Non-Realizable Minimal Vertex Triangulations of Surfaces: Showing Non-Realizability using Oriented Matroids and Satisfiability Solvers

NON-REALIZAB LE MINIMAL VER TEX TRIANGULA TIONS OF SURF A CES: SHO WING NON-REALIZAB I LITY USING ORI ENTED MA TR OIDS AND SA TISFIABILITY SOL VER S LARS SCHEWE Abstract. W e show that no minimal v ertex triangulation of a closed, connected, orientable 2-manifold of g enus 6 a dmits a p oly- hedral embedding in R 3 . W e also pr ovide examples of minimal vertex triangulations o f clo sed, connected, orien table 2-manifolds of genus 5 that do no t admit any p olyhedr al embeddings. W e construct a new infinite fa mily o f non-realiza ble tria ngulations of surfaces. Thes e results w ere ac hiev ed by transforming the problem of finding suitable o riented matroids in to a s atisfiability problem. This metho d can b e a pplied to other geometric rea lizability prob- lems, e.g. for face la ttices of p o lytop es. Gr ¨ un baum conjectured [1 5, Exercise 13.2.3] that all triangula t ed sur- faces (compact, orien table, connected, 2 - dimensional manifolds without b oundary) admit p o lyhedral embeddings in R 3 . This conjecture was sho wn to b e false by Bok o wski and G uedes de Oliv eira [5]. They sho w ed that one sp ecial triangulation with 12 v ertices of a surface of genus 6 do es not admit a p o lyhedral embedding in R 3 . Recen tly , Arc hdeacon et al. [2] settled the case of gen us 1 b y sho wing that all triangulations of the t o rus admit a p olyhedral em b edding. Still, triangulated surfaces with p olyhedral em b eddings can b e quite complicated. McMullen, Sc h ulz, and Wills constructed polyhedral em- b eddings of triangulated surfaces with n ve rtices of g en us Θ( n log n ) ([23], see also [30]). Ho wev er, a gap remains: Jun german a nd Ringel [18, 27] sho w ed that n v ertices suffice to triangulate a surface of gen us Θ( n 2 ) and explicitly constructed suc h triangulations. So, can w e construct polyhedral embeddings of triangulated surfaces with few ve rtices? In the case of 2-spheres the com binator ial b ound is sharp; this is a consequence of Steinitz’s Theorem [28]. It is known that all v ertex minimal triangula t ions of surfaces up to gen us 4 admit p olyhedral em b eddings (gen us 1 w as first done b y Cs´ asz´ ar [11], the 2000 Mathe matics Subje ct Classific ation. P rimary 52 B70; Seconda ry 5 2C40. The author w a s supported b y a sc ho larship of t he Deutsc he T elekom F oundatio n. 1 2 LARS SCHEWE cases of gen us 2 and 3 we re solv ed by Lut z and Bok o wski [19], Lutz [19] and Hougardy , Lutz, and Zelk e [17]). Our ma in result is that none of the vertex minimal triangulations of a surface of genu s 6 admits a realization in R 3 . Moreo ve r, three minimal triangulations of a surfa ce of genus 5 do not admit realizations either. A small modificatio n of one of triangulations help us to construct a new infinite class of non- realizable tria ngulated surfaces. F or all results w e use an impro ved metho d to construct orien ted matroids tha t are admissible for t he surface in ques tion. The metho d can also b e applied to em b edding problems for general simplical complexes in arbitrary dimensions. A small mo dification of the method allows us to also treat immersions of simplical complexes. Using this mo dification w e can rule out for all but one triangulation of the surface of gen us 6 with 12 v ertices that it can b e imme rsed in to R 3 . The new metho d we prop ose to generate oriented matroids reduces the generation problem to an instance of the satisfiabilit y problem. This allo ws us to use we ll-tuned soft ware and sp eeds up the c hec k- ing pro cess immensly . As orien ted matroids ha ve b een used t o tac kle other g eometric realizabilit y problems, our metho d giv es more effectiv e algorithms for these problems as w ell. 1. Resul ts Using the algorithm giv en b elo w, it w as p ossible to show the follo wing theorems: Theorem 1.1. No triangulation of a surfac e o f genus 6 with 12 vertic es admits a p olyhe dr al r e alization in R 3 . The theorem is a consequence of the follo wing prop osition. A key step is the classification of comb inatorial surfaces with 12 v ertices of gen us 6 b y Altsh uler, Boko wski and Sc huc hert [1]. Prop osition 1.2. None of the 59 c ombinatorial surfac es w i th 12 ver- tic es of genus 6 admits an acyclic, uniform oriente d matr oid. The situation is more difficult in the case of gen us 5. T o triangulate a surface of gen us 5 w e also need at least 12 v ertices. Ho wev er, there are far more p ossibilities (7 51 593 as en umerated b y Lutz and Sulanke [21]) than in the case of gen us 6. Nev ertheless, the next theorem shows that the case of g en us 5 lo oks also more in teresting. Theorem 1.3. Ther e exist at le ast thr e e c ombinatorial ly di s tinct tri- angulations of a surfac e of genus 5 with 12 v ertic es that do n ot a dmit NON-R EALIZABLE MINIMAL VER TEX TRI ANGULA TIONS OF SURF A CES 3 T able 1. Number of com binato rial triangulations g n min # 0 4 1 1 7 1 2 10 865 3 10 20 4 11 821 5 12 751 593 6 12 59 T able 2. T riangulatio n 2 12 1 1 of [20] 1 2 3 1 2 12 1 3 6 1 4 8 1 4 11 1 5 9 1 5 10 1 6 9 1 8 10 1 11 12 2 3 4 2 4 7 2 5 9 2 5 12 2 6 10 2 6 11 2 7 10 2 9 11 3 4 5 3 5 8 3 6 10 3 7 11 3 7 12 3 8 11 3 10 12 4 5 6 4 6 9 4 7 11 4 8 12 4 9 12 5 6 7 5 7 10 5 8 12 6 7 8 6 8 11 7 8 9 7 9 12 8 9 10 9 10 11 10 11 12 T able 3. T riangulatio n 2 12 1 2 of [20] 1 2 3 1 2 12 1 3 6 1 4 9 1 4 11 1 5 8 1 5 9 1 6 10 1 8 10 1 11 12 2 3 4 2 4 7 2 5 10 2 5 12 2 6 9 2 6 10 2 7 11 2 9 11 3 4 5 3 5 8 3 6 11 3 7 10 3 7 11 3 8 12 3 10 12 4 5 6 4 6 9 4 7 12 4 8 11 4 8 12 5 6 7 5 7 10 5 9 12 6 7 8 6 8 11 7 8 9 7 9 12 8 9 10 9 10 11 10 11 12 a p olyhe dr al r e alization in R 3 . Howev e r, ther e exists at le ast one tri- angulation of a surfac e of genus 5 with 12 vertic es that admits m any oriente d matr oids. Sp ecifically no admissible oriented matroids exist f o r the manifolds 2 12 1 1 , 2 12 1 2 , and 2 12 1 6 described in the dissertation of F ra nk Lutz [20]. Ho wev er, more than 100 000 admissible orien ted matroids exist for the manifold 2 12 5 1 . A facet description of the non-realizable manifolds can b e found in the T a bles 2, 3, 4. Another interes ting question w as dealt with b y Bok owski a nd Guedes de Olive ira [5]: Are there infinite classes of surfaces of a fixed g enus that cannot b e realized? Bok owsk i and Guedes de Oliv eira tried to answ er 4 LARS SCHEWE T able 4. T riangulatio n 2 12 1 6 of [20] 1 2 4 1 2 6 1 3 6 1 3 12 1 4 11 1 5 9 1 5 12 1 8 9 1 8 10 1 10 11 2 3 5 2 3 7 2 4 7 2 5 12 2 6 10 2 9 10 2 9 11 2 11 12 3 4 6 3 4 8 3 5 8 3 7 11 3 10 11 3 10 12 4 5 7 4 5 9 4 6 9 4 8 12 4 11 12 5 6 8 5 6 10 5 7 10 6 7 9 6 7 11 6 8 11 7 8 10 7 8 12 7 9 12 8 9 11 9 10 12 T able 5. Surfa ce no.1 of [1] 1 2 11 1 2 12 1 3 4 1 3 10 1 4 9 1 5 6 1 5 11 1 6 9 1 7 8 1 7 1 0 1 8 12 2 3 6 2 3 8 2 4 10 2 4 12 2 5 9 2 5 10 2 6 7 2 7 11 2 8 9 3 4 6 3 5 7 3 5 12 3 7 8 3 9 10 3 9 11 3 11 12 4 5 8 4 5 12 4 6 8 4 7 10 4 7 11 4 9 11 5 6 10 5 7 9 5 8 11 6 7 12 6 8 9 6 10 11 6 11 12 7 9 12 8 10 11 8 10 12 9 10 12 this question b y taking a non-realizable surface a nd cutting out a tri- angle suc h that the remaining manifold stays no n- realizable. W e found that the remaining manifold given b y Bok o wski and Guedes de Oliv eira admits chirotop es after all. Their ar g umen t for non-realizability de- p ends crucially on the symmetry of t he surface t o reduce the searc h space. Ho wev er, the symmetry g r o up of the manifold in question is smaller than the symmetry gro up of the whole surface. Still, o ur algorit hm yields an eve n strong er statement: Theorem 1.4. F or e ach g e n us g ≥ 5 ther e exist infinite classes of surfac es that have no p olyhe dr al emb e dde ding in R 3 . The main idea to construct suc h an infinite family w as already give n b y Bok o wski and Guedes de Oliv eira [5 ]. W e tak e the connected sum of suitable surfaces; w e can ensure that the result is non-realizable if one of the summands stay ed non-realizable after the remov al of one triangle. As w e do no t need to imp ose an y conditions on the sec ond summand, w e can then construct surfaces of arbitrary gen us g as long as g is greater or equal than the gen us of the first summand; b y additionally adding triangulations of spheres with arbitrar y num b ers o f v ertices w e can construct t he infinite families w e a r e after. The cons truction is summarized in the follo wing Lemma. W e o mit the straigh t-forward pro of. NON-R EALIZABLE MINIMAL VER TEX TRI ANGULA TIONS OF SURF A CES 5 Lemma 1.5. Given two triangulations S and T of surfac es and a tri- angle T ∈ S such that S \ { T } is non-r e aliz able, then ther e exists a triangulation X with V ( X ) = V ( S ) + V ( T ) − 3 vertic es of the surfac e of genus g S + g T that is non-r e alizable as wel l. The follow ing prop osition sho ws that the conditions of the Lemma can b e satisfied. Prop osition 1.6. a) L et O b e the surfac e 2 12 1 1 as ab ove and let M := O \ {{ 1 , 2 , 3 }} . Then M do es not admi t an acyclic uniform orien te d matr oid. b) L et P b e the surfac e no. 1 in the enumer ation of Altshuler and Bokowski [1] (se e T a b le 5) and l e t N := P \ {{ 1 , 2 , 11 }} . Then N do es not admit an acyclic uniform oriente d matr oid. Pr o of of The or em 1.4. As a first step w e will sho w a construction that yields for an y surface X a no n- realizable surface S . W e then exhibit suitable sequences of surfaces to sho w the Theorem. T ak e a triang ulated surface X of gen us g with n vertices . After ren umbering w e ma y assume that the v ertices are 13 , . . . n + 12 and that [ n + 10 , n + 11 , n + 12] is a triangle in X . No w w e tak e the connec ted sum of X and O where w e iden tify the pairs of v ertices (1 , n + 10), (2 , n + 11), and (3 , n + 12 ). W e call this complex S . It follo ws from the construction tha t S is a surface. F urthermore, S has gen us g + 5 and n + 9 v ertices. W e claim that S cannot b e realizable: it contains M as a sub complex. As w e hav e seen M is not realizable, so t he claim follo ws. No w, let g ≥ 5. Then let X 0 b y an y tr iangulated surface of gen us g − 5 a nd let X i b e the connected sum of X 0 with a triangulated sphe re with i + 3 p oints . W e see that the sequence S 0 , S 1 , . . . constructed as ab ov e is an infinite sequ ence of surfaces of gen us g all of whic h are not realizable.  Our results dep end on the following m etho d to generate orien ted matroids. W e first give an o vervie w b efore w e deal with the tec hnical details. 2. The Main Algorithm W e w an t to treat the em b eddability problem algorithmically . T o do so, w e need a combinatorial mo del of a po in t set in R n , whic h captures in teresting prop erties (for instance, conv exit y). Oriented matroids are a go o d c ho ice for this purp ose. Ex amples o f suc h applications can b e found for instance in the bo ok b y Bok ow ski and Sturmfels [6] . In the realizable case the circuits of an orien ted matroid correspond to minimal R a don partitions of the corresp onding elemen ts. W e can use 6 LARS SCHEWE this correspondence to c hec k whether t w o simplices inters ect each other. If F and G are simplices suc h tha t F ∩ G = ∅ , they in tersect if and only if F ∪ G con tains a circuit C suc h that C + ⊆ F and C − ⊆ G . W e sa y that an orien ted matroid M = ( E , χ ) is admissible for a simpli c ial c omplex K if E = | K | and for all F, G ∈ K with F ∩ G = ∅ there do es not exist an y circuit C suc h that C + ⊆ F and C − ⊆ G . If w e consider only uniform o r ien ted matroids of rank 4 and our simplices are f a ces of a su rface, w e only need to consider the case that F is a triangle and G is an edge. Additionally , w e use a kno wn fact ab out orien ted matroids tha t a re derive d from p oint sets: no circuit of suc h an o r iented matroid is totally p ositiv e. Orien ted matroids with this prop ert y are called acyclic . W e can r estrict our problem ev en further: p o lyhedral embeddings of triangulated surfaces are “nice” ; w e can p erturb the v ertices b y a small amoun t without creating an y in tersections of the triangles. This mak es o ur task of finding oriented matroids comparat ively easy . W e can restrict our atten tion to uniform oriented matroids. So, for a giv en simplicial complex, w e can deduce that K cannot b e em b edded in R d , if K do es not admit an y acyclic, uniform orien ted matroid of rank d + 1. W e will no w c hec k for this condition by trans- forming it into an instance of SA T. Luc kily , this transformation is quite straigh t forw a rd. Ho w eve r, we first review some oriented matroid ter- minology and fix the notatio n for instances of SA T. The main part con- sists of the enco ding the oriented matroid axioms, i.e. t he three-term Grassmann-Pl ¨ uc ke r relations, and of enco ding the “ forbidden” circuits. 2.1. Simplicial Complexes. W e no w giv e a rough sk etch how ori- en ted matroids can b e used to ta c kle realizabilit y questions. Assume w e hav e a realization of a triangulated surface S , i.e. a map f : S → R 3 suc h that for all ∆ 1 , ∆ 2 ∈ S holds that conv( f (∆ 1 ∩ ∆ 2 )) = f (con v(∆ 1 )) ∩ f (con v (∆ 2 )). If w e w an t that f is an em b edding, we need to mak e sure that the image of tw o simplices has non-trivial in- tersection if and only if the simplices themself intersec ted non-trivially . Definition 2.1 (Em b edding) . Giv en a triangula tion K of a surface, w e sa y that a mapping f : K → R d induces a n em b edding if for no tw o simplices that are disjoin t in K their images under f in tersect in R d . When w e w ant to c hec k whether a mapping is an embedding, we can restrict our atten tion to simplices whose dimension sum to d . In our case this means we only need to c hec k in tersections of one triangle with an edge that is disjoint from the triangle. NON-R EALIZABLE MINIMAL VER TEX TRI ANGULA TIONS OF SURF A CES 7 2.2. Orien ted Ma troids. The following discussion of orien ted ma- troids is extremely brief, w e recommend the monograph [3], esp ecially Section 3.5 f o r the missin g details. W e only consider uniform oriented matro ids and a ssume these are giv en b y their c hirotop es. W e also assume that the gr o und set E of the orien ted matro ids is { 1 , . . . , n } . W e use the following axioms for orien ted matroids: Definition 2.2. Let E = { 1 , . . . , n } , r ∈ N , and χ : E r → {− 1 , +1 } . W e call M = ( E , χ ) a uniform orien ted matroid of rank r , if the fol- lo wing conditions are satisfied : (B1) The mapping χ is alternating. (B2) F or all σ ∈  n r − 2  and a ll subsets { x 1 , . . . , x 4 } ⊆ E \ σ the fo llo wing holds: { χ ( σ , x 1 , x 2 ) χ ( σ , x 3 , x 4 ) , − χ ( σ, x 1 , x 3 ) χ ( σ , x 2 , x 4 ) , χ ( σ , x 1 , x 4 ) χ ( σ , x 2 , x 3 ) } ⊇ { − 1 , +1 } R emark 2.3 . The mapping χ is called the chir otop e of the orien ted matroid. As a first conse quence of these axioms we can restrict our atten t ion to the v alues that χ attains on the ordered r -subsets of E . The other v alues are then determined by (B1). The class of oriented matroids w e ar e interes ted in is s till smaller than t he class of uniform orien ted matroids. W e also w ant our orien ted matroids t o b e acyclic , that means the should contain no circuit in whic h ev ery elemen t has p ositive signature. Orien ted matroids with a p ositiv e circuit are called cyclic . Giv en a unifo rm oriented matroid M = ( E , χ ) the circuit signatures of M can b e computed from the c hirotop e: Let C = [ c 1 , . . . , c r +1 ] ( c 1 < · · · < c r +1 ) b e the unoriented circuit, then the t wo p o ssible signatures C + and C − of C are giv en by C i = ( − 1) i χ [ c 1 , . . . , b c i , . . . , c r +1 ] and its negativ e (for a pro of, see [3 , Lemma 3.57]). Recalling the discus sion in the section ab ov e, the circuit signatures giv e us the p ossibilit y to c hec k whether tw o simplices of complemen ta ry dimensions in tersect. 2.3. SA T. Before we giv e our transformatio n, we first fix our notation for instances of SA T. T ak e a Bo olean function Φ : { 0 , 1 } n → { 0 , 1 } , w here 0 stands for false and 1 for true. W e c all the e lemen ts of { 0 , 1 } n valuations . A v aluation is satisfying if Φ( v ) = 1. W e transform our problem, whether there exists an a dmissible ori- en ted mat r oid for a g iven simplicial complex, in to a n instance of SA T. 8 LARS SCHEWE An instance of SA T consists of a bo olean function give n in conjunc- tiv e normal form (CNF). T hat is, given the v ariables p 1 , . . . , p n the function Φ is of the form Φ( p ) = V m i =0 C i where the C i are o f the form C i = W j ∈ I i p j ∨ W j ∈ I i p j . A SA T solv er answ ers the question whether Φ is satisfiable. In that case it returns a v aluation v suc h that Φ( v ) = 1. The follo wing observ a tion go es back to P eirce [25]. It giv es us a wa y to write an arbitrary b o olean function in CNF. Lemma 2.4. L et Φ b e a b o ole an function Φ : { 0 , 1 } n → { 0 , 1 } . Then we c an write Φ as: Φ( x ) = ^ σ ∈{ 0 , 1 } n ¬ Φ( σ )   _ i ∈{ j | σ j =1 } x i   ∨   _ i ∈{ j | σ j =0 } x i   2.4. Enco ding. W e are now ready to giv e t he transformation of our problem: Giv en a simplicial complex K on n p oin ts and a dimension d , w e w an t to decide whether there exists a n acyclic, uniform orien ted matroid of rank d + 1 on n p oints that is admissible for K . T o en co de the c hirotop e w e in tro duce a v a r iable fo r each ordered r -subset B of { 1 , . . . , n } whic h w e denote b y [ B ]. Giv en a v aluation v w e construct a c hirotop e χ v as follows : If v [ B ] = 1, w e set χ v ( B ) = +1 and if v [ B ] = 0 then we set χ v ( B ) = − 1. W e start by enco ding the oriented matroid axioms. W e do not deal explicitly with the axiom (B1) as w e only fix t he signs f or the ordered subsets. The follow ing prop osition allo ws us to deal with a xiom (B2). It follows directly from L emma 2.4. Prop osition 2.5. L et α, β , γ , δ, ǫ, ζ b e or der e d r -subsets of E , v ∈ { 0 , 1 } ( | E | r ) , and χ v define d as ab ov e . T hen the fol lowing two c onditions ar e e quivalent: (1) { χ v ( α ) χ v ( β ) , − χ v ( γ ) χ v ( δ ) , χ v ( ǫ ) χ v ( ζ ) } ⊇ { +1 , − 1 } (2) v s a tisfi e s GP ( α, β , γ , δ, ǫ, ζ ) as d e fine d in T able 6. So, t hr ee-term Gra ssmann-Pl ¨ uc ke r relation is enco ded with 16 clauses with 6 literals eac h. As we ha v e  n r − 2  n − r +2 4  differen t Grassmann- Pl ¨ uc ke r relations to consider, w e get 16  n r − 2  n − r +2 4  man y clauses of length 6 in our resulting SA T instance. These clauses guara ntee the prop erty that eac h satisfiable v a luation of the instance will correspond to a c hirotop e. T o complete the mo del w e need a condition that excludes all orien ted matroids that hav e a giv en circuit signature. As a sp ecial case w e w ant to exclude all cyclic orien ted matroids. The follo wing pro p osition NON-R EALIZABLE MINIMAL VER TEX TRI ANGULA TIONS OF SURF A CES 9 GP ( α , β , γ , δ , ǫ, ζ ) = ( ¬ [ α ] ∨ ¬ [ β ] ∨ ¬ [ γ ] ∨ [ δ ] ∨ ¬ [ ǫ ] ∨ ¬ [ ζ ]) ∧ ( ¬ [ α ] ∨ ¬ [ β ] ∨ ¬ [ γ ] ∨ [ δ ] ∨ [ ǫ ] ∨ [ ζ ]) ∧ ( ¬ [ α ] ∨ ¬ [ β ] ∨ [ γ ] ∨ ¬ [ δ ] ∨ ¬ [ ǫ ] ∨ ¬ [ ζ ]) ∧ ( ¬ [ α ] ∨ ¬ [ β ] ∨ [ γ ] ∨ ¬ [ δ ] ∨ [ ǫ ] ∨ [ ζ ]) ∧ ( ¬ [ α ] ∨ [ β ] ∨ ¬ [ γ ] ∨ ¬ [ δ ] ∨ ¬ [ ǫ ] ∨ [ ζ ]) ∧ ( ¬ [ α ] ∨ [ β ] ∨ ¬ [ γ ] ∨ ¬ [ δ ] ∨ [ ǫ ] ∨ ¬ [ ζ ]) ∧ ( ¬ [ α ] ∨ [ β ] ∨ [ γ ] ∨ [ δ ] ∨ ¬ [ ǫ ] ∨ [ ζ ]) ∧ ( ¬ [ α ] ∨ [ β ] ∨ [ γ ] ∨ [ δ ] ∨ [ ǫ ] ∨ ¬ [ ζ ]) ∧ ([ α ] ∨ ¬ [ β ] ∨ ¬ [ γ ] ∨ ¬ [ δ ] ∨ ¬ [ ǫ ] ∨ [ ζ ]) ∧ ([ α ] ∨ ¬ [ β ] ∨ ¬ [ γ ] ∨ ¬ [ δ ] ∨ [ ǫ ] ∨ ¬ [ ζ ]) ∧ ([ α ] ∨ ¬ [ β ] ∨ [ γ ] ∨ [ δ ] ∨ ¬ [ ǫ ] ∨ [ ζ ]) ∧ ([ α ] ∨ ¬ [ β ] ∨ [ γ ] ∨ [ δ ] ∨ [ ǫ ] ∨ ¬ [ ζ ]) ∧ ([ α ] ∨ [ β ] ∨ ¬ [ γ ] ∨ [ δ ] ∨ ¬ [ ǫ ] ∨ ¬ [ ζ ]) ∧ ([ α ] ∨ [ β ] ∨ ¬ [ γ ] ∨ [ δ ] ∨ [ ǫ ] ∨ [ ζ ]) ∧ ([ α ] ∨ [ β ] ∨ [ γ ] ∨ ¬ [ δ ] ∨ ¬ [ ǫ ] ∨ ¬ [ ζ ]) ∧ ([ α ] ∨ [ β ] ∨ [ γ ] ∨ ¬ [ δ ] ∨ [ ǫ ] ∨ [ ζ ]) T able 6. D efinition of the function GP giv es the exact condition; again the prop osition follo ws directly from Lemma 2.4: Prop osition 2. 6. L et M = ( E , χ ) b e a uniform oriente d m a tr oid of r ank r , v the c orr esp onding valuation and C = ( s 1 c 1 , . . . , s r +1 c r +1 ) b e a signe d ( r + 1) -tuple ( s i ∈ { +1 , − 1 } , c i ∈ { 1 , . . . , n } , c i 6 = a j ). Then C is not a cir cuit of χ if and only if v satisfies Γ( C ) : Γ( C ) = ^ i ∈ I + [ c 1 , . . . , b c i , . . . , c r +1 ] ∧ ^ i ∈ I − ¬ [ c 1 , . . . , b c i , . . . , c r +1 ] ∨ ^ i ∈ I + ¬ [ c 1 , . . . , b c i , . . . , c r +1 ] ∧ ^ i ∈ I − [ c 1 , . . . , b c i , . . . , c r +1 ] I + = { i | ( − 1) i s i = + 1 } I − = { i | ( − 1) i s i = − 1 } 10 LARS SCHEWE Th us, w e add for ev ery forbidden circuit tw o clauses consisting of r + 1 literals eac h. With these clauses we ha v e completed our SA T- mo del. In the next section we will see how this giv es us an effectiv e w ay to solv e our problem. If w e w an t to use this method to treat other realizabilit y problems, other res trictions are of in terest. In the case of the a lg orithmic Steinitz problem, i.e. whether a lattice is a face lattice of a conv ex p olytop e, w e need to generate orien ted matroids with prescrib ed co circuits. The nece ssary clauses can b e deriv ed in the same manner as described ab ov e. 3. Imple ment a tion W e wrote a Hask ell [26] program that do es the translation describ ed in the preceding section. W e then used the SA T-solv ers ZChaff [22] and Minisat [12] to solve the resulting SA T instances. T o v erify the data en try of the 59 Altsh uler examples w e c heck ed the resulting surfaces using P o lymak e [14]. W e tested our programs on known examples . W e computed all chi- rotop es that are admissible for the M ¨ obius torus. W e f ound 2 7 72 chiro- top es (in less than 20 seconds) whic h is the same n umber that Bok owsk i and Eggert [4] found. F urthermore, we tested all triangulated surfaces with up to 9 v ertices (including the non-orien ta ble ones). In that case our progra m correctly fo und out whic h surfaces (all the o r ientable ones) admitted c hirotop es and whic h did not. Additionally , w e used o ur pro gram to v erify that all 821 minimal v ertex triangulations of a surface of genu s 4 as classified b y Lutz and Sulank e [21] admit a c hirot op e. There are quite a num b er of soft ware pac k ages to generate orien ted matroids (for instance [5, 8, 13, 1 6]). These pac k ages use one of t w o dif- feren t approac hes: The programs b y Bok owsk i and Guedes de O liveira and b y Finsc hi construct orien ted matroids b y using single elemen t extensions, whereas the ot her programs try t o construct the orien ted matroids globally b y filling in the c hiroto p es. Our approa c h is of the second t yp e. W e w ant to men tion that Da vid Br emner rep orted on his transformation of the problem into a 0/1 integer program. Ho w ever, his b enc hmark results sho wed that his bac ktracking program is faster in the instance s he used. T o giv e an impression of the efficiency of our program, w e state some running times: F or the genus 6 examples the transformation to ok appro ximately 3 0 seconds p er instance. Solving the SA T instances to ok b et wee n 22 and 98 minutes . All times w ere take n on a machine with tw o Pen tium I I I pro cessors (1 GHz) and 2 GB RAM. F or a ll NON-R EALIZABLE MINIMAL VER TEX TRI ANGULA TIONS OF SURF A CES 11 computations o nly one pro cessor was used. These r esults show that our program is m uc h fa ster than the program of [5]. W e think t he most inte resting comparison would b e to the program MPC of Dav id Bremner. Ho w ev er, he do es not implemen t the p ossibilit y to exclude orien ted matroids with ce rtain circuits. One of the adv an tages of o ur metho d lies in the fact tha t one can use a v ariety of SA T solv ers to c hec k the results. The transformatio n is simple enough to be che c ked b y hand. Man y SA T solv ers allow the p ossibilit y to giv e a “pro o f ” that an instance is unsatisfi able. The y output how to deriv e a con tr a diction from the giv en input. How ev er, this do es not impro v e our situation: the pr o ofs generated t his w ay are so large tha t they can only b e c hec k ed with the help o f a computer. Adv a nces in the dev elopmen t of pro of assistan ts might mak e it p ossible to give a full f ormal ve rification of our results in the near future. 4. Immersions W e hav e seen that w e cannot hop e t o find em b eddings for all trian- gulations of orien table surfaces. Ho we v er, one could hop e for w eak er results. In the context of non-orien table surfaces, where em b eddings cannot b e found for top ological reasons, one tries instead to find immer- sions of these surfaces. Th us, we could hop e to find some immersions for the surface s w e found not to b e em b eddable. W e men tion that Cerv one [10] sho w ed there are non- immersable tri- angulations with eight ve rtices of the Klein b ottle, whereas one can find an imme rsion of a triangulation with nine v ertices. Brehm had earlier sho wn that there is no ga p b etw een the the necessary v ertex n um b ers for immersions of the real pro j ective plane [7]. Definition 4.1 (Immersion) . Given a triangulation K of a surface, w e sa y that a mapping f : K → R d induces an immersion if for no tw o triangles in the star of a v ertex v ∈ K their images under f interse ct in R d . R emark 4.2 . The star of a v ertex is the smallest simplicial complex, that contains all faces that con ta in the giv en v ertex. This definition directly leads to an adaptation of the notion o f an admissible oriente d matroid. W e sa y t hat an orien ted matro id M = ( E , χ ) is admissible with r es p e ct to an immersion of a simplicial c omplex K if the follo wing conditions hold: • E = | K | , • for all F , G ∈ star( v ) with F ∩ G = ∅ there do es not exist any circuit C suc h that C + ⊆ F and C − ⊆ G . 12 LARS SCHEWE Using a suitably mo dified v ersion of the algorithm ab ov e (one just needs to te st few er p ossible in tersections), w e can sho w that all but one of the 59 surfaces of genus 6 do not admit an orien ted matro id that is admissible with resp ect to an immersion of that surface. The exception is surface n um b er 15 (a gain using the n um b ering sc heme used b y Altsh uler et al. [1]). 5. Conclusion Our r esults giv e additional insight in the prop erties of minimal vertex triangulations o f surfaces. Still, the main problems r emain: How can w e c har a cterize non- realizabilit y? Are all triangulated surfaces of small gen us (i.e. g ≤ 4) realizable? The infinite class of non-realizable surfaces giv en abov e hints that there will b e no easy answ er to the first question. F or genus 5 and 6 w e can construct non-realizable triangula t io ns f o r an y n um b er of v ertices. W e conjecture that this holds also for an y genus larger than 6. Ho w ev er, w e think it should b e p o ssible to prov e t hat for eve ry g en us greater than 4 w e need strictly more ve rtices for a p o lyhedral embedding of a surface than for a com binatorial triangulation. How ev er, one of the main obstacles fo r such an inv estigation is the lac k o f go o d construction metho ds f o r “interes ting” com binatorial su rfaces. The metho d w e used is in teresting in its o wn right. It helps tremen- dously in the study of small examples. How ev er, w e hop e that the small examples giv en here will help in the solution of the general problem. One p oint that needs improv emen t is the fact that w e cannot use effec- tiv ely use the information we gain if w e find o rien ted matroids in the course of our search. The metho ds f or finding realizatio ns of oriented matroids are no t go o d enough to yield practical results. As an op en pr o blem remains the question ho w strong orien ted ma- troid metho ds a r e compared to the metho ds described b y Novik [24] and Timmrec k [29]. W e conjecture that using orien ted matroids will giv e as stro ng results a s the metho d prop osed b y Timmrec k. W e are lead to this conjecture by the result o f Carv alho and Guedes de Oliv eira [9]. They show ed that the linking n um b er argumen ts give n by Br ehm as incorp orated by Timmrec k hold also in the setting of orien ted ma- troids. That means tha t these arguments ar e subsumed by the orien ted matroid tech nique. The tec hnique used in this article can b e applied to other g eometric problems. It has already b een used to treat realizabilit y of p oin t-line NON-R EALIZABLE MINIMAL VER TEX TRI ANGULA TIONS OF SURF A CES 13 configurations. Another a pplicatio n could b e in tac kling the Algorith- mic Steinitz problem (cf. [3]). W e hop e that this tec hnique prov es itself to b e a useful building blo c k in these and other applicatio ns. A cknowledgements I would lik e to thank J ¨ urgen Boko wski for all the advice and en- couragemen t receiv ed while undertaking this w ork. I had a n um b er of helpful discussions with a n um b er of p eople: I lik e to men tion P eter Lietz, F rank Lutz (who also generously provided me lots of in teresting examples for testing m y softw are), and Dagmar Timmrec k. Thanks also to Mic hael Joswig who ga v e a n umber of v aluable commen ts on an earlier draft of this art icle. Reference s [1] Amos Altshuler, J ¨ urgen Bo ko wsk i, a nd Peter Sch uchert, Neighb orly 2 - manifolds with 1 2 vertic es , J. Combin. T heo ry Ser. 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Ziegler, Polyhe dr al sur fac es of high genus , av ailable at arXiv: math. MG/0412093 . Lars Schewe, F achberei ch Ma thema tik, AG Optimier ung, TU Darm- st adt, Schlossgar tenstraße 7, 64 289 Darmst adt, Germany E-mail add r ess : schewe@ mathe matik.tu-darmstadt.de