partially.In dimensions 1 and 2, every lattice polytope contains a unimodular simplex, i. e.,a lattice basis, and is hence lattice-spanning. In dimension 3 it is easy to constructinfinitely many lattice polytopes of width 1 and of any index q ∈ N , generalizingWhite’s empty tetrahedra ([9]). Indeed, for any positive integers p, q, a, b withgcd( p, q ) = 1 the lattice tetrahedron T p,q ( a, b ) := conv { (0 , , , ( a, , , (0 , , , ( bp, bq, } has index q , width 1, size a + b +2 and volume abq (see a depiction of it in Figure 1).Here and in the rest of the paper we consider the volume of lattice polytopes (0 , , , ,
1) ( bp, bq,
1) ( a, , T p,q ( a, b ) Figure 1.
A polytope T p,q ( a, b ). An empty tetrahedron in it is highlighted.normalized to the lattice, so that it is always an integer and the normalized volumeof a simplex conv( v , . . . , v d ) equals its determinant (cid:12)(cid:12)(cid:12)(cid:12) det (cid:18) v · · · v d · · · (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . Lemma 1.1 (Corollary 3.3) . Every non-spanning -polytope of width one is iso-morphic to some T p,q ( a, b ) . For larger width, the complete enumeration of lattice 3-polytopes of width largerthan one up to size 11 shows that their index is always in { , , , } . The numbersof them for each index and size are as given in Table 1 (copied from Table 6in [4]). This data seems to indicate that apart from a few small exceptions there ON-SPANNING LATTICE 3-POLYTOPES 3 size 5 6 7 8 9 10 11index 1 index 2 index 3 index 5
Table 1.
Lattice 3-polytopes of width > n ) polytopes of index three, about Θ( n ) of index two, and none oflarger indices. Infinite families of lattice 3-polytopes of indices two and three thatmatch these asymptotics are given in the following statement: Lemma 1.2 (See Section 4) . Let S a,b := conv { (0 , , a ) , (0 , , b ) } be a lattice seg-ment, with a, b ∈ Z , a ≤ < b . Then the following are non-spanning lattice -polytopes of width > (see pictures of them in Figure 2): • (cid:101) F ( a, b ) := conv ( S a,b ∪ { ( − , − , , (2 , − , , ( − , , − } ) , of index . • (cid:101) F ( a, b ) := conv ( S a,b ∪ { ( − , − , , (1 , − , , ( − , , } ) , with ( a, b ) (cid:54) =(0 , , of index . • (cid:101) F ( a, b, k ) := conv ( S a,b ∪ { ( − , − , , (1 , − , , ( − , , , (1 , , k − } ) ,for any k ∈ { , . . . , b } and ( a, b, k ) (cid:54) = (0 , , , of index . • (cid:101) F ( a, b ) := conv ( S a,b ∪ { ( − , − , , (1 , − , , ( − , , , (3 , − , − } ) , ofindex 2. Observe that in all cases of Lemma 1.2 the lattice points in the polytope arethose in the segment S a,b plus three or four additional points. We call S a,b (or,sometimes, the line containing it) the spike of (cid:101) F i ( a, b, ), because it makes thesepolytopes be very closely related to (some of) the spiked polytopes introduced in [4](see the proof of Theorem 5.2 for some properties of them).For all polytopes in the same family (cid:101) F i the projection of the lattice points alongthe direction of the spike is the same lattice point configuration, the F i depictedin the bottom of Figure 2. The spike projects to the origin, and the other three orfour points in each (cid:101) F i ( a, b, ) project bijectively to three or four points in F i .The number of polytopes in the families of Lemma 1.2, taking redundancies intoaccount, is computed in Corollary 4.2. Table 2 shows the result for sizes 5 to 11. size 5 6 7 8 9 10 11index 2 index 3 Table 2.
The number of polytopes of size ≤
11 in the families of Lemma 1.2Comparing those numbers to Table 1 we see that the only discrepancies are sixmissing polytopes in sizes five to eight. Our main result in this paper is that indeed,Lemma 1.2 together with six exceptions gives the full list of non-spanning lattice3-polytopes of width larger than one.
M ´ONICA BLANCO AND FRANCISCO SANTOS (cid:101) F ( a, b ) (cid:101) F ( a, b ) (cid:101) F ( a, b, k ) (cid:101) F ( a, b ) (0 , , b ) (0 , , b )(0 , , a ) (0 , , a )( − , , −
1) (1 , − , − , − , , , − ,
1) ( − , − ,
1) ( − , , F F (0 , , b )(0 , , a )(1 , − , − , − ,
1) ( − , , , − , − F , , b )(0 , , a )(1 , − , − , − ,
1) ( − , , , , k − F Figure 2.
The non-spanning 3-polytopes of Lemma 1.2. Blackdots in the top figures are the lattice points in (cid:101) F i ( a, b, ). Black seg-ments represent segments of at least three collinear lattice points,and the gray triangles or tetrahedron are the convex hulls of thelattice points not in the spike. The lattice point configurations F i arise as the projections of the configurations (cid:101) F i ( a, b, ) ∩ Z inthe direction of the spike. Black dots are points of F i (projectionof lattice points of (cid:101) F i ( a, b, )), and white dots are lattice points inconv( F i ) \ F i . Theorem 1.3 (Classification of non-spanning 3-polytopes) . Every non-spanninglattice -polytope either has width one (hence it is isomorphic to a T p,q ( a, b ) byLemma 1.1) or is isomorphic to one in the infinite families of Lemma 1.2, or isisomorphic to one of the following six polytopes, depicted in Figure 3: • Two tetrahedra of sizes and , with indices and respectively: E (5 , := conv { (0 , − , , (1 , , − , (1 , , , ( − , , − } .E (6 , := conv { (1 , , , ( − , − , , (1 , , , ( − , , − } . • A pyramid and two double pyramids over the following triangle of size : B := conv ( { ( − , − , , (2 , , , (1 , , } ) . Namely, the following polytopes of sizes 7 and 8, all of index : E (7 , := conv( B ∪ { (0 , − , } ) E , := conv( B ∪ { (0 , − , , (0 , , − } ) ,E , := conv( B ∪ { (0 , − , , ( − , − , − } ) . • A tetrahedron of size and index : E , := conv { (0 , − , − , (2 , , , (1 , , − , ( − , , } . ON-SPANNING LATTICE 3-POLYTOPES 5 (1 , , − , − ,
1) ( − , , − , ,
0) (1 , ,
1) (1 , , − , − ,
0) ( − , , − , ,
0) (0 , , , , E (5 , E (6 , (1 , , , , − , − ,
0) (0 , − , , , , ,
0) (1 , ,
0) (1 , , , , − , − ,
0) (0 , − , , , −
2) (1 , , , , , ,
0) (1 , , , , − , − , −
2) (0 , − , − , − ,
0) (1 , , , , , , E (7 , E , E , ( − , ,
2) (1 , , − , , , − , −
1) (0 , , , ,
0) (1 , , , , E , Figure 3.
The six non-spanning 3-polytopes of Theorem 1.3.
Corollary 1.4 (Corollary 4.2) . For any n ≥ all non-spanning -polytopes ofsize n and width larger than one belong to the families of Lemma 1.2. There are (cid:98) ( n − n + 1) / (cid:99) of index two, (cid:100) n − (cid:101) of index three, and none of larger index. (cid:3) As a consequence of our classification for non-spanning 3-polytopes we have thefollowing statement. Recall that an empty tetrahedron is a lattice tetrahedronwhose only lattice points are its vertices.
Corollary 1.5.
Let P be a non-spanning lattice -polytope. Then the collection ofall empty tetrahedra in P is a triangulation, and all such tetrahedra have volumeequal to the index of P . Observe that the triangulation in the statement is necessarily the unique trian-gulation of P with vertex set equal to P ∩ Z . M ´ONICA BLANCO AND FRANCISCO SANTOS
Proof. • For T p,q ( a, b ), since all lattice points lie on two lines, every emptytetrahedron has as vertices two consecutive lattice points on each of thelines. These tetrahedra have volume equal to the index (in this case q ) and,together, form a triangulation: the join of the two paths along the lines. • Every polytope in the families (cid:101) F i has the property that the triangle formedby any three lattice points outside the spike meets the spike at a latticepoint. Thus, every empty tetrahedron has two vertices along the spikeand two outside the spike. The collection of all of such tetrahedra formsa triangulation: the join of the path along the spike and a path (for thepolytopes in the family (cid:101) F ) or a cycle (for the rest) outside the spike. Also,these tetrahedra have the volume of a triangle with vertices in the projection F i that has the origin as a vertex. Any such triangle in F i has the samevolume as the index of the polytopes in the family (cid:101) F i . • For the six exceptions of Theorem 1.3, the statement can be checked withgeometric arguments or with computer help. Details are left to the reader. (cid:3)
Corollary 1.5 is not true in dimension 4, as the following example shows:
Example 1.6.
Let P = conv ( 1 , , , , ( 0 , , , , ( 0 , , , , ( − , − , − , , ( 1 , , , This is a lattice 4-simplex with six lattice points: its five vertices plus the origin,which lies in the relative interior of the facet given by the first four vertices. Thus,the volumes of the empty tetrahedra in P are the absolute values of the 4 × P has index two. We thank Gabriele Balletti for providing (a variation of) thiscounterexample.In the same vein, one can ask whether all spanning 3-polytopes have a unimod-ular tetrahedron. The answer is that only two do not. Theorem 1.7 (See Section 6) . The only lattice-spanning -polytopes not containinga unimodular tetrahedron are the following two tetrahedra of size five: • E , := conv { (1 , , , (0 , , , (2 , , , ( − , − , − } with four empty tetra-hedra of volumes , , and . • E , := conv { (1 , , , (0 , , , (3 , , , ( − , − , − } with four empty tetra-hedra of volumes , , and . The structure of the paper is as follows. After a brief introduction and remarksabout the sublattice index in Section 2, Section 3 proves the (easy) classification ofnon-spanning 3-polytopes of width one (Lemma 1.1), and Section 4 is devoted tothe study of the infinite families of non-spanning 3-polytopes of Lemma 1.2.Section 5 proves our main result, Theorem 1.3. This is the most complicated partof the paper, relying substantially in our results from [4]. A sketch of the proof is asfollows: For small polytopes, Theorem 1.3 follows from comparing Tables 1 and 3(together with the easy observation that the six polytopes described in Theorem 1.3
ON-SPANNING LATTICE 3-POLYTOPES 7 are not isomorphic to one another or to the ones in Lemma 1.2). For polytopes oflarger size we use induction on the size (taking the enumeration of size ≤
11 as thebase case) and we prove the following: • In Section 5.1 we look at polytopes that cannot be obtained merging twosmaller ones, in the sense of [4]. These polytopes admit quite explicitdescriptions that allow us to prove that the only non-spanning ones (forsizes ≥
8) are those of the form (cid:101) F i (0 , b ) for i ∈ { , } (Theorem 5.2). Sincemerging can only decrease the index, this immediately implies that all non-spanning 3-polytopes of width > ≥ • In Sections 5.2 and 5.3 we look at merged polytopes of indices three andtwo, respectively, and prove that, with the five exceptions mentioned inTheorem 1.3, they all belong to the families of Lemma 1.2.Section 6 proves Theorem 1.7.2.
Sublattice index
By a lattice point configuration we mean a finite subset A ⊂ Z d that affinelyspans R d . We denote by (cid:104) A (cid:105) Z the affine lattice generated by A over the integers: (cid:104) A (cid:105) Z := (cid:40)(cid:88) i λ i a i | a i ∈ A, λ i ∈ Z , (cid:88) i λ i = 1 (cid:41) Since A is affinely spanning, (cid:104) A (cid:105) Z has finite index as a sublattice of Z d . Definition 2.1.
The sublattice index of A is the index of (cid:104) A (cid:105) Z in Z d . We say that A is lattice-spanning if its sublattice index equals . That is, if (cid:104) A (cid:105) Z = Z d . Remark 2.2. If A = P ∩ Z d for P a lattice d -polytope, we call sublattice index of P the sublattice index of A , and say that P is lattice-spanning if A is. Lemma 2.3.
Let A ⊂ Z d be a lattice point configuration.(1) The sublattice index of A divides the sublattice index of every subconfigura-tion B of A .(2) Let π : Z d (cid:16) Z s , for s < d , be a lattice projection. Then the sublatticeindex of π ( A ) divides the sublattice index of A .Proof. In part (1), the injective homomorphism (cid:104) B (cid:105) Z → (cid:104) A (cid:105) Z induces a surjectivehomomorphism Z d / (cid:104) B (cid:105) Z → Z d / (cid:104) A (cid:105) Z . In part (2), the lattice projection π inducesa surjective homomorphism Z d / (cid:104) A (cid:105) Z → Z s / (cid:104) π ( A ) (cid:105) Z . (cid:3) It is very easy to relate the index of a lattice polytope or point configuration tothe volumes of (empty) simplices in it.
Lemma 2.4.
Let A be a lattice point configuration of dimension d . Then, thesublattice index of A equals the gcd of the volumes of all the lattice d -simplices withvertices in A . Remark 2.5.
Observe that the gcd of volumes of all simplices in A equals the gcd of volumes of simplices empty in A , by which we mean simplices T such that T ∩ A = vert( T ) . Indeed, if T is a non-empty simplex, then T can be triangu-lated into empty simplices T , . . . , T k . Since vol( T ) = (cid:80) vol( T i ) , we have that gcd(vol( T ) , . . . , vol( T n )) = gcd(vol( T ) , . . . , vol( T n ) , vol( T )) . M ´ONICA BLANCO AND FRANCISCO SANTOS
Proof.
Without loss of generality assume that the origin is in A . Then, the sublat-tice index of A equals the gcd of all maximal minors of the d × | A | matrix havingthe points of A as columns. These minors are the (normalized) volumes of lattice d -simplices with vertices in A . (cid:3) Non-spanning -polytopes of width one Lemma 3.1.
Let P be a lattice -polytope of width one. Then P either contains aunimodular tetrahedron or it equals the convex hull of two lattice segments lying inconsecutive parallel lattice planes.Proof. Since P has width one, its lattice points are distributed in two consecutiveparallel lattice planes. If P has three non collinear lattice points in one of theseplanes we can assume without loss of generality that they form a unimodular tri-angle. Then, these three points together with any point in the other plane (thereexists at least one) form a unimodular tetrahedron.If P does not have three non-collinear points in one of the two planes then allthe points in each of the planes are contained in a lattice segment. (cid:3) Proposition 3.2.
The convex hull of two lattice segments lying in consecutiveparallel lattice planes is equivalent to T p,q ( a, b ) := conv { (0 , , , ( a, , , (0 , , , ( bp, bq, } for some ≤ p < q with gcd( p, q ) = 1 , and a, b ≥ .The sublattice index of T p,q ( a, b ) is q and its size is a + b + 2 . See Figure 1 for an illustration of T p,q ( a, b ). Proof.
Let P be the convex hull of two lattice segments lying in consecutive parallellattice planes. Without loss of generality we assume that the segments are containedin the planes { z = 0 } and { z = 1 } , respectively. Let a and b be the lattice lengthof the two segments, so that the size of P is indeed a + b + 2.By a unimodular transformation there is no loss of generality in assuming thefirst segment to be conv { (0 , , , ( a, , } and the second one to contain (0 , , bp, bq,
1) for some coprime p, q ∈ Z and with q (cid:54) = 0 in order for P to be full-dimensional. We can assume q > q < x, y, z ) (cid:55)→ ( x ± y, y, z ) fixes (0 , , a, ,
0) and(0 , ,
1) and sends ( p, q, (cid:55)→ ( p ± q, q, ≤ p < q .Since all lattice points of P lie in the lattice y ≡ q ) its index is a multipleof q . Since P contains the tetrahedron conv { (0 , , , , , (0 , , , ( p, q, } , ofdeterminant q , its index is exactly q . (cid:3) Corollary 3.3.
Let P be a lattice -polytope of width one. Then • P is lattice-spanning if, and only if, it contains a unimodular tetrahedron. • P has index q > if, and only if, P ∼ = T p,q ( a, b ) for some ≤ p < q with gcd( p, q ) = 1 and a, b ≥ . (cid:3) ON-SPANNING LATTICE 3-POLYTOPES 9 Four infinite non-spanning families
In this section we study the infinite families of non-spanning 3-polytopes intro-duced in Lemma 1.2, whose definition we now recall: (cid:101) F ( a, b ) := conv (cid:0) S a,b ∪ { ( − , − , , (2 , − , , ( − , , − } (cid:1) , (cid:101) F ( a, b ) := conv (cid:0) S a,b ∪ { ( − , − , , (1 , − , , ( − , , } (cid:1) , (cid:101) F ( a, b, k ) := conv (cid:0) S a,b ∪ { ( − , − , , (1 , − , , ( − , , , (1 , , k − } (cid:1) , (cid:101) F ( a, b ) := conv (cid:0) S a,b ∪ { ( − , − , , (1 , − , , ( − , , , (3 , − , − } (cid:1) . In all cases, S a,b := conv { (0 , , a ) , (0 , , b ) } with a, b ∈ Z and a ≤ < b , and in thethird family, k ∈ { , . . . , b } .The statement of Lemma 1.2 is that all these polytopes are non-spanning andhave width >
1, with the exceptions of (cid:101) F (0 ,
1) and (cid:101) F (0 , , S a,b the spike , and that apart of the b − a + 1 lattice points inthe spike these polytopes only have three (in the families (cid:101) F and (cid:101) F ) or four (inthe families (cid:101) F or (cid:101) F ) other lattice points. Proof of Lemma 1.2.
The lattice points in (cid:101) F ( a, b ) generate the sublattice { ( x, y, z ) ∈ Z : x − y ≡ } , of index 3. Those in (cid:101) F ( a, b ), (cid:101) F ( a, b, k ) and (cid:101) F ( a, b ) gen-erate the sublattice { ( x, y, z ) ∈ Z : x + y ≡ } , of index 2.For the width we consider two cases: Functionals constant on S a,b (that is,not depending on the third coordinate) project to functionals in Z on one of theconfigurations F i of Figure 2, of which F has width 3 and the others width 2.Functionals that are non-constant along S a,b produce width at least b − a , so theonly possibility for width one would be a = 0 and b = 1. It is clear that (cid:101) F (0 , (cid:101) F (0 , ,
1) have width one with respect to the functional z but it is easy tocheck (and left to the reader) that the rest have width at least two even when( a, b ) = (0 , (cid:3) The list in Lemma 1.2 contains some redundancy, but not much.
Proposition 4.1.
The only isomorphic polytopes among the list of Lemma 1.2 are:(1) For i ∈ { , , } we have that (cid:101) F i ( a, b ) is isomorphic to (cid:101) F i ( − b, − a ) .(2) (cid:101) F ( a, b, k ) is isomorphic to (cid:101) F ( k − b, k − a, k ) .(3) In size six: (cid:101) F (0 , ∼ = (cid:101) F (0 , and (cid:101) F ( − , ∼ = (cid:101) F (0 , , . (4) In size seven: (cid:101) F (0 , , ∼ = (cid:101) F ( − , . Proof.
The following maps show the isomorphisms in parts (1) and (2) amongpolytopes within each family:( x, y, z ) (cid:55)→ ( y, x, − z ) ⇒ (cid:101) F ( a, b ) ∼ = (cid:101) F ( − b, − a ) . ( x, y, z ) (cid:55)→ ( x, y, − x − y − z ) ⇒ (cid:101) F ( a, b ) ∼ = (cid:101) F ( − b, − a ) . ( x, y, z ) (cid:55)→ ( y, − x, k + y ( k − − z ) ⇒ (cid:101) F ( a, b, k ) ∼ = (cid:101) F ( k − b, k − a, k ) . ( x, y, z ) (cid:55)→ ( x, y, − x − y − z ) ⇒ (cid:101) F ( a, b ) ∼ = (cid:101) F ( − b, − a ) . Whenever b − a ≥ • The spike contains b − a + 1 ≥ b − a is an invariant (modulounimodular equivalence). • Polytopes in different families (cid:101) F i cannot be isomorphic: the polytopes inthe family (cid:101) F are the only ones of index three; those in (cid:101) F are the only onesof index two with only three points outside the spike; and those in (cid:101) F arethe only ones with three collinear lattice points outside the spike. • For i = 1 , , {− a, b } is an invariant since the plane spanned bylattice points not in the spike intersects the spike at the point (0 , , − a and b of the end-points of the spike. • For i = 3 the pair {− a, b − k } is an invariant since the midpoints of thepairs of opposite lattice points around the spike have as midpoints (0 , , , , k ), which are on the spike and at distances − a and b − k from theend-points.It only remains to see that, when b − a ≤
2, the only isomorphisms thatappear are those of parts (3) and (4) of the statement. By parts (1) and (2)we can assume that − a ≤ b in all cases and that − a ≤ b − k in the family (cid:101) F . That is, ( a, b ) ∈ { (0 , , (0 , , ( − , } in (cid:101) F , (cid:101) F and (cid:101) F , and ( a, b, k ) ∈{ (0 , , , (0 , , , (0 , , , (0 , , , ( − , , } for (cid:101) F (remember that (cid:101) F (0 ,
1) and (cid:101) F (0 , ,
1) are excluded). Let us separate the cases by index and size: • In the family (cid:101) F , the only one of index 3, the three possibilities are distin-guished by the fact that (cid:101) F (0 ,
1) is has size five, (cid:101) F (0 ,
2) is a tetrahedronof size six, and (cid:101) F ( − ,
1) is a triangular bipyramid of size six. • For index 2 and size six there are only (cid:101) F ( − , (cid:101) F (0 , , (cid:101) F (0 ,
2) and (cid:101) F (0 , x, y, z ) (cid:55)→ ( x + z, y + z, − x − y − z ). The last two are tetrahedra, isomorphic via( x, y, z ) (cid:55)→ ( − x + y − z + 1 , z − , − y ). • For index 2 and size seven there are four possibilities in the family (cid:101) F andtwo in the family (cid:101) F . All of them happen to have three collinear triplesof lattice points. Looking at the distribution of the seven lattice points incollinear triples is enough to distinguish among five of the six possibilities,as the following diagram shows. (Each diagram shows what lattice pointsform collinear triples but also the relative order of points along each triple): (cid:101) F (0 , , (cid:101) F (0 , , (cid:101) F (0 , , (cid:101) F ( − , , (cid:101) F (0 , x, y, z ) (cid:55)→ ( − x + y − z +1 , z − , x ) maps (cid:101) F (0 , ,
1) to the remaining polytope of size seven, (cid:101) F ( − , (cid:3) ON-SPANNING LATTICE 3-POLYTOPES 11
Corollary 4.2.
The number of isomorphism classes of polytopes in the families (cid:101) F i are as given in Table 3. size 5 6 7 8 9 10 11 general n family (cid:101) F (cid:100) n − (cid:101) total index 3 1 2 2 3 3 4 4 (cid:100) n − (cid:101) family (cid:101) F ∗ (cid:100) n − (cid:101) family (cid:101) F ∗ (cid:106)(cid:0) n − (cid:1) (cid:107) family (cid:101) F (cid:4) n − (cid:5) total index 2 0 2 ∗∗ ∗∗
11 15 19 24 (cid:106) ( n − n +1)4 (cid:107) Table 3.
The number of non-spanning 3-polytopes in the infinitefamilies of Lemma 1.2. The two entries marked with ∗ do notcoincide with the formula for general n because in each of themone of the configurations counted by the formula has width one.The entries marked ∗∗ are less than the sum of the three abovethem (and, in particular, do not coincide with the general formula)because of the isomorphisms in parts (3) and (4) of Proposition 4.1. Proof.
For size up to seven, the counting is implicit in the proof of parts (3) and(4) of Proposition 4.1. Thus, in the rest of the proof we assume size n ≥ (cid:101) F and (cid:101) F we have b − a + 1 = n − n − (cid:100) ( n − / (cid:101) . For (cid:101) F we have the same count except thatnow b − a + 1 = n −
4, so we get (cid:100) ( n − / (cid:101) = (cid:98) ( n − / (cid:99) .For (cid:101) F we have b − a + 1 = n − (cid:0) n − (cid:1) possibilities. To mod out symmetric choices wedivide that number by two but then have to add one half of the self-symmetricchoices, of which there are (cid:100) ( n − / (cid:101) = (cid:98) ( n − / (cid:99) . The count, thus, is12 (cid:18)(cid:18) n − (cid:19) + (cid:22) n − (cid:23)(cid:19) = 12 (cid:22) n − n − (cid:23) = (cid:22) ( n − (cid:23) . (cid:3) Proof of Theorem 1.3
Merged and non-merged -polytopes. For a lattice d -polytope P ⊂ R d of size n and a vertex v of P we denote P v := conv( P ∩ Z d \ { v } ) , which has size n − d or d −
1. We abbreviate ( P u ) v as P uv . Thefollowing definition is taken from [4]. Definition 5.1.
Let P be a lattice -polytope of width > and size n . We say that P is merged if there exist at least two vertices u, v ∈ vert( P ) such that P u and P v have width larger than one and such that P uv is -dimensional. Loosely speaking, we call a polytope of size n merged if it can be obtained merging two subpolytopes Q ∼ = P u and Q ∼ = P v of size n − > P uv . This merging operationis the basis of the enumeration algorithm in [4] and to make it work, a completecharacterization of the polytopes that are not merged was undertaken. Combiningseveral results from [4] we can prove that: Theorem 5.2.
Let P be a lattice -polytope of size n ≥ and suppose that P isnot merged. Then, one of the following happens:(1) P contains a unimodular tetrahedron, and in particular it is spanning.(2) P has sublattice index and is isomorphic to (cid:101) F (0 , n −
4) = conv { (1 , − , , ( − , , , ( − , − , , (0 , , n − } . (3) P has sublattice index and is isomorphic to (cid:101) F (0 , n −
4) = conv { (2 , − , , ( − , , − , ( − , − , , (0 , , n − } . Proof.
This follows from Theorems 2.9, 2.12, 3.4 and 3.5 in [4], but let us brieflyexplain how. • [4, Theorem 2.12] shows that every non-merged 3-polytope of size ≥ v such that P v has width larger than one. • [4, Theorem 2.9] says that every quasi-minimal 3-polytope is either spiked or boxed . • By [4, Proposition 4.6], boxed polytopes of size ≥ n − ≥ P is spiked. Theorems 3.4and 3.5 in [4] contain a very explicit description of all spiked 3-polytopes, which inparticular implies that every spiked 3-polytope P of size n projects to one of theten 2-dimensional configurations A (cid:48) , . . . , A (cid:48) of Figure 4 (notation taken from [4]),where the number next to each point in A (cid:48) i indicates the number of lattice pointsin P projecting to it.An easy inspection shows that all A (cid:48) i except A (cid:48) and A (cid:48) have a unimodulartriangle T with the property that (at least) one vertex of T has at least two latticepoints of P in its fiber. Then T is the projection of a unimodular tetrahedron in P . It thus remains only to show that projections to A (cid:48) and A (cid:48) correspond exactlyto cases (2) and (3) in the statement. For this we use the coordinates shownin Figure 5. Observe that A (cid:48) and A (cid:48) are exactly the F and F of Lemma 1.2,respectively. • Suppose that P projects to F . By the assumption on the number oflattice points of P projecting to each point in F , P has exactly threelattice points outside the spike { (0 , , t ) : t ∈ R } , and they have coordinates(1 , − , h ), ( − , , h ) and ( − , − , h ). In order not to have lattice points ON-SPANNING LATTICE 3-POLYTOPES 13 n − A A A A A A A A A A
911 1 1 11 11 1111 11 1 1 1111 1 1 11 n − n − n − n − n − n −
51 11 1 1 1 11 11 1 n − n − n − k − k Figure 4.
Possible projections of a non-merged 3-polytope P ofsize n ≥ j is theprojection of exactly j lattice points in P . White dots are latticepoints in the projection of P which are not the projection of anylattice point in P . (0 , A = F A = F − , − , −
1) (1 , −
1) ( − , − − ,
2) (2 , − , Figure 5.
Possible projections of non-merged non-spanning 3-polytopes of size ≥ P projecting to the white dots in the figure for F , h and h must be ofthe same parity and h of the opposite parity. There is no loss of generalityin assuming h and h even, in which case the unimodular transformation( x, y, z ) → (cid:18) x, y, h − h − x + h − h − y + z − h + h (cid:19) sends P to (cid:101) F ( a, b ) for some a ≤ < b and b − a = n − a and b are non-zero, then (0 , , a ) and (0 , , b ) are vertices of P and P is merged from P (0 , ,a ) and P (0 , ,b ) (of width > n ≥ a or b is zero, and by the isomorphism inpart (1) in Proposition 4.1 there is no loss of generality in assuming it tobe a . • For F the arguments are essentially the same, and left to the reader. (cid:3) Corollary 5.3.
With the only exception of the tetrahedron E (5 , of Theorem 1.3(of size and index ) every lattice -polytope of width > has index at most .Proof. Let P be a lattice 3-polytope of width >
1. If P has size at most 7 (orat most 11, for that matter), the statement follows from the enumerations in [4], as seen in Table 1. For size n ≥ n = 7 as the basecase. Either P is non-merged, in which case Theorem 5.2 gives the statement, or P is merged, in which case it has a vertex u such that P u still has width >
1. Byinductive hypothesis P u has index at most three, and by Lemma 2.3(1), the indexof P divides that of P u . (cid:3) Lattice -polytopes of index .Theorem 5.4. Let P be a lattice -polytope of width > and of index three. Then P is equivalent to either the tetrahedron E (6 , of Theorem 1.3 (of size six) or to apolytope in the family (cid:101) F of Lemma 1.2.Proof. Let n be the size of P . The statement is true for n ≤
11 by the enumerationsin [4], as seen comparing Tables 1 and 3. For size n >
11 we use induction, taking n = 11 as the base case.Let n >
11. If P is not merged, then the result holds by Theorem 5.2. Sofor the rest of the proof we assume that P is merged. Then there exist vertices u, v ∈ vert( P ) such that P u and P v (of size n −
1) have width >
1, and such that P uv (of size n −
2) is full-dimensional. By Lemma 2.3(1) the sublattice indices of P u and P v are multiples of 3 and by Corollary 5.3 they equal 3. Thus, by inductionhypothesis, both P u and P v are in the family (cid:101) F .Since both P u and P v have a spike with n − > P u and P v have their spikesalong the same line. If u ∈ P v lies along the spike of P v (resp. if v lies along thespike of P u ), then P is obtained from P u (resp. P v ) by extending its spike by onepoint, which implies P is also in the family (cid:101) F .Hence, we only need to study the case where both u and v are outside the spikeof P v and P u , respectively.In this case, P uv consists of a spike of length n − w and w be these two lattice points. We use the followingobservation about the polytopes in the family (cid:101) F : the barycenter of the three latticepoints outside the spike is a lattice point in the spike . That is, we have that both ( u + w + w ) and ( v + w + w ) are lattice points in the spike. In particular, u − v is parallel to the spike and of length a multiple of three, which is a contradictionto the fact that u and v are the only lattice points of P not in P uv . See Figure 6. (cid:3) spike w w uv Figure 6.
The setting at the end of the proof of Theorem 5.4
ON-SPANNING LATTICE 3-POLYTOPES 15
Lattice -polytopes of index . The case of index 2 uses the same ideasas in index 3 but is a bit more complicated, since there are more cases to study.As a preparation for the proof, the following two lemmas collect properties of thepolytopes (cid:101) F i ( a, b, ) of index two. Lemma 5.5. (1) Let Q = (cid:101) F ( a, b, k ) and w be any vertex of Q not in the spike.Then Q w is isomorphic to either (cid:101) F ( a, b ) or (cid:101) F ( k − b, k − a ) .(2) Let Q = (cid:101) F ( a, b ) and w ∈ { ( − , − , , (3 , − , − } . Then Q w ∼ = (cid:101) F ( a, b ) .Proof. For part (2) simply observe that ( x, y, z ) (cid:55)→ ( − x − y, y, x + y + z ) is anautomorphism of F ( a, b ) that swaps ( − , − ,
1) and (3 , − , − Q ( − , − , ∼ = Q (3 , − , − = (cid:101) F ( a, b ).In part (1), the automorphism of (cid:101) F ( a, b, k )( x, y, z ) (cid:55)→ ( − y, − x, − ( k − x − ( k − y + z )swaps ( − , − ,
1) and (1 , , k − Q ( − , − , ∼ = Q (1 , , k − = (cid:101) F ( a, b ) . But we have also the isomorphism in part (2) of Proposition 4.1, which sends Q to (cid:101) F ( k − b, k − a, k ) and maps { ( − , − , , (1 , , k − } to { (1 , − , , ( − , , } .Thus Q (1 , − , ∼ = Q ( − , , ∼ = (cid:101) F ( k − b, k − a ) . (cid:3) Lemma 5.6.
Let Q be of size ≥ and isomorphic to a polytope in one of thefamilies (cid:101) F or (cid:101) F of Lemma 1.2. Let w be a vertex of Q such that Q w = (cid:101) F ( a, b ) .Then, w is one of (3 , − , − , ( − , , − , or (1 , , j − for some j ∈ { a, . . . , b } . (3 , − , − e F ( a, b ) (1 , , j − − , , − Figure 7.
Illustration of Lemma 5.6.
Proof.
Since Q has four lattice points outside the spike and Q w has three, theirspikes have the same size. Also, since the size of Q is ≥
8, the spike has at leastfour lattice points. In particular Q and Q w have the same spike. The four pointsof Q outside the spike are (1 , − , − , , − , − ,
1) and w . The fact that(1 , − , − , ,
0) have their midpoint on the spike implies that: • If Q is in the family (cid:101) F then the midpoint of w and ( − , − ,
1) must also bea lattice point in the spike. That is, w = (1 , , j −
1) with j ∈ { a, . . . , b } . • If Q is in the family (cid:101) F then w and ( − , − ,
1) have as midpoint one of(1 , − ,
0) and ( − , , w ∈ { (3 , − , − , ( − , , − } . (cid:3) Theorem 5.7.
Let P be a lattice -polytope of width > and of index two. Then P is equivalent to either one of the four exceptions of index two in Theorem 1.3 orto one of the polytopes in the families (cid:101) F , (cid:101) F or (cid:101) F of Lemma 1.2.Proof. With the same arguments as in the proof of Theorem 5.4 we can assumethat P has size >
11, is merged from P u and P v , that P u and P v are in the families (cid:101) F , (cid:101) F or (cid:101) F and have the same spike, and that u and v lie outside the spike. Inparticular, P u and P v have the same number of lattice points outside the spike.This number can be three or four, and we look at the two possibilities separately: • If P u and P v have three lattice points not on the spike then they are bothin the family (cid:101) F . The lattice points in P uv are those in the spike (which iscommon to P u and P v ) together with two extra lattice points w and w .Let φ u : (cid:101) F ( a, b ) → P u and φ v : (cid:101) F ( a (cid:48) , b (cid:48) ) → P v be isomorphisms. Then φ u ( − , − , (cid:54) = v and φ v ( − , − , (cid:54) = u , since this would imply P uv to be2-dimensional. Thus, φ u ( − , − , , φ v ( − , − , ∈ { w , w } . This implies { φ u ( − , , , φ u (1 , − , } = { v, w i } , and { φ v ( − , , , φ v (1 , − , } = { u, w j } for some i, j ∈ { , } . In particular the segments vw i and uw j have as mid-points the points φ u (0 , ,
0) and φ v (0 , , i = j (top part of the figure) then uv is parallel to thespike and of even length, which gives a contradiction: the midpoint of u and v would be a third lattice point of P outside P uv . If i (cid:54) = j then P u ∪ P v are the lattice points in a polytope isomorphic to (cid:101) F ( a, b, k ) (bottom rowin the figure), so the statement holds. P uv ⊂ P u P uv ⊂ P v P u ∪ P v OUTCOME vw w uw w u, vw w contradiction vw w uw w w w u v (cid:101) F ( a, b, k ) Figure 8.
The two cases in the first part of the proof of Theorem 5.7. • If P u and P v have four lattice points not on the spike then they belong tothe families (cid:101) F or (cid:101) F .Our first claim is that there is no loss of generality in assuming that P uv is in the family (cid:101) F . Indeed, if P u is in the family (cid:101) F then this isautomatically true by Lemma 5.5(1). If P u = (cid:101) F ( a, b ) then at least one ofthe vertices ( − , − ,
1) and (3 , − , −
1) of P u is also a vertex of P (e.g.,because these two vertices are the unique minimum and maximum on P u ON-SPANNING LATTICE 3-POLYTOPES 17 of the linear functional f ( x, y, z ) = x + y ). Call that vertex v (cid:48) , and consider P as merged from P u and P v (cid:48) instead of the original P u and P v . Then P uv (cid:48) is in the family (cid:101) F by Lemma 5.5(2).Thus, we can assume for the rest of the proof that P uv = (cid:101) F ( a, b ).Lemma 5.6 (applied first with Q = P u and w = v and then with Q = P v and w = u ) implies that both u and v belong to { (3 , − , − , ( − , , − } ∪ { (1 , , j −
1) : j = a, . . . , b } which is contained in { ( x, y, z ) : x + y = 2 , x ≡ y ≡ z ≡ } . Butthen the segment uv , lying in the hyperplane { x + y = 2 } , contains at leastan extra lattice point, namely ( u + v ). See Figure 7. This implies P hasat least three more lattice points than P uv , a contradiction. (cid:3) (Almost all) spanning -polytopes have a unimodular tetrahedron The following statement follows from Corollary 1.5.
Lemma 6.1.
Let P be a non-spanning lattice -polytope and let v be a vertex of itsuch that P v is still -dimensional. Then P and P v have the same index. (cid:3) With this we can now prove Theorem 1.7: The only spanning 3-polytopes that donot contain a unimodular tetrahedron are the E , and E , from the statementof Theorem 1.7. Proof of Theorem 1.7.
Let P be a spanning 3-polytope. If P has width one, thestatement is true by Corollary 3.3. So assume P to be of width > n beits size. If n ≤ P isnot merged and of size n ≥ n ≥ P ismerged. That is, there exist u, v ∈ vert( P ) such that P u and P v have width > P uv is 3-dimensional.If P u or P v are spanning, then by inductive hypothesis they contain a unimodulartetrahedron, and so does P . To finish the proof we show that it is impossible for P u and P v to both have index greater than one. If this happened, then Lemma 6.1tells us that both P u and P v have the same index as P uv . That is, u and v lie inthe affine lattice spanned by P uv ∩ Z , which implies the index of P being the same(and bigger than one), a contradiction. (cid:3) The h ∗ -vectors of non-spanning -polytopes Part of our motivation for studying non-spanning 3-polytopes comes from re-sults on h ∗ -vectors of spanning lattice polytopes recently obtained by Hofscheieret al. in [6, 7]. Recall that the h ∗ -polynomial of a lattice d -polytope P , first intro-duced by Stanley [8], is the numerator of the generating function of the sequence(size( tP )) t ∈ N . That is, it is the polynomial h ∗ + h ∗ z + · · · + h ∗ s z s := (1 − z ) d +1 ∞ (cid:88) t =0 size( tP ) z t . Its coefficient vector h ∗ ( P ) := ( h ∗ , . . . , h ∗ s ) is the h ∗ -vector of P . All entries of h ∗ ( P ) are known to be nonnegative integers, and the degree s ∈ { , . . . , d } of the h ∗ -polynomial is called the degree of P . See [1] for more details. For a lattice d -polytope with n lattice points in total, n of them in the interior,and with normalized volume V , one has h ∗ = 1, h ∗ = n − d − h ∗ d = n and (cid:80) i h ∗ i = V . In particular, in dimension three the h ∗ -vector can be fully recoveredfrom the three parameters ( n, n , V ) as follows:(1) h ∗ = 1 , h ∗ = n − , h ∗ = V + 3 − n − n, h ∗ = n . From this we can easily compute the h ∗ -vectors of all non-spanning 3-polytopes.They are given in Table 4. For their computation, in the infinite families (cid:101) F i we usethat: • The total number of lattice points equals b − a + 4 in (cid:101) F ( a, b ) and (cid:101) F ( a, b )and it equals b − a + 5 in (cid:101) F ( a, b, k ) and (cid:101) F ( a, b ). • The number of interior lattice points is zero in (cid:101) F ( a, b ) and b − a − • The volume always equals the volume of the projection F i (see Figure 2)times the length b − a of the spike. This follows from Corollary 1.5. P ( h ∗ , h ∗ , h ∗ , h ∗ ) T p,q ( a, b ) (1 , a + b − , abq − a − b + 1 , (cid:101) F ( a, b ) (1 , n − , n − , n − (cid:101) F ( a, b ) (1 , n − , n − , (cid:101) F ( a, b, k ) (1 , n − , n − , n − (cid:101) F ( a, b ) (1 , n − , n − , n − P ( h ∗ , h ∗ , h ∗ , h ∗ ) E (5 , (1 , , , E (6 , (1 , , , E (7 , (1 , , , E , (1 , , , E , (1 , , , E , (1 , , , Table 4.
The h ∗ -vectors of non-spanning 3-polytopes. In the in-finite families (top table) n is the size of the polytope. Rememberthat: in T p,q ( a, b ) we have a, b ≥ n = a + b + 2 and q >
1; in (cid:101) F ( a, b ) we have n ≥ n ≥ Theorem 7.1 ([6, Theorem 1.3], [7, Theorem 1.2]) . Let h ∗ = ( h ∗ , . . . , h ∗ s ) be the h ∗ -vector of a spanning lattice d -polytope of degree s . Then:(1) h ∗ has no gaps, that is, h ∗ i > for all i ∈ { , . . . , s } .(2) for every i, j ≥ with i + j < s one has h ∗ + · · · + h ∗ i ≤ h ∗ j +1 + · · · + h ∗ j + i . ON-SPANNING LATTICE 3-POLYTOPES 19
In dimension three the only nontrivial inequality in part (2) is h ∗ ≤ h ∗ for latticepolytopes of degree s = 3 (that is, for lattice polytopes with interior lattice points),which is true by Hibi’s Lower Bound Theorem [5, Theorem 1.3]. This inequalityfails for spanning polytopes without interior lattice points, as the prism [0 , × [0 , k ]shows (its h ∗ -vector equals (1 , k, k − Proposition 7.2.
Let P be a lattice -polytope of index q > . Then, h ∗ ( P ) ≥ ( q − h ∗ ( P )) .Proof. Let Λ (cid:48) ⊂ Z be the lattice spanned by P ∩ Z and let P (cid:48) be the polytope P considered with respect to Λ (cid:48) (equivalently, let P (cid:48) = φ ( P ) where φ : R → R is anaffine map extending a lattice isomorphism Λ (cid:48) ∼ = → Z ).Let V (cid:48) and V = qV (cid:48) be the volumes of P (cid:48) and P . Observe that the set of latticepoints (in particular, the parameters n and n in Equation (1)) does not dependon whether we look at one or the other lattice. Then, using the expression for h ∗ in Equation (1) we get: h ∗ ( P ) = h ∗ ( P (cid:48) ) − V (cid:48) + V == h ∗ ( P (cid:48) ) + ( q − V (cid:48) == h ∗ ( P (cid:48) ) + ( q − (cid:88) i h ∗ i ( P (cid:48) ) ≥≥ ( q − h ∗ ( P (cid:48) )) = ( q − h ∗ ( P )) . (cid:3) Concerning gaps (part (1) of Theorem 7.1), empty tetrahedra of volume q > h ∗ -vectors are (1 , , q − h ∗ = n − > h ∗ > References [1] M. Beck, S. Robins,
Computing the Continuous Discretely: Integer-Point Enumeration inPolyhedra , Springer, New York, 2007.[2] M. Blanco, F. Santos,
Lattice -polytopes with few lattice points , SIAM J. Discrete Math. 30(2) (2016) 669–686.[3] M. Blanco, F. Santos, Lattice -polytopes with six lattice points , SIAM J. Discrete Math. 30(2) (2016) 687–717.[4] M. Blanco, F. Santos, Enumeration of lattice -polytopes by their number of lattice points ,Discrete Comput. Geom. 60 (3) (2018) 756–800.[5] T. Hibi, A lower bound theorem for Ehrhart polynomials of convex polytopes , Adv. Math.105 (2) (1994) 162–165.[6] J. Hofscheier, L. Katth¨an, B. Nill,
Ehrhart theory of spanning lattice polytopes , Int. Math.Res. Not. (2017) rnx065, https://doi.org/10.1093/imrn/rnx065 , arXiv:1608.03166 .[7] J. Hofscheier, L. Katth¨an, B. Nill, Spanning lattice polytopes and the uniform position prin-ciple , Preprint, arXiv:1711.09512 , November 2017.[8] R. P. Stanley,
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Lattice tetrahedra , Canadian J. Math. 16 (1964) 389–396.(M. Blanco)
Mathematical Sciences Research Institute, 17 Gauss Way, CA 94720Berkeley, U. S. A.
E-mail address : [email protected] (F. Santos) Departamento de Matem´aticas, Estad´ıstica y Computaci´on, Universidadde Cantabria, Av. de Los Castros 48, 39005 Santander, Spain
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