Proc. Indan Acad. Sc. (Math. Sc.) Vo. 6, No., November 06, pp. 69 65. DOI 0.007/s0-06-00-7 An agorthmc approach to construct crystazatons of -manfods from presentatons of fundamenta groups BIPLAB BASAK Department of Mathematcs, Indan Insttute of Scence, Bangaore 560 0, Inda. E-ma: bpab0@math.sc.ernet.n MS receved 9 October 0; revsed May 05 Abstract. We have defned the weght of the par ( S R,R) for a gven presentaton S R of a group, where the number of generators s equa to the number of reatons. We present an agorthm to construct crystazatons of -manfods whose fundamenta group has a presentaton wth two generators and two reatons. If the weght of ( S R,R) s n, then our agorthm constructs a the n-vertex crystazatons whch yed ( S R,R). As an appcaton, we have constructed some new crystazatons of -manfods. We have generazed our agorthm for presentatons wth three generators and a certan cass of reatons. For m andm n k, our generazed agorthm gves a (m + n + k 6 + δn + δ k )-vertex crystazaton of the cosed connected orentabe -manfod M m, n, k havng fundamenta group x,x,x x m = xn = xk = x x x. These crystazatons are mnma and unque wth respect to the gven presentatons. If n = or k andm then our crystazaton of M m, n, k s vertex-mnma for a the known cases. Keywords. Pseudotranguatons of manfods; crystazatons of manfods; spherca and hyperboc -manfods; presentatons of groups. 00 Mathematcs Subect Cassfcaton. Prmary: 57Q5; Secondary: 05C5, 57N0, 57Q05.. Introducton For d, a (d + )-coored graph (Ɣ, γ ) represents a pure d-dmensona smpca ce compex K(Ɣ) whch has Ɣ as dua graph. For a certan cass of such graphs, the underyng space K(Ɣ) s a cosed connected d-manfod. In such a case, the (d + )- coored graph (Ɣ, γ ) s caed a crystazaton of the d-manfod K(Ɣ). In[7], Pezzana showed the exstence of crystazatons for each cosed connected PL manfod. In [], Gagard ntroduced an agorthm to fnd a presentaton of the fundamenta group of a cosed connected d-manfod M from a crystazaton of M. The components of the graph restrcted over two coors gve the reatons, and the components of the graph restrcted over the remanng coors gve the generators of the presentaton. In [0], Epsten proved that the fundamenta group of a -manfod has a presentaton whch has the number of reatons s ess than or equa to the number of generators. For a par ( S R,R) wth #S = #R, we have defned ts weght λ( S R,R) n Defnton.. If (Ɣ, γ ) s a crystazaton of a cosed connected orentabe -manfod and yeds a presentaton ( S R,R)then, from Lemma.7, #V(Ɣ) λ( S R,R). Gven a presentaton S R wth c Indan Academy of Scences 69
60 Bpab Basak two generators and two reatons, our am s to construct a crystazatons whch yed ( S R,R)and have λ( S R,R)vertces. For such a presentaton S R of a group, we have presented an agorthm (Agorthm n subsecton.) whch gves a crystazatons such that the crystazatons yed the par ( S R,R)and are mnma (cf. Defnton.8) wth respect to the par ( S R,R). In partcuar, the agorthm determnes whether such a crystazaton exsts or not. Let M = L(p, q) be a cosed connected orentabe prme -manfod and the fundamenta group of M has a presentaton S R wth two generators and two reatons. Usng Agorthm, we have constructed a possbe crystazatons of M whch yed ( S R, R) and are mnma wth respect to the par ( S R, R) (cf. Theorem.). If M = L(p, q) then the agorthm gves a such crystazatons of L(p, q ) for some q {,...,p }. As an appcaton of Agorthm, we have constructed such crystazatons of some cosed connected orentabe -manfods, namey, M m, n, for am, n, ens spaces and a hyperboc -manfod. (Here M m, n, k s as n subsecton..) We have aso generazed ths agorthm for presentatons wth three generators and a certan cass of reatons (Agorthm n subsecton 5.). As an appcaton of ths, we have constructed a (m + )-vertex crystazaton of the generazed quaternon space S /Q m and a (m + n + k )-vertex crystazaton of the -manfod M m, n, k for m, n, k. (Here S /Q m s as n subsecton..) For (m,n,k) = (,, ), these crystazatons are vertex-mnma, when the number of vertces are at most 0. In fact, there are no known crystazatons of these manfods whch have ess number of vertces than our constructed ones (cf. Remark 5.5). We have aso constructed a (m + n )- vertex crystazaton of the -manfod M m, n, for m, n. The crystazatons of the -manfods S /Q m, M m, n, and M m, n, k for m, n, k are mnma and unque wth respect to the gven presentatons (cf. Theorems 5., 5. and 5.).. Premnares. Coored graphs A mutgraph Ɣ = (V (Ɣ), E(Ɣ)) s a fnte connected graph whch can have mutpe edges but no oops, where V(Ɣ) and E(Ɣ) denote the sets of vertces and edges of Ɣ respectvey. For n, an n-path s a tree wth (n + ) dstnct vertces and n edges. If a and a + are adacent n an n-path for n then the n-path s denoted by P n (a,a,...,a n+ ).Forn, an n-cyce s a cosed path wth n dstnct vertces and n edges. If vertces a and a + are adacent n an n-cyce for n (addton s moduo n) then the n-cyce s denoted by C n (a,a,...,a n ).BykC n, we mean a graph conssts of k dsont n-cyces. The dsont unon of the graphs G and H s denoted by G H. A graph Ɣ s caed (d + )-reguar f the number of edges adacent to each vertex s (d + ). Frst we ca d ={0,,...,d} the coor set. A edge coorng wth (d + ) coors on the graph Ɣ = (V (Ɣ), E(Ɣ)) s a map γ : E(Ɣ) d such that γ(e) = γ(f)whenever e and f are adacent (.e., e and f are adacent to a common vertex). A (d + )-coored graph s a par (Ɣ, γ ), where Ɣ s a mutgraph and γ s a edge coorng on the graph Ɣ wth (d + ) coors. Two vertces are caed -adacent to each other f they are oned by an edge of coor. Let (Ɣ, γ ) be a (d + )-coored connected graph wth coor set d.ifb d wth k eements then the graph (V (Ɣ), γ (B)) s a k-coored graph wth coorng γ γ (B).
An agorthmc approach to construct crystazatons 6 Ths coored graph s denoted by Ɣ B.IfƔ d \{c} s connected for a c d then (Ɣ, γ ) s caed contracted. For standard termnoogy on graphs, see [5].. Spherca and hyperboc -manfods A -manfod M s caed a spherca -manfod f M = S /Ɣ, where Ɣ s a fnte subgroup of SO() actng freey by rotatons on the -sphere S. Therefore, spherca - manfods are prme, orentabe and cosed. Spherca -manfods are sometmes caed eptc-manfods or Cfford Ken -manfods. In Chapter of [8], Thurston conectured that a cosed -manfod wth fnte fundamenta group s spherca, whch s aso known as eptzaton conecture. In[6], Pereman proved the eptzaton conecture. Consder the -sphere S = {(z,z ) C : z + z = }. Let p and q be reatvey prme ntegers. Then the acton of Z p = Z/pZ on S generated by [] (z,z ) := (e π/p z, e πq/p z ) s free and hence propery dscontnuous. Therefore the quotent space L(p, q) := S /Z p s a -manfod whose fundamenta group s somorphc to Z p. The -manfods L(p, q) are caed the ens spaces. It s a cassca theorem of Redemester that L(p, q ) s homeomorphc to L(p, q) f and ony f q ±q ± (mod p). A -manfod s caed a hyperboc -manfod f t s equpped wth a compete Remannan metrc of constant sectona curvature. In other words, t s the quotent of three-dmensona hyperboc space by a subgroup of hyperboc sometres actng freey and propery dscontnuousy. From Theorem. of [], we know the foowng. PROPOSITION. Let M and N be two orentabe, cosed, prme -manfods and et ϕ : π (M, ) π (N, ) be an somorphsm. () If M and N are not ens spaces then M and N are homeomorphc. () If M and N are not spherca then there exsts a homeomorphsm whch nduces ϕ.. Weghts of presentatons of groups Gven a set S, etf(s)denote the free group generated by S. So, any eement w of F(S) s of the form w = x ε xε m m, where x,...,x m S and ε =±for m and (x +,ε + ) = (x, ε ) for m. For R F(S),etN(R) be the smaest norma subgroup of F(S)contanng R. Then, the quotent group F(S)/N(R)s denoted by S R. For a presentaton P = S T wth N(T) = N(R), the par (P, R) denotes the presentaton P wth the reaton set R. So, f T = R and N(T) = N(R)then S T = S R but as a par ( S T,T) = ( S R,R). Two eements w,w F(S)are sad to be ndependent (resp., dependent) fn({w }) = N({w }) (resp., N({w }) = N({w })). For a fnte subset R of F(S),et R := {w N(R) : N((R \{r}) {w}) = N(R) for each r R}. (.) Observe that = and f R = s a fnte set then w := r R r R and hence R =. For w = x ε xε m m F(S), m, et { 0 f m =, ε(w) := ε ε + + ε m ε m + ε m ε f m.
6 Bpab Basak Consder the map λ: F(S) Z + defned nductvey as foows: f w =, λ(w) := m ε(w) f w = x ε xε m m,(x m,ε m ) = (x, ε ), λ(w ) f w = x ε w x ε. (.) Snce ε ε =0or,ε(w) s an even nteger and hence λ(w) s aso even. For w F(S), λ(w) ssadtobetheweght of w. Observe that λ(w w ) = λ(w w ) for w, w F(S).In[0], Epsten proved that the fundamenta group of a -manfod has a presentaton where the number of reatons s ess than or equa to the number of generators. Here, we are nterested n those presentatons S R for whch #S = #R <. DEFINITION. Let S ={x,...,x s } and R ={r,...,r s } F(S).Letr s+ be an eement n R of mnmum weght. Then, the number λ( S R,R):= λ(r ) + +λ(r s ) + λ(r s+ ). s caed the weght of the par ( S R,R). Let w = α ε αε αε m m F(S := {x,...,x s }), where ε {+, } for m. Then, we defne () w () := tota number of appearances of x x,x S and < s, () w () := tota number of appearances of x x x,x S and < s, () w () (s+) := tota number of appearances of x for x,x S and = s, (v) w () (s+) := tota number of appearances of x for x,x S and = s. x and x x n α ε m m α ε αε αε m m,for and x x x x n α ε m m α ε αε αε m m,for and x x n α ε m m α ε αε αε m m, and x x n α ε m m α ε αε αε m m, Observe that λ(w) = w (c) s the sum over < s + and c.. Bnary poyhedra groups and generazed quaternon spaces A group s caed a bnary poyhedra group f t has a presentaton of the form x,x,x x m = x n = xk = x x x for some nteger m, n, k. Ths group s denoted by m, n, k. Ths group s known to be the fundamenta group of the -manfod L/(m, n, k), where L s the connected Le group of orentaton preservng sometres of a pane P (cf. [, 5]) and (m,n,k) = x,x,x x m = x n = x k =. Snce m, n, k s not a free product and not somorphc to Z p, any -manfod whch has a fundamenta group m, n, k, s prme and not homeomorphc to the ens space. Therefore, by Proposton., any two cosed connected orentabe manfods wth the same fundamenta group m, n, k, are homeomorphc. In ths artce, we w denote such a -manfod by M m, n, k. Observe that M m, ñ, k = M m, n, k for every permutaton mñ k of mnk. Thus, we can assume that m n k. Ceary, the group x,x
An agorthmc approach to construct crystazatons 6 x mx n,xnk n k x k,x x x x s somorphc to the abeanzed group of m, n, k. Therefore, (m,n,k) = (5,, ) or (7,, ) mpes that abeanzaton of m, n, k s trva. Thus, M 5,, and M 7,, are homoogy spheres, n fact, M 5,, s the Poncaré homoogy sphere. Snce m,, ( = Q m ), P :=,,, P 8 :=,, and P 0 := 5,, are fnte groups, by the proof of eptzaton conecture of Pereman, M m, n, k s spherca,.e., M m, n, k = S / m, n, k for these groups m, n, k. It s not dffcut to prove that, the abeanzaton of m, n, k = Z H for some group H f and ony f (m,n,k)= (6,, ), (,, ) or (,, ). Therefore, n these three cases, the -manfod M m, n, k has a hande and n a the other cases, M m, n, k s hande-free. A group s caed a generazed quaternon group or dcycc group f t has a presentaton of the form x,x x m = x =,xm = x,x x x = x for some nteger m. Ths group has order m and s denoted by Q m. Cam. For m, Q m has a presentaton S R, where S ={x,x,x } and R = {x m x x,x x x,x x x }. Observe that x x x = = x x x mpes x x = x = x x,.e., x = x x x.agan,x m x x = and x x x = mpes that x m = x.now, x = x x x = x (x x x )x = x x x = = xm x x m = x x x = x5 mpes x =. Snce xm = x, xm = x =. Thus, S R = x,x x m = x =,xm = x,x x x = x. Ths proves the cam. The -manfod M m,, s caed generazed quaternon space. Then, by the proof of eptzaton conecture of Pereman, M m,, s spherca and homeomorphc to S /Q m..5 Crystazatons A CW-compex X s sad to be reguar f the attachng maps whch defne the ncdence structure of X are homeomorphsms. Gven a reguar CW-compex X, etx be the set of a cosed ces of X together wth the empty set. Then, X s a poset, where the parta orderng s the set ncuson. Ths poset X s sad to be the face poset of X. Ceary,fX and Y are two fnte reguar CW-compexes wth somorphc face posets then X and Y are homeomorphc. A reguar CW-compex X s sad to be smpca f the boundary of each ce n X s somorphc (as a poset) to the boundary of a smpex of same dmenson. A smpca ce compex K of dmenson d s a poset, somorphc to the face poset X of a d- dmensona smpca CW-compex X. The topoogca space X s caed the geometrc carrer of K and s aso denoted by K. If a topoogca space M s homeomorphc to K, then K s sad to be a pseudotranguaton of M. Let K be a smpca ce compex wth parta orderng. Ifβ α K then we say β s a face of α. If a the maxma ces of a d-dmensona smpca ce compex K are d-ces then t s caed pure. Maxma ces n a pure smpca ce compex K are caed the facets of K. The 0-ces n a smpca ce compex K aresadtobethevertces of K. If u s a face of α and u s a vertex then we say u s a vertex of α.ceary,ad-dmensona smpca ce compex K has at east d + vertces. If a d-dmensona smpca ce compex K has exacty d + vertces then K s caed contracted. Let K be a pure d-dmensona smpca ce compex. Consder the graph (K) whose vertces are the facets of K and whose edges are the ordered pars ({σ,σ },γ), where σ, σ are facets, γ s a (d )-ce and s a common face of σ, σ. The graph
6 Bpab Basak (K) s sad to be the dua graph of K. Observe that (K) s n genera a mutgraph wthout oops. On the other hand, for d, f (Ɣ, γ ) s a (d + )-coored graph wth coor set d ={0,...,d} then we defne a d-dmensona smpca ce compex K(Ɣ) as foows. For each v V(Ɣ),wetakead-smpex σ v and abe ts vertces by 0,...,d. If u, v V(Ɣ)are oned by an edge e and γ(e)=, then we dentfy the (d )-faces of σ u and σ v opposte to the vertces abeed by, so that equay abeed vertces are dentfed together. Snce there s no dentfcaton wthn a d-smpex, ths gves a smpca CW-compex W of dmenson d. So, the face poset (denoted by K(Ɣ)) ofw s a pure d-dmensona smpca ce compex. We say that (Ɣ, γ ) represents the smpca ce compex K(Ɣ). Ceary, the number of -abeed vertces of K(Ɣ) s equa to the number of components of Ɣ d \{} for each d. Thus, the smpca ce compex K(Ɣ) s contracted f and ony f Ɣ s contracted (cf. []). A crystazaton of a connected cosed d-manfod M s a (d + )-coored contracted graph (Ɣ, γ ) such that the smpca ce compex K(Ɣ) s a pseudotranguaton of M. Thus, f (Ɣ, γ ) s a crystazaton of a d-manfod M then the number of vertces n K(Ɣ) s d +. On the other hand, f K s a contracted pseudotranguaton of M then the dua graph (K) gves a crystazaton of M. Ceary, f (Ɣ, γ ) s a crystazaton of a cosed d-manfod M then, ether Ɣ has two vertces (n whch case M s S d ) or the number of edges between two vertces s at most d. In [7], Pezzana showed the foowng. PROPOSITION. [7] Every connected cosed PL manfod admts a crystazaton. Thus, every connected cosed PL d-manfod has a contracted pseudotranguaton,.e., a pseudotranguaton wth d + vertces. From [9], we know the foowng. PROPOSITION. [9] Let (Ɣ, γ ) be a crystazaton of a PL manfod M. Then M s orentabe f and ony f Ɣ s bpartte. Let d = {0,...,d} be the coor set of a (d + )-coored graph (Ɣ, γ ). For0 = d, g denote the number of connected components of the graph Ɣ {,}.In[], Gagard proved the foowng. PROPOSITION.5 [] Let (Ɣ, γ ) be a contracted -coored graph wth the coor set. Then,(Ɣ,γ) s a crystazaton of a connected cosed -manfod f and ony f () g = g k for {,, k, } =, and () g 0 + g 0 + g 0 = + #V(Ɣ). Let (Ɣ, γ ) be a crystazaton (wth coor set d ) of a connected cosed d-manfod M. So, Ɣ s a (d + )-reguar graph. Choose two coors, say, and from d.let {G,...,G s+ } be the set of a connected components of Ɣ d \{,} and {H,...,H t+ } be the set of a connected components of Ɣ {,}. Snce Ɣ s reguar, each H p s an even cyce. Note that, f d = then Ɣ {,} s connected and hence H = Ɣ {,}. Consder a set
An agorthmc approach to construct crystazatons 65 S ={x,...,x s,x s+ } of s + eements. For k t +, consder the word r k n F( S) as foows. Choose a startng vertex v n H k.leth k = v e v e v e v e v e v, where ep and e q are edges wth coors and respectvey. Defne r k := x k + xk x k + x k + xk, (.) where G kh s the component of Ɣ d \{,} contanng v h.for k t +, et r k be the word obtaned from r k by deetng x ± s+ s n r k. So, r k s a word n F(S), where S = S \{x s+ }.In[], Gagard proved the foowng. PROPOSITION.6 [] For d, et (Ɣ, γ ) be a crystazaton of a connected cosed PL manfod M. Fortwo coors,, et s, t, x p,r q be as above. If π (M, x) s the fundamenta group of M at a pont x, then { π (M, x) x,x,...,x s r f d =, = x,x,...,x s r,...,r t f d. In ths case, we w say (Ɣ, γ ) yeds ( S R,R), where S = {x,...,x s } and R ={r,...,r t }. From Proposton.5, t s cear that, f (Ɣ, γ ) s a crystazaton of a -manfod then s = t. Note that, there may have a reaton r R such that r R \{r} and n that case S R = S R \{r}. Lemma.7. If (Ɣ, γ ) s a crystazaton of a -manfod such that (Ɣ, γ ) yeds ( S R,R)then #V(Ɣ) λ( S R,R). Proof. Snce the crystazaton yeds ( S R,R), from the above dscusson, we know the crystazaton yeds the reatons n R {w}, where w R. Thus, the emma foows from the constructon of r as n eq. (.) and Defnton.. DEFINITION.8 A crystazaton (Ɣ, γ ) of a -manfod s caed mnma wth respect to the par ( S R, R) f#v(ɣ)= λ( S R,R).. Constructons of crystazatons and Agorthm Here we are nterested n orentabe -manfods. Thus, by Proposton., correspondng crystazatons are bpartte. We use back dots for vertces n one part and whte dots for vertces n the other part of the crystazatons.. Constructons Let S R = x,x r,r be a presentaton of a group. Now we construct a possbe crystazatons from the par ( S R,R)by the foowng steps. Step. If a crystazaton yeds ( S R,R)then the crystazaton yeds the reatons n R {w}, where w s an eement R. Snce we are nterested n λ( S R, R)-vertex crystazatons, w R s of mnmum weght. Let {w R, k} be the set of a
66 Bpab Basak ndependent words wth mnmum weght (as there are ony fnte number of ndependent words n R wth mnmum weght). Let R = R {w } for k. For each R {R,...,R k }, we w construct a possbe crystazatons whch yed the reatons n R. Choose a R {R,...,R k }. Step. If possbe, et (Ɣ, γ ) be a crystazaton of a -manfod whch yeds the reatons n R. Wthout oss of generaty, et G, G, G be the components of Ɣ {0,} such that G represents the generator x for and G represents x (cf. eq. (.) for constructon of r). Let n be the tota number of appearance of x n the three reatons n R for and n = λ( S R, R) (n + n ). Then, the tota number of vertces n G s n and et G = C n (x (),..., x (n ) )for. Ceary, each n s even and n + n + n = #V(Ɣ). Wthout oss of generaty, we can assume that x ( ) x () for n / and. γ () and x () x (+) γ (0) wth x (n +) = x () Here and after, the addtons and subtractons at the pont nx ( ). are moduo n for Step. From the fact #V(Ɣ) = λ( S R,R), we know that #V(Ɣ)s aways even and there s no -cyce n Ɣ {,} for 0 and. If #V(Ɣ) s of the form n for some n N, then g 0 + g 0 + g 0 = n +. Ths mpes, g 0 + g 0 = n. Wthout oss of generaty, consder g 0 = g = n and g 0 = g = n. Therefore, a the components of Ɣ {0,} (resp., Ɣ {,} ) are -cyces. But, there are two choces for Ɣ {0,} (resp., Ɣ {,} ). Ether Ɣ {0,} has one 8-cyce and remanng -cyces or Ɣ {0,} has two 6- cyces and remanng -cyces. Smar arguments hod for Ɣ {,}. On the other hand, f #V(Ɣ)s of the form n + forsomen N, then g 0 + g 0 + g 0 = n +. Ths mpes, g 0 + g 0 = n and hence g 0 = g = g 0 = g = n. Therefore, Ɣ {,} has one 6-cyce and remanng a -cyces for 0 and. Step. Snce components of Ɣ {,} yed the reatons n R, wthout oss of generaty, et the coors and be the coors and respectvey as n the constructon of r for r R (cf. eq. (.)). Let m (c) := w R w(c) for < and c, where w (c) as n subsecton.. Then, the number of edges of coor c between G and G s m (c) for c and <. Therefore, (m (c) + m(c) + m(c) ) = #V(Ɣ)for c. Thus, g 0 + g 0 + g 0 = #V(Ɣ)/ + = m (c) + m(c) + m(c) +. Snce g 0 =, g 0 = g and g 0 = g,wehaveg 0c + g c = m (c) + m(c) + m(c) for c. Step 5. Snce Ɣ {0,,c} s connected for c, #{m (c), < }. Case. Let #{m (c), < } =,.e., m (c), where < for some c {, }. Then, the maxmum number of b-coored -cyces n Ɣ {0,,c} wth two edges of coor c s (m (c) ) + (m(c) ) + (m(c) ) = m(c) + m(c) + m(c). Snce g 0c + g c = m (c) + m(c) + m(c), from the arguments n Step, Ɣ {0,,c} must have m (c) + m(c) + m(c) b-coored -cyces and two 6-cyces wth some edges of coor c. Therefore, from the arguments n Step, f #V(Ɣ) = n for some n N then Ɣ {0,} (resp., Ɣ {,} ) s a unon of -cyces and Ɣ {c,c} s of the form C 6 (n )C.But,f #V(Ɣ)= n + forsomen N then Ɣ {,c} s of the form C 6 (n )C for 0.
An agorthmc approach to construct crystazatons 67 Case. Let #{m (c), < } = forsomec {,}. Then, assume {,, } = {,, } such that m (c) = 0. Then, the maxmum number of b-coored -cyces n Ɣ {0,,c} wth two edges of coor c s (m (c) ) + (m (c) ) = m (c) + m (c). Snce #V(Ɣ)= λ( S R,R)and m (c) = 0, Ɣ {0,,c} does not have a b-coored 6-cyce wth some edges of coor c. Therefore, g 0c +g c = m (c) +m(c) and the arguments n Step mpes that Ɣ {0,,c} must have m (c) + m (c) b-coored -cyces and one 8-cyce wth some edges of coor c. Thus, Ɣ {0,} (resp., Ɣ {,} ) s a unon of -cyces and Ɣ {c,c} s of the form C 8 (n )C. In ths case, #V(Ɣ)= n for some n N. Therefore, t s cear that, m (c) edges of coor c between G and G yed m (c) b-coored -cyces for < and c. Step 6. Now, we w construct Ɣ {0,,} and w show that Ɣ {0,,} s unque up to an somorphsm. Choose {n,n,n }={n,n,n } such that n n and n n. Ceary, there are edges of coor between each of the pars (G,G ) and (G,G ). Wthout oss of generaty, et x () x (),x (n ) x () γ (). Case. If m () = 0 then the path P 5 (x (n ),x (),x (n ),x (),x (),x (n ) ) must be a part of the 8-cyce of Ɣ {0,}. Snce the m () (resp., m () ) edges of coor c between the par (G,G ) (resp., (G,G )) yed m () (resp., m () ) b-coored -cyces, we have x () x (),...,x (m() ) x (m() ) and x (n ) x (),...,x (n + m () ) x (m() ) γ (). In ths case, m () = n, m () = n and n = n + n. Case. If m (), then x (n ) x (n ) γ () competes the 6-cyce n Ɣ {0,}. Therefore, by smar reasons as above, {x () x (),..., x (m() ) x (m() ) }, {x (n ) x (),..., x (n + m () ) x (m() ) } and {x (n ) x (n ),..., x (n + m () ) x (n + m () ) } are the sets of edges of coor. In ths case, m () + m () = n, m () + m () = n and m () + m () = n. Thus, Ɣ {0,,} s unque up to an somorphsm. We use whte dot for vertex x () and back dot for vertex x (n ). Snce Ɣ {0,,} s bpartte graph, x (p ),x (p ), x (p ) are denoted by whte dots and x (p ),x (p ),x (p ) are denoted by back dots for p n /, p n / and p n /. Step 7. Now, we are ready to construct a crystazaton (Ɣ, γ ) whch yeds the reatons n R. For a gven set of reatons R, we constructed Ɣ {0,,} unquey. By smar arguments as above, we have m (), m(). Choose an edge x (q ) x (q ) γ (). Snce q n /, there are n / choces for such an edge. Now, choose two edges x (q ) x (q ) respectvey. Then, ether x (q ) x (q ),x (q ) x (q ), x (q ) x (q ) x (q ) γ (). Thus, ether P 5 (x (q ) and x (q ) x (q ) γ () n G and G γ () or x (q ), x (q ), x (q ), x (q ) x (q ),, x (q ) )
68 Bpab Basak or P 5 (x (q ), x (q ), x (q ), x (q ), x (q ), x (q ) ) s a path n Ɣ {,}. Therefore, by smar arguments as n Step 6, there s a unque way to choose the remanng edges of coor. Snce q n /, q n / and q n /, we have n n n = n n n choces for the -coored graph. Step 8. For a set R of reatons there are (n n n )/ choces for the -coored graph. If there s a choce, for whch (Ɣ, γ ) yeds the reatons n R then (Ɣ, γ ) s a reguar bpartte -coored graph whch satsfes a the propertes of Proposton.5. Therefore, (Ɣ, γ ) s a crystazaton of a cosed connected orentabe -manfod M whose fundamenta group s ( S R,R). Now, we choose a dfferent R {R,...,R k } and repeat the process from Step to fnd a possbe λ( S R,R)-vertex crystazatons whch yed ( S R,R). If there s no such -coored graph for each R {R,...,R k } then there s no crystazaton of a cosed, connected orentabe -manfod, whch yeds ( S R,R) and s mnma wth respect to ( S R,R). Theorem.. Let M be a cosed connected orentabe prme -manfod wth fundamenta group S R, where #S = #R =. Let (Ɣ, γ ) be a crystazaton constructed from the par ( S R,R)by usng the above constructon. Then, we have the foowng. () If S R = Zp, then K(Ɣ) = M. () If S R = Z p, then K(Ɣ) = L(p, q ) for some q {,...,p }. () (Ɣ, γ ) s mnma wth respect to the par ( S R,R). Proof. Snce (Ɣ, γ ) yeds the presentaton S R, by Proposton.6, π ( K(Ɣ), ) = S R. Snce Ɣ s reguar and bpartte, K(Ɣ) s a cosed connected orentabe - manfod. Snce M s prme -manfod, the fundamenta group of M s not a free product of two groups. Snce M and K(Ɣ) have the same fundamenta group, t foows that K(Ɣ) s aso prme -manfod. Part () now foows from Proposton.. If π ( K(Ɣ), ) = π (M, ) = Z p then, by the proof of eptzaton conecture, K(Ɣ) s spherca and hence K(Ɣ) = S /Z p = L(p, q ) for some q {,...,p }. Ths proves part (). Part () foows from the constructon.. Agorthm Now, we present our agorthm whch fnds crystazatons of -manfods for a presentaton ( S R,R)wth two generators and two reatons, such that the crystazatons yed the par and have the number of vertces s equa to the weght of the par ( S R,R). () Fnd the set {w R, k} of ndependent words such that λ(w ) s mnmum. Let R = R {w } and consder a cass of graphs C whch s empty. () For R, (a) fnd m (c) for c and <, (b) fnd n,n,n and (c) choose,, such that n n and n n. () Consder three b-coored cyces G = C n (x () x ( ) x () has coor and x () x (+) x () for n / and.,...,x (n ) ) for such that has coor 0 wth the consderaton x (n +) =
An agorthmc approach to construct crystazatons 69 (v) The edges x () x (),...,x (m() ) x (m() ) and x (n ) x (),...,x (n + m () ) x (m() ) have coor. If n + n = n, then the edges x (n ) x (n ),...,x (n + m () ) x (n + m () ) have aso coor. (v) For q n /, q n / and q n /, choose a b-coored path of coors and from the two paths P 5 (x (q ), x (q ), x (q ), x (q ), x (q ),x (q ) ) and P 5 (x (q ),x (q ),x (q ),x (q ),x (q ),x (q ) ) and on the remanng vertces by edges of coor as there s an unque way to choose these edges wth the chosen path. There are (n n n )/ choces for -coored graphs. If some graphs yed ( S R,R) then put them n cass C. (v) If R = R {w },forsome {,...,k }, choose R = R {w + } and go to step (). If R = R {w k }, then C s the coecton of a crystazatons whch yed ( S R,R) and are mnma wth respect to ( S R,R).IfC s empty then such a crystazaton does not exst.. Appcatons of agorthm From a gven presentaton S R wth two generators and two reatons, usng our agorthm, we construct a the possbe λ( S R,R)-vertex crystazatons whch yed ( S R,R). We aso dscuss the cases, where no such crystazaton exsts.. Constructons of some crystazatons Here we consder the cases where Agorthm gves crystazatons for a par ( S R, R). Exampe. (Crystazatons of M m, n, for m, n ). Reca that the bnary poyhedra group m, n, k has a presentaton x,x,x x m = xn = xk = x x x for some nteger m, n, k. If k =, then x = x x and hence x m = x x x x and x n = x x x x. Therefore, m, n, has a presentaton S R, where S ={x,x } and R = {x m x x x,xn x x x }. It s not dffcut to prove that xm x n s the ony ndependent eement n R of mnmum weght. Therefore, R = R {x mx n }. Thus, m(c) =, m(c) = m, m(c) = n for c, (n,n,n ) = (m+, n+, m+n ) and G = C n (x (),...,x (n ) ) for as n fgure. Choose (n,n,n ) = (n,n,n ). Thus, {x () x(),...,x(n ) x (n ) }, {x (m+n ) x (),...,x(n) x (m ) } and {x (m+) x (n+),x (m+) x (n+),x (m) x (n) } are the sets of edges of coor. Here the ony choce for the trpet (q,q,q ) s (,n,n ) and for the path P 5 s P 5 (x (),x(),x(n ),x (n ),x (n 5),x (n ) ). They gve a reguar bpartte -coored graph (Ɣ, γ ) whch yeds ( S R,R) (cf. Lemma.). Therefore, by Theorem., (Ɣ, γ ) (cf. fgure ) s a crystazaton of the cosed connected orentabe -manfod M m, n, (cf. subsecton.) and mnma wth respect to ( S R,R). Observe that () #V(Ɣ)= m + n + = (m + ) + (n + ) + (m + n ) = λ( S R,R), () Ɣ {,} = (m + n )C C 6 for 0 and as#v(ɣ)= (m + n ) +, and () the m (c) edges of coor c between G and G yed m (c) b-coored -cyces for < and c.
60 Bpab Basak Fgure. Crystazaton of M m, n, for m, n. Lemma.. Let the presentaton ( S R,R) and q,q,q,p 5 be as n Exampe.. Then, the choce of the trpet (q,q,q ) and the path P 5 for whch the -coored graph (Ɣ, γ ) yeds ( S R,R),s unque. Proof. Snce x m x x x (resp., x mx n ) s a reaton whch contans xm (resp., x m), m (resp., m ) edges of coor between G and G are nvoved to yed x m (resp., x m ). Snce Ɣ s bpartte, m() = m and G has m + vertces, ether whte dots vertces (resp., back dots vertces) or back dots vertces (resp., whte dots vertces) n G are nvoved to yed x m (resp., x m ). Agan, the fact that m () = and {x(n+), x (n) } s a set of whte dots vertces mpes that one of the vertces x (n+), x (n) s the startng vertex for one of the two reatons above. Thus, the back dot vertexx (n+) s the startng vertex for the other reaton. Up to an automorphsm, we can assume that x (n) and x (n+) are the startng vertces for the two reatons above. Therefore, x (m) and x (m+) are oned wth vertces of G or by edges of coor. Snce the reaton wth startng vertex x (n+) contans x m x m, ether {x(m+) x (n), x (m ) x (n+),..., x (7) x(m+n 6) }or{x (m+) x (m+n ), x () x(m+n ),..., x (m 7) x (n+) } s the set of edges of coor. But, n the atter case, x (m) x () γ () as x (m) and x (m+) are oned wth vertces of G by edges of coor. Then, the reaton wth startng vertex x (n) contans x x wx for some w F(S), whch s a contradcton. Therefore, x (m+) x (n), x (m ) x (n+),..., x (7) x(m+n 6) γ () and hence x (m) x (n+), x (m ) x (n+),..., x (6) x(m+n 5) γ (). Snce one reaton contans x m and other contans xm, x (5) x(m+n ) γ () and hence x (), x (), x() are oned wth G wth edges of coor. Thus, x (m+) x (n ), x () x(n ) γ () as m () = m. Therefore, x(n+) s the startng vertex for the reaton x m x n. In other words, we can say that x (m+) s the startng vertex for the reaton x n x m.
An agorthmc approach to construct crystazatons 6 Therefore, x (m+) s the startng vertex for the reaton x n x x x. Thus, by smar arguments as above, x (n ) x (m+n ), x (n) x (m+n ),..., x (n 5) x (n ) γ (). Therefore, x () x(n ), x () x(n ), x () x(n ) γ (). Thus, the emma foows. Exampe. (Crystazaton of L(kq,q), for k, q ). We know Z kq has a presentaton S R, where S ={x,x } and R ={x q x,xk x }. It s not dffcut to prove that x q x k s the ony ndependent eement n R of mnmum weght. Therefore, R = R {x q x k }. Thus, m (c) =, m(c) = q, m(c) = k for c, (n,n,n ) = (q,k, q + k ) and G = C n (x (),...,x (n ) ) for as n fgure. Choose (n,n,n ) = (n,n,n ). Thus, {x () x(),...,x(k ) x (k ) }, {x (q+k ) x (),...,x(k ) x (q ) } and {x (q ) x (k ),x (q) x (k) } are the sets of edges of coor. Here the ony choce for the trpet (q,q,q ) s (q,k,k ) and for the path P 5 s P 5 (x (q ),x (q ), x (k ),x (k ),x (k ),x (k) ). They gve a reguar bpartte -coored graph (Ɣ, γ ) whch yeds ( S R,R)(cf. Lemma.). Therefore, by Theorem., (Ɣ, γ ) (cf. fgure ) s a crystazaton of a cosed connected orentabe -manfod and mnma wth respect to ( S R,R).Here(Ɣ, γ ) s a crystazaton of L(kq,q) as (Ɣ, γ ) s somorphc to the graph M k,q (cf. subsecton 5. of []) and M k,q s a crystazaton of L(kq,q). Observe that () #V(Ɣ)= (q + k ) = q + k + (q + k ) = λ( S R,R), and () Ɣ {0,} (resp., Ɣ {,} ) s a unon of (q + k ) -cyces and Ɣ {0,} (resp., Ɣ {,} )s of type (q + k )C C 6 as #V(Ɣ)= (q + k ) and m (c) for c and <. Lemma.. Let the presentaton ( S R,R) and q,q,q,p 5 be as n Exampe.. Then, the choce of the trpet (q,q,q ) and the path P 5 for whch the -coored graph (Ɣ, γ ) yeds ( S R,R),s unque. Fgure. Crystazaton of L(kq, q) fork, q.
6 Bpab Basak Proof. Ceary, ether x (k) or x (k ) s the startng vertex for the reaton x q x.uptoan automorphsm, we can assume that x (k) s the startng vertex for the reaton x q x. Snce Ɣ s bpartte, a the back dots vertces n G the reaton x q x {x (q) x (q+k 5),x () x(q+k 7),...,x (q ) x (k ) are nvoved to yed. Thus, ether {x(q) x (k ),x (q ) x (k+),..., x () x(q+k 5) } or } s the set of edges of coor. Snce the q edges of coor between G and G form q b-coored -cyces n Ɣ {0,,}, n the frst case, x (q ) x (k),x (q ) x (k+),...,x (5) x(q+k 6) γ (). Ths gves a reaton x q wx other than x q x for some w F(S). But, there does not exst such a reaton. So, x (q) x (q+k 5),x () x(q+k 7),...,x (q ) x (k ) γ () and s the hence x () x(q+k 6),x () x(q+k 8),..., x (q 5) x (k) γ (). Snce x (k) startng vertex for the reaton x q x, we have x(q ) x (k) γ () and hence x (q ) x (),x(q ) x (k ) γ () as a components of Ɣ {0,} are -cyce. By sm- γ () as ar arguments as above, x (k ) x (k ), x (k ) x (k ),..., x () x(q+k ) s the startng vertex for the reaton x kx. Thus, the emma foows. x (q ) Exampe.5 (Crystazaton of L((k )q +,q) for k, q ). We know Z (k )q+ has a presentaton S R, where S ={x,x } and R ={x q x,xq x k }. It s not dffcut to prove that x x k s the ony ndependent eement n R of mnmum weght. Therefore, R = R {x x k }. Thus, m(c) =, m(c) = q, m(c) = k for c, (n,n,n ) = (q,k, q + k ). Choose (n,n,n k ) = (n,n,n ). Therefore, the -coored graph wth coors 0,, as n prevous exampe. Here, the ony choce for the trpet (q,q,q ) s (q,k,) and for the path P 5 s P 5 (x (q ),x (q ),x (),x(),x(k),x (k ) ). They gve a reguar bpartte -coored graph (Ɣ, γ ) whch yeds ( S R,R) (cf. Lemma.6). Therefore, by Theorem., (Ɣ, γ ) (cf. fgure ) s a crystazaton of a cosed connected orentabe -manfod and mnma wth respect to ( S R,R).Here(Ɣ, γ ) s a (k + q )-vertex crystazaton of L((k )q,q)as (Ɣ, γ ) s somorphc to the graph N k,q (cf. subsecton 5. of []) and N k,q s a crystazaton of L((k )q,q). Lemma.6. Let the presentaton ( S R,R) and q,q,q,p 5 be as n Exampe.5. Then, the choce of the trpet (q,q,q ) and the path P 5 for whch the -coored graph (Ɣ, γ ) yeds ( S R,R),s unque. Proof. Wthout oss of generaty, we choose x (k ) as the startng vertex for the reaton x q x. Thus, ether {x(q ) x (k), x (q ) x (k+),..., x () x(q+k ) } or {x (q ) x (q+k ), x () x(q+k 6),..., x (q 5) x (k) } s the set of edges of coor. By smar arguments as n the proof of Lemma., we get a reaton x kx n the frst case. Therefore, we have to choose the second case. Agan, by smar arguments as n the proof of Lemma., the emma foows. Exampe.7 (Crystazaton of a hyperboc -manfod). Let S R be a presentaton of a group, where S = {x,x } and R = {x x x x x x x x x x, x x x x x x x x x x }. It s not dffcut to prove that x x x x x x R s the ony ndependent eement of mnmum weght.
An agorthmc approach to construct crystazatons 6 Fgure. Crystazaton of L((k )q +,q)fork, q. Therefore, R = R {x x x x x x }. Thus,m(c) =,m(c) = 7,m(c) = 7 for c, (n,n,n ) = (8, 8, ) and G = C n (x (),..., x (n ) ) for as n fgure. Choose (n,n,n ) = (n,n,n ). Thus, {x () x(),..., x () x () }, {x (8) x (),...,x() x (7) } and {x () x (8),...,x (8) x () } are the sets of edges of coor. Here the ony choce for the trpet (q,q,q ) s (8, 5, 0) and for the path P 5 s P 5 (x (0),x (9),x (6),x (5),x (9),x(0) ). They gve a bpartte - coored graph (Ɣ, γ ) (cf. fgure ) whch yeds ( S R,R) (cf. Lemma.8). Snce S R s not a free product of two non trva groups, K(Ɣ) s prme. Now, S R = x,x x x x x x x x x x x,x x x x x x = x,x x x x x x x x x x x,x x x x x x whch s the presentaton of the fundamenta group of a cosed, connected orentabe prme hyperboc -manfod (see the presentaton n http://www.dms.umontrea.ca/~math/logces/magma/text.htm). Therefore, by Proposton., K(Ɣ) s homeomorphc to the hyperboc -manfod as n the st. Thus, usng the agorthm we get a crystazaton of the hyperboc -manfod from a gven presentaton. Observe that () #V(Ɣ) = 70 = 8 + 6 + 6 = λ( S R,R) and () Ɣ {,} = 6C C 6 for 0, as#v(ɣ)= 70 = 7 +. Lemma.8. Let the presentaton ( S R,R) and q,q,q,p 5 be as n Exampe.7. Then, the choce of the trpet (q,q,q ) and the path P 5 for whch the -coored graph (Ɣ, γ ) yeds ( S R,R),s unque. Proof. From the dscussons n the proofs of prevous emmas t s cear that, f x ε x±m x ε s a part of a reaton for some ε,ε {, }, m and = then to yed x ε x±m x ε, there are exacty m vertces of G have both the -adacent vertces and the -adacent vertces n G. There are three dfferent words of type x ε x xε and three dfferent words of type x ε x xε n the reatons n R. Therefore, exacty
6 Bpab Basak Fgure. Crystazaton of a hyperboc -manfod. (( ) + ( ) =) 5 vertces of G have both the -adacent vertces and the - adacent vertces n G. By smar arguments as above, (( ) + ( ) =) 5 vertces of G have both the -adacent vertces and the -adacent vertces n G. Snce there are eght dfferent words of type x x x and one word of type x x x n the reatons n R, exacty (8 + =) 9 vertces of G have both the -adacent vertces and the -adacent vertces n G. Snce m () = m() = 7 and 5 vertces of G have both the -adacent vertces and the -adacent vertces n G, x (+m),...,x (5+m),0 m are the choces of these vertces. If m = then, ether x () or x (7) s oned to G wth an edge of coor as m () = 7. Snce exacty 5 vertces of G are oned to G wth both edges of coors and, m =. Up to an automorphsm, we can assume x () s not such a vertex,.e., m =. Therefore, x () and x () are oned to G wth edges of coor. Let x () x(6+n),...,x (9) x (0+n) γ () for some nteger n. If n, then there s a reaton contanng x 5 and f n 8orn 8, then there s no reaton contanng x and therefore both cases are not possbe. Therefore, n =±6. If m = and
An agorthmc approach to construct crystazatons 65 n = 6, then x () x(0),...,x (9) x () γ (). Ths contradcts that 9 vertces of G have both the -adacent vertces and the -adacent vertces n G. Therefore, m = and n = 6 and hence x () x(),x() x(),...,x(9) x (6) γ (). Snce x (8) and x (9) are aready oned to G wth edges of coor, the remanng vertces x (0),...,x () are oned to G wth both edges of coors and. Snce the reatons wth startng vertces x (8) and x (9) yed x w for some w F(S), x(0) x (+k),...,x () x (8+k) γ () for k =. So, x (5) x (9) γ (). Thus, the emma foows. Remark.9 If (Ɣ, γ ) s a crystazaton of a -manfod then the reguar genus of Ɣ s the nteger ρ(ɣ) = mn{g 0,g 0,g 0 } (cf. secton of []). The crystazatons constructed n Exampe. for n = k = and n Exampes. and.5 are crystazatons of hande-free manfods. Thus, by Proposton of [6] and from the cataogue n [], these crystazatons are vertex-mnma reguar genus two crystazatons when the number of vertces of the crystazatons are at most. The crystazatons constructed n Exampes. and.5 are vertex-mnma for a known cases. In fact, the crystazatons of L((k )q +,q)are vertex-mnma when (k )q + are even (cf. [, 8, 9]).. Non exstence of some crystazatons Here we consder the cases where Agorthm determnes the non exstence of any crystazaton for a par ( S R,R). Exampe.0 (For a presentaton of Z 6 ). Let ( S R,R)be a presentaton of the cycc group Z 6, where S ={x,x } and R ={x x,x x }. Ceary, x s the ony ndependent eement n R of mnmum weght. Therefore, R = R {x } and et (Ɣ, γ ) be a crystazaton reazng the above presentaton. Thus, m (c) =, m(c) = 5, m(c) = for c and (n,n,n ) = (6,, 8). Therefore, by choosng (n,n,n ) = (n,n,n ),wehave Ɣ {0,,} as n fgure 5. If x (q ) x (q ) = x (7) x(8),x() x() or x (5) x(6) then, for each of the two choces of the path P 5, ether x () or x () s oned wth G wth edges of coor. Snce the graph s Fgure 5. The graph Ɣ {0,,}.
66 Bpab Basak bpartte, nether x () x() nor x () x() can be edge of coor n Ɣ. Therefore, no components of Ɣ {,} yed the reaton x. For the same reason, we can not choose x(q ) x (q ) = x () x() and the path P 5 (x (q ),x (q ),x (),x(),x(q ),x (q ) ). Therefore, the ony remanng choce s x (q ) x (q ) = x () x() and the path P 5 (x (q ), x (q ), x (), γ () (snce x (), x(q ), x (q ) ). Thus, x () s oned wth G and hence x () x(),x() ) s part of a com- x x s not a part of reatons n R, ths choce aso s not possbe. Thus, there s no crystazaton of a cosed connected -manfod whch yeds ( S R,R)and s mnma wth respect to ( S R,R). Ɣ s bpartte and has no doube edge). Then, P (x (6),x(),x() ponent of Ɣ {,}, whch yeds the word x. Snce x Exampe. (For a presentaton of Z mn+n+, m, n ). Let ( S R, R) be the presentaton of Z mn+n+, where S = {x,x } and R = {x m+ x, x x n } and m, n. It s not dffcut to prove that {x m x xm x n : m + m = m, m, m } s the set of a ndependent eements n R of mnmum weght. So, R = R {x m x xm x n } and et (Ɣ, γ ) be a crystazaton reazng the above presentaton. Thus, m (c) =, m(c) = m, m (c) = n for c and (n, n, n ) = (m +, n +, m + n ). By choosng (n,n,n ) = (n,n,n ), we have Ɣ {0,,} as n fgure. From the dscussons n the proof of Lemma.8, t s cear that, snce there s a reaton x m+ x and there are two words x xm, x xm n another reaton n R, exacty m + m + x x m = m vertces of G have both the -adacent vertces and the -adacent vertces n G. But, to yed the reaton x m+ x, exacty m vertces of G have both the -adacent vertces and the -adacent vertces n G. Snce Ɣ s bpartte, {x (n), x (n+),..., x (m+n ) } s the ony possbe set of those m vertces. Therefore, x (n+),x (n+),...,x (m+n ) are aso oned to G wth edges of coor as a the m () edges of coor between G and G yed m () b-coored -cyces n Ɣ {0,,}. Thus, we get m + m = m vertces of G have both the -adacent vertces and the -adacent vertces n G, whch s a contradcton. Therefore, there s no choce for the trpet (q,q,q ) whch yeds the reatons. Thus, there s no crystazaton of a cosed connected -manfod whch yeds ( S R, R) and s mnma wth respect to ( S R, R). 5. Generazaton of Agorthm In, we have computed crystazatons of -manfods from a gven presentaton ( S R,R)wth two generators and two reatons. For such a presentaton, Ɣ {0,,} and Ɣ {0,,} were unque up to an somorphsm. But, f the gven presentaton ( S R,R) has the number of generators and reatons greater than two then Ɣ {0,,} and Ɣ {0,,} may have many choces. But, there are some casses of presentatons, for whch Ɣ {0,,} and Ɣ {0,,} are unque up to an somorphsm. In ths secton, we generaze Agorthm for a presentaton ( S R,R) wth three generators and a certan cass of reatons. Let r R be an eement of mnmum weght and R = R {r}. Let m (c) := w R w(c) for < and c, where w (c) s as n subsecton.. LetC R := {w,...,w k } be the set of a ndependent eements n
An agorthmc approach to construct crystazatons 67 R such that () the weght of w s mnmum and () for each R {w },wehavem (c) w R {w } w(c) =, where < and c. Let (Ɣ, γ ) be a crystazaton of a -manfod such that (Ɣ, γ ) s mnma wth respect to ( S R,R)and yeds R = R {w}, where w C R. Wthout oss of generaty, et Ɣ {0,} = = G such that G represents the generator x for and G represents x (cf. eq. (.) for constructon of r). Let n be the tota number of appearances of x n the four reatons n R for and n = λ( S R,R) (n + n + n ). Then, the tota number of vertces n G shoud be n. Assume G = C n (x (),...,x (n ) ) for. Ceary, each n s even and n + n + n + n = #V(Ɣ). Wthout oss of generaty, we can assume that x ( ) x () γ () and x () x (+) γ (0) wth x (n +) = x () for n / and. Here and after, the addtons and subtractons at the pont nx ( ) are moduo n for. Let the coors and be the coors and respectvey as n constructon of r for r R (cf. eq. (.)). Then, the number of edges of coor c between G and G s m (c) for c and <. Therefore, m := < m(c) = #V(Ɣ)/. Now, the maxmum number of b-coored -cyces n Ɣ {0,,c} wth two edges of coor c s < (m(c) ) = m 6. Agan, by Proposton.5, + g 0c + g c = #V(Ɣ)/ +,.e., g 0c + g c = m. Snce G and G are connected by an edge of coor c for < and c, we have at east four dstnct b-coored paths P 5 wth some edges of coor c n Ɣ {0,,c} whch touch G,G,G for a dstnct,, {,,, }. Therefore, we must have m 6 b-coored -cyces and four b-coored 6-cyces wth some edges of coor c n Ɣ {0,,c}. Thus, m (c) edges of coor c between G and G yed m (c) b-coored -cyces n Ɣ {0,,c} for < and c. Therefore, two b-coored 6-cyces wth some edges of coor c n Ɣ {0,,c} gve unque choces for the remanng edges of coor c. Wthout oss, we can assume that Ɣ {0,} has a 6-cyce C 6 (x (),x(n ),x (n ),x (),x(n ) on x (),...,x(m() x() ) ) x (m(),x () ). Then, by edges of coor. Wthout oss of generaty, choose x (p) G such that C 6 (x (m() ),x (m() +),x (p),x (p+),x (m() +),x (m() ) ) s a b-coored cyce wth three edges of coor. Therefore, we have a unque choce for Ɣ {0,,} up to an somorphsm. The choces of two 6-cyces wth three edges of coor n Ɣ {0,,} gve a possbe -coored graphs. If some graphs yed ( S R,R), then these satsfy a the propertes of Proposton.5 and hence they are crystazatons of some -manfods. By smar arguments as n the proof of Theorem., f M s a cosed connected prme manfod wth fundamenta group ( S R,R)and (Ɣ, γ ) s a crystazaton, constructed from the par ( S R,R), then (Ɣ, γ ) s a crystazaton of M. 5. Agorthm We now present an agorthm for a presentaton ( S R,R) wth #S = #R = and C R =. Ths agorthm gves a crystazatons whch yed the reaton set R {w}, where w C R and are mnmum wth respect to ( S R,R). () Fnd the set {w R, k} of ndependent words such that λ(w ) s mnmum and for each R {w },wehavem (c) = w R {w } w(c), where < and c. Let R = R {w } and consder a cass of graphs C whch s empty.
68 Bpab Basak () For R, (a) fnd m (c) for c and < and (b) fnd n,n,n,n. () Consder four b-coored cyces G = C n (x (),...,x (n ) ) for such that x ( ) x () has coor and x () x (+) has coor 0 wth the consderaton x (n +) = x () for n / and. (v) The sets {x (),...,x(m() x() ) x (m() ) }, {x (n ) x (n ),...,x (n + m () ) x (n + m () ) } and {x (n ) x (),..., x(n + m () ) x (m() ) } contan edges of coor. Wthout oss of generaty, choose x (p) G such that C 6 (x (m() ),x (m() +),x (p),x (p+),x (m() +), x (m() ) ) s a b-coored cyce wth three edges of coor. Therefore, the edges of the sets {x (m() +) x (p+),..., x (m() +m() ) x (p+m() ) }, {x (p+m() +) x (m() +),..., x (p+m() +m() ) x (m() +m() ) } and {x (m() +),...,x (m() +m() ) x (p+ m() ) }haveaso x (p) coor. (v) For each q n, choose x (q ) G such that {x (q ) x (q ),..., x (q +m () ) x (q +m () ) } γ (). Then, choose x (q ) G and x (q ) G such that, ether {x (q ) x (q ),x (q +m () ) x (q ) }or{x (q ) x (q ),x (q +m () ) x (q ) } contans edges of coor. There are n n n n = n n n n choces for choosng these vertces and edges. Then, for each choce, on the remanng vertces by edges of coor as there s an unque way to choose the remanng edges wth the known m () + edges of coor. If some graphs yed ( S R,R)then put them n the cass C. (v) If R = R {w },forsome {,...,k }, choose R = R {w + } and go to step (). If R = R {w k } then C s the coecton of a crystazatons whch yed ( S R,R) and are mnma wth respect to ( S R,R).IfC s empty, then such a crystazaton does not exst. 5. Constructons of crystazatons of M m, n, k Reca that M m, n, k s the cosed connected orentabe -manfod wth the fundamenta group m, n, k whch has a presentaton ( S R mnk,r mnk ), where S ={x,x,x } and R mnk ={x m x x, xn x x,xk x x }. It s not dffcut to prove that x m x xn x xk x s the ony ndependent eement n R mnk of mnmum weght. Observe that m f k = n =,m, λ(x m x xn x xk x )= m+n 6 f k =,m,n, m+n+k f m, n, k. Therefore, (m + ) f k = n =,m, λ( S R mnk,r mnk ) = (m + n) f k =,m,n, (m + n + k ) f m, n, k. Snce m (c) for the set R mnk {x m x xn x xk x }, where < and c, we have x m x xn x xk x C Rmnk. Thus, we can appy Agorthm. Reca that M m,, = S /Q m.
An agorthmc approach to construct crystazatons 69 Theorem 5.. For m,s /Q m has a crystazaton wth (m + ) vertces whch s unque and mnma wth respect to ( S R m,r m ). Proof. Snce C Rm {x m x x x = for m, we can appy Agorthm. Let R = R m }. Thus, m() = m() = m() = m() = m() = m() =, m() = m () = m() = m() = and m() = m() = m. Observe that, (n,n,n,n ) = (m,,, m) and G = C n (x (),...,x (n ) ) for asnfgure6. Choose x (p) = x () as n Agorthm, then the -coored graph wth the coor set {0,, } s as n fgure 6, whch s unque up to an somorphsm. For the choces (q,q,q,q ) = (5,,, ) and (x (q ) x (q ),x (q +m () ) x (q ) ) = (x () x(),x(6) x() ), we get a -coored graph whch yeds ( S R m,r m ). Therefore, for each m, we get a crystazaton (Ɣ, γ ) of the -manfod S /Q m. Now, we show that the crystazaton (Ɣ, γ ) s unque. Here we choose the par of coors (, ) = (, ) as n the constructon of r for r R (cf. eq. (.)). From the constructon of r for r R, t s cear that, ether x () or x () s the startng vertex v of the component of Ɣ {,}, whch yeds the reaton x m x x (resp., x m x x x ). If possbe, et x () be the startng vertex to yed the reaton x m x x then x () s the startng vertex Fgure 6. Crystazaton of S /Q m for m.
650 Bpab Basak to yed the reaton x m x x x and x () x(m ),x (m ) x (m ),...,x (7) x(5) γ () (as Ɣ s bpartte). Snce m () edges of coor yed m() b-coored - cyce n Ɣ {0,,},wehavex (m) x (m ), x (m ) x (m ),...,x (8) x(6) γ (). Snce x () x(m) γ (), the component of Ɣ {,} wth startng vertex x () yeds a reaton x m wx for some w F(S), whch s not possbe. Thus, x () s the startng vertex γ () and to yed the reaton x m x x and x () x(m),x () x(m ),..., x (8) x(6) x (7) x(5) γ (). Therefore, x () x(),x() x() γ () as m () = m. To yed the reaton x m x x,wehavex(6) x(),x() x() γ (). Snce x () x() γ (), we have x () x() γ () and hence C (x (),x(),x(),x() ) s the component of Ɣ {,} whch yeds the reaton x x x wth startng vertex v = x (). Now, there s an unque way to choose the remanng edges of coor as n fgure 6. Snce x m x ony ndependent eement n R m of mnmum weght, the theorem foows. x x s the Remark 5.. For m =, there s no crystazaton of S /Q m wth (m + ) = 6 vertces (cf. []). Theorem 5.. For m, n, M m, n, has a crystazaton wth (m + n) vertces whch s unque and mnma wth respect to ( S R mn,r mn ). Proof. Snce C Rmn {x m x xn x x n, m () = m() = m() = m() = for m, n, we can appy Agorthm. Let R = R mn }. Thus, m() = m() = m() = m() =, m() = n,m() = =, m() = m and m() = m. Observe that (n,n,n,n ) = (m, n,, (m + n )) and G = C n (x (),...,x (n ) ) for as n fgure 7. Choose x (p) = x () as n Agorthm, then the -coored graph wth the coor set {0,, } s as n fgure 7, whch s unque up to an somorphsm. For the choces (q,q,q,q ) = (, n,, ) and (x (q ) x (q ), x (q +m () ) x (q ) ) = (x () x(), x(6) x() ), we get a -coored graph whch yeds ( S R mn,r mn ). Therefore, for each m, n, we get a crystazaton (Ɣ, γ ) of the -manfod M m, n,. Now, we show that the crystazaton (Ɣ, γ ) s unque. Here we choose the par of coors (, ) = (, ) as n the constructon of r for r R (cf. eq. (.)). By smar arguments as n the proof of Theorem 5., x () s the startng vertex to yed the reaton γ () and x m x x and hence x () x(m+n 6),x () x(m+n 5),...,x (8) x(n) x (7) x(n ) γ (). Therefore, x () x() γ () as m () = m. Snce m() =, to, we have x(6) x(),x() x(n ) γ (). Smary, x () s yed the reaton x m x x the startng vertex to yed the reaton x n x x and hence x () x(n ),x () x(n 5),...,x (n ) x () γ () and x (n ) x () γ (). Therefore, x () x(n ), x () x(n ) γ () as m () = n and hence x(7) x(), x() x(n ), x () x(n ) γ () to yed the reaton x m x xn x x. Now, there s an unque way to choose the remanng edges of coor as n fgure 7. Snce x m x xn x x s the ony ndependent eement n R mn of mnmum weght, the theorem foows. Theorem 5.. For m, n, k, M m, n, k has a crystazaton wth (m + n + k ) vertces whch s unque and mnma wth respect to ( S R mnk,r mnk ).