MICC: A tool for computing short distances in the curve complex
Paul Glenn, William W. Menasco, Kayla Morrell, Matthew Morse
aa r X i v : . [ m a t h . G T ] A ug MICC: A TOOL FOR COMPUTING SHORT DISTANCES IN THECURVE COMPLEX
PAUL GLENN, WILLIAM W. MENASCO, KAYLA MORRELL, AND MATTHEW MORSE
Abstract.
The complex of curves C ( S g ) of a closed orientable surface of genus g ≥ C ( S g ), are isotopy classes of essential simpleclosed curves in S g . Two vertices co-bound an edge of the 1-skeleton, C ( S g ), if there aredisjoint representatives in S g . A metric is obtained on C ( S g ) by assigning unit length toeach edge of C ( S g ). Thus, the distance between two vertices, d ( v, w ), corresponds to thelength of a geodesic—a shortest edge-path between v and w in C ( S g ). In (4), Birman,Margalit and the second author introduced the concept of efficient geodesics in C ( S g ) andused them to give a new algorithm for computing the distance between vertices. In thisnote, we introduce the software package MICC ( Metric in the Curve Complex ), a partialimplementation of the efficient geodesic algorithm. We discuss the mathematics underlyingMICC and give applications. In particular, up to an action of an element of the mappingclass group, we give a calculation which produces all distance 4 vertex pairs for g = 2 thatintersect 12 times, the minimal number of intersections needed for this distance and genus. Introduction
Let S or S g denote a compact, connected, orientable surface of genus g , where g ≥ S is essential if does not bound a disk in S . The complex ofcurves, introduced by Harvey (6), is the simplicial complex, C ( S ), whose vertices (or 0-skeleton), C ( S ), are isotopy classes of essential simple closed curves; and, whose edges ofthe 1-skeleton, C ( S ), connect vertices that have disjoint representatives. For the remainderof this note, “curve” will mean “simple closed curve”. By declaring that each edge of C ( S )has length 1, we endow C ( S ) with a metric. Specifically, an edge path is a sequence ofvertices { v = v , v , · · · , v n = w } such that d ( v i , v i +1 ) = 1. A geodesic path joining v and w is a shortest edge-path. The distance , d ( v, w ), between arbitrary vertices is the length of ageodesic path. Since it is known that the complex of curves is connected, which was statedby Harvey (6) and followed from a previous argument of Lickorish (12), the value d ( v, w )is well-defined for all vertex pairs.We note that if d ( v, w ) = 2, there is a vertex ¯ γ ∈ C ( C ) and curve representatives in S , α ∈ v , β ∈ w and γ ∈ ¯ γ , such that α ∩ β = ∅ and γ ⊂ S \ ( α ∪ β ). The generic situation(when some component of S \ ( α ∩ β ) has Euler characteristic less than zero) is that thereare infinitely many isotopically distinct choices for γ ⊂ S \ ( α ∩ β ) and, thus, infinitely manypossible geodesics for distance 2. In this case, the existence of infinitely many geodesics atdistance 2 forces infinitely many geodesics for all distances. It is this infinite local pathologywhich makes finding an effective distance computing algorithm challenging.The curve complex was first introduced by Harvey (6). Its coarse geometric propertieswere first studied extensively by Masur–Minsky (13; 14). The complex of curves has proveda useful tool for the study of hyperbolic 3-manifolds, mapping class groups and Teichm¨ullertheory. In particular, in (14), it was established that Teichm¨uller space is quasi-isometric to the complex of curves and is therefore δ -hyperbolic. Here, δ -hyperbolic means thatgeodesic triangles in C ( S ) are δ -thin: any edge is contained in the δ -neighborhood of theunion of the other two edges. In (1), Aougab established uniform hyperbolicity— δ canbe chosen independent of genus (for g ≥ C ( S ) ((11), Corollary 3.2.6). Later, other algorithms were discov-ered by Shackleton (17) and Webb (19), but none of these algorithms were studied seriouslyfrom the viewpoint of doing explicit computations, and all seem unsuitable for that purpose.Recently Birman, Margalit and the second author (4) have given a new algorithm— the efficient geodesic algorithm —and we have developed an implementation of it called the Metric in the Curve Complex (MICC). Applications of MICC we will present in this noteinclude:(i) establishing that the minimal geometric intersection number for vertices of C ( S )with distance four is 12,(ii) listing of all vertex pairs (up to an action of an element of the mapping class group)of C ( S ) with distance four and having minimally positioned representatives withintersection number at most 25, and(iii) producing an explicit example of two vertices of C ( S ) that have distance four andintersection number 29The key idea in (4) is the introduction of a new class of geodesics, efficient geodesics .They are not the same as the ‘tight geodesics’ that have dominated almost all publishedwork on the curve complex following their introduction in (13; 14), however they share withtight geodesics the nice property that there are finitely many efficient geodesics betweenany two fixed vertices in C ( S ).For convenience, for a pair of curves, ( α, β ), we will refer to a component of ( α ∪ β ) \ ( α ∩ β ) as a segment . We will use a slightly weaked definition for efficient geodesic thanthat given in (4). Definition 1.1.
Let v, w ∈ C ( S ) with d ( v, w ) ≥
3. An oriented path v = v , . . . , v n = w, n ≥
3, in C ( S ) is initially efficient if there are representatives α ∈ v , α ∈ v and α n ∈ v n such that | α ∩ b | ≤ n − b ⊂ α n \ α . We say v = v , · · · , v n = w is efficient if v k , · · · , v n is initially efficient for each 0 ≤ k ≤ n − v n , v n − , v n − , v n − is also initially efficient.The efficient path algorithm is a consequence of the following. Theorem 1.2. (Theorem 1.1 of (4))
Let g ≥ , and let v and w be two vertices of C ( S g ) with d ( v, w ) ≥ . There exists an efficient geodesic from v to w , and in fact there are finitelymany. When n = 3, notice that an efficient geodesic v = v , v , v , v = w yields an oppositelyoriented efficient geodesic, w = v , v , v , v = v . That is, distance 3 vertices have non-oriented efficient geodesics. Thus, for corresponding representatives α , α , α , α , we havethat α (respectively α ) will intersect segments of α \ α (respectively of α \ α ) at mostonce. From this observation, we will establish the following test for distance ≥ ICC: A TOOL FOR COMPUTING SHORT DISTANCES IN THE CURVE COMPLEX 3
Theorem 1.3. ( Distance ≥ Test ) Let v, w be vertices with d ( v, w ) ≥ . Let Γ ⊂ C ( S ) be the collection of all vertices such that the following hold: (1) for ¯ γ ∈ Γ , we have d ( v, ¯ γ ) = 1 ; and (2) for ¯ γ ∈ Γ , there exist representatives α, β, γ of v, w, ¯ γ respectively, such that for eachsegment b ⊂ β \ α we have | γ ∩ b | ≤ .Then d ( v, w ) ≥ if and only if d (¯ γ, w ) ≥ for all ¯ γ ∈ Γ . Moreover, the collection Γ isfinite.Remark . Keeping with our previous observation regarding non-oriented efficient geodesicsat distance 3, we can flip the roles of v and w . Thus, the test can also be stated in termsof d ( v, ¯ γ ′ ) ≥ γ ′ ∈ Γ ′ where:1. If ¯ γ ′ ∈ Γ ′ then d (¯ γ ′ , w ) = 1.2. If ¯ γ ′ ∈ Γ ′ , there exists representatives, α ∈ v , β ∈ w and γ ′ ∈ ¯ γ ′ such that for allsegments, a ⊂ α \ β , we have | a ∩ γ ′ | ≤ ≥ α, β ⊂ S be a pair of curves such that | α ∩ β | is minimal with respectto isotopies of β . That is, α and β are minimally positioned . Determining when α and β are minimally positioned is straightforward due to the bigon criterion (Propostion 1.7,(5))—no disc component of S \ ( α ∪ β ) has exactly two segments of ( α ∪ β ) \ ( α ∩ β ) inits boundary. We say α and β (or ( α, β )) is a filling pair if S \ ( α ∪ β ) is a collection of2-discs. It is readily seen that a pair is filling on S if and only if their corresponding verticesin C ( S ) are at least distance 3 apart. When a minimally positioned pair of curves is notfilling but still intersects, some component of S \ ( α ∪ β ) contains an essential curve. Thus,the corresponding vertices are distance 2 apart. Algorithmically determining whether aminimally positioned pair is filling, or not, requires simple tools coming from classicaltopology. For α ∩ β = ∅ , let N ( α ∪ β ) ⊂ S g be a regular neighborhood. The genus ofthis neighborhood, genus ( N ( α ∪ β )), can be algorithmically computed as discussed in § genus (Σ) = 1 − χ (Σ)+ | ∂ Σ | , where χ (Σ) isthe Euler characteristic and χ ( N ( α ∪ β )) = −| α ∩ β | .) If genus ( N ( α ∪ β )) < g , then acomponent of S g \ ( α ∪ β ) contains an essential curve of S g and the vertices that α and β represent are distance 2 apart. If genus ( N ( α ∪ β )) = g , those vertices are distance at least 3apart. We will see in § filling calculation can be readily implemented. However,if one is handed a nice enough presentation of α and β in S , determining whether they area filling pair can be done by inspection. For example, we will do such filling determinationsin Example 1.5.We now give the proof of Distance ≥ Proof:
From the above discussion we see that the assumption, d ( v, w ) ≥
3, translatesinto considering only minimally positioned filling pairs in S . To determine whether theassociated vertices in C ( S ) of a filling pair are at distance ≥
4, we need only determinethat they are not at distance 3.Thus, suppose α and β represent classes v and w such that d ( v, w ) ≥
3. Assume thereexists a length 3 path v = v , v , v , v = w . From Theorem 1.2, we can further assumethat this path is initially efficient. In particular, for representative α = α , α , α , α = β of the vertices of this path, respectively, we can assume α intersects segments of α \ α at most once. Thus, v is an element of the set Γ. But, since d ( v , v (= w )) = 2, we need PAUL GLENN, WILLIAM W. MENASCO, KAYLA MORRELL, AND MATTHEW MORSE only establish that d (¯ γ, w ) ≥ γ ∈ Γ to contradict the assumption that there was adistance 3 path.The fact that the set Γ is finite is due to S \ ( α ∪ α ) being a collection of 2-discsand representatives of any vertex of Γ having bounded intersection with any such 2-disccomponent. The stated test for ≥ (cid:3) Example 1.5.
We consider an example of a pair of curves, α and β , on a genus 2 surfacewhich represent classes that are distance 4 apart. (See Figure 1.) This is an example of J.Hempel and appears in the lecture notes (16). These notes assert distance 4 for the pairwithout proof. As an application of the Distance ≥ arcs of αβ βγ γ ′ Figure 1.
An example due to J. Hempel
In Figure 1, the surface S is represented as a rectangular region minus two discs.The gray sides of the rectangle are identified, left-to-right and top-to-bottom, to form atorus minus two discs. The genus 2 surface is obtained by identifying the two oriented redboundary curves, and the resulting single curve is β (The identification is initiated by liningup the six colored dots on the α, γ, γ ′ curves). These identifications induce identificationsof the endpoints of the dark blue arcs, so as to form the curve α . By inspection, one cansee that ( α, β ) is a minimally positioned filling pair.We now apply the Distance ≥ γ , that represent vertices,¯ γ ∈ Γ. Such a γ will be in the complement of α , intersecting any segment of β \ α at most ICC: A TOOL FOR COMPUTING SHORT DISTANCES IN THE CURVE COMPLEX 5 once. Three such γ ′ s can be immediately identified. This is because the complement of α ∩ β in S is a collection of some number of 4-gon regions and one single 12-gon region.The boundary of any one of these regions is an alternating joining of segments in α and β .Thus, any 4-gon boundary has two segments in β ; and, the boundary of the single 12-gonhas 6 segments in β . Requiring that any γ intersect segments of β at most once forces itto either not intersect a 4-gon, or intersect each of the two β segments of a 4-gon once.However, there six different ways a γ can exit/enter the 12-gon, giving us three possible γ ′ sthat intersect the 12-gon region once. In Figure 1, the dashed green and purple curves γ and γ ′ illustrate two of the three curves generated by the exit/enter possibilities. It is readilyapparent that both S \ ( β ∪ γ ) and S \ ( β ∪ γ ′ ) is a collection of 2-discs, none of which arebigons. Thus, the corresponding vertex pairs are at least distance 3 apart. The remainingpossibilities for a γ can be dealt with in a similar straightforward manner (theoretically,there are also γ ′ s that intersect the 12-gon region 2 and 3 times). ⋄ The Γ-calculation above illustrates the primary computing capabilities of the MICCsoftware package (15). MICC is a computational tool that can determine whether thedistance between two vertices in C ( S g ≥ ) is 2, 3, or ≥
4. Its input is readily produced fromany representation of two curves on a closed surface. It has functionality that can be usedto search for new curve pairs or manipulate existing examples. Its output can be used toconstruct geodesic paths between curves of short distances.As such, MICC is an additional tool scholars can utilize in answering a number ofbasic questions about the local pathology of the complex of curves. As an illustration, weconsider the relationship between distance and minimal intersection number. It is knownthat the theoretical minimal intersection number for a filling pair on a S g is 2 g − g = 2, this theoretical minimum is notrealizable and the realizable minimum is in fact 4. Recent work of Aougab and Huang (2)has given a construction for realizing the theoretical minimum for g ≥
3. Additionally,they show that all such minimum filling pairs are distance 3. For fixed g ≥
2, using hisuniform hyperbolicity result, Aougab proved that the theoretical minimum intersectionnumber grows exponentially as a function of distance (Theorem 1.2, (1)). Also, Aougaband Taylor (3) give a recipe for producing filling pairs at a given distance whose intersectionnumbers are close to the minimum in an asymptotic sense; see their paper for the precisestatement. Ido, Jang and Kobayashi (10) also have a construction for producing fillingpairs of a prescribed distance. The arguments in these last three citations employs the highpower machinery of Masur and Minsky, including the
Bounded geodesic image theorem (14).Thus, the growth bounds and constructed examples inherit a “coarse geometry” quality,which so far in the literature has not been used to produce the exact minimal intersectionnumber with accompanying filling pairs for a specified distance and genus.In contrast, MICC can be used to find explicitly all minimum intersecting filling pairsof distance 4. Using the Distance ≥ C ( S ) up to homeomorphism.We can next use MICC to calculate distance for curve pairs of increasing intersection numberstarting at this minimum. The result of this calculation is the following theorem. Theorem 1.6.
The minimal intersection for a filling pair, α, β ⊂ S , representing vertices v, w ⊂ C ( S ) , respectively, with d ( v, w ) = 4 is . Combining this theorem with the Distance ≥ PAUL GLENN, WILLIAM W. MENASCO, KAYLA MORRELL, AND MATTHEW MORSE
Corollary 1.7.
Let d ( v, w ) ≥ for two vertices in C ( S ) . Let α, β ⊂ S be curves inminimal position representing v and w , respectively. Let γ ⊂ S be a curve. If (1) γ ∩ β = ∅ and | γ ∩ α | < , or (2) γ ∩ α = ∅ and | γ ∩ β | < ,then d ( v, w ) = 4 . The proof of concept calculation for Theorem 1.6 involves finding all solutions to aninteger linear programing problem so as to identify all potential candidates for minimallyintersecting distance four curve pairs. However, such a comprehensive search is not neces-sarily needed to find examples of distance ≥ C ( S ). In particular, we have the following result: Theorem 1.8.
The minimal intersection number for a pair of filling curves in S thatrepresent distance vertices in C ( S ) is less than or equal to . The outline for our paper is as follows. In §
2, we discuss a method of representing afilling pair on S g ≥ . In §
3, we give the proof of Theorem 1.6. In particular, the proof canbe viewed as giving a general strategy for calculating theoretical minimal intersections ofdistance 4 filling pairs for any higher genus. This strategy employed in a limited mannerallowed us to verify that the explicit example given in (4) (cf. §
2) of a distance 4 vertexpair in C ( S ) establishes Theorem 1.6. We finish § §
4, we discuss the complete functionality ofMICC. To make this discussion concrete, we illustrate the range of MICC commands witha running example.At the end of this manuscript, we attach the current known spectrum of pairs ofdistance 4 or greater in C ( S ) with up to 14 intersections. The full distance 4 or greaterspectrum in C ( S ) with up to 25 intersections is available at (15). An expository remark —Throughout we will continue to use α and β as representatives of v and w , respectively. Similarly, indexed α i curves will be used as representatives of indexed v i vertices. This is meant to be consistent with the notation used in (4). For all othercurves in the surface, we will use various “flavors” of γ , and ¯ γ will denote the correspondingvertex. We will always assume that any pairing of curves are in minimal position.2. Representations of pairs of curves.
Disc with handles.
In deciding how to represent curves on surfaces, we must firstchoose how we will represent closed oriented surfaces. Representing surfaces with boundaryas disc with handles (or
DW H , example in Figure 2) is a well known method among workinggeometers and topologists, and it can be readily adapted to closed surfaces in our situation.For an essential curve α ∈ S g ≥ , we can split S along α to produce a surface ˆ S having twoboundary curves, ∂ + and ∂ − . If α is separating, ˆ S will have two connected components,i.e. ˆ S = ˆ S ∪ ˆ S . The genus of each component will be less than g and their sum would begenus( ˆ S ) + genus( ˆ S ) = g . As such, a DW H representation will have ˆ S (respectively, ˆ S )being a single 2-disc with 2 × genus( ˆ S ) (respectively, 2 × genus( ˆ S )) 1-handles attached.We recover S by gluing the boundary of these two components together. ICC: A TOOL FOR COMPUTING SHORT DISTANCES IN THE CURVE COMPLEX 7
When α is non-separating ˆ S will be a g − ∂ + and ∂ − . A DW H representation of ˆ S would be a single 2-disc with 2 × genus( ˆ S ) + 11-handles attached. As before, S is recovered by giving a gluing of its two boundary curves.Now suppose ( α, β ) is a filling pair in minimal position on S . We obtain a DW H representation of S by splitting along α . Since α and β are in minimal position, we knowthat β ∩ ˆ S will be a collection of properly embedded essential arcs, { ω , · · · , ω k } . We requirethat each 1-handle in our DW H representation contain at least one ω -arc. For example,skipping ahead to Figure 2, the arcs with labels w , w , w are the needed arcs. We referto such an arc as a co-cores of a 1-handle. Skipping ahead to Figure 5, the reader willfind an example of a genus 1 DW H surface with two boundary components. The properlyembedded essential black arcs are examples of ω -arcs and each 1-handle contains at leastone such arc.Since we must be able to recover both β and S by a gluing of ∂ + and ∂ − , we musthave | ( ∪ ≤ j ≤ k ω j ) ∩ ∂ + | = | ( ∪ ≤ j ≤ k ω j ) ∩ ∂ − | . More precisely, for ∂ω = p ∪ p there arethree possible configurations: ω is a ++ arc (respectively, −− arc) when p ∪ p ⊂ ∂ + (respectively, p ∪ p ∈ ∂ − ); and, ω is a + − arc when p ∈ ∂ + and p ∈ ∂ − . Thus we canhave any number of + − ω -arcs but the number of ++ arcs must equal the number of −− arcs. Finally, observe that the ω -arcs divide both ∂ + and ∂ − into k intervals. In order tospecify a gluing of ∂ + and ∂ − , we orient and cyclically label these k intervals, by convention,0 through k −
1. Again, referring to Figure 5, we illustrate a gluing of ∂ + and ∂ − by the 0through 11 labels, shown in red and blue respectively.2.2. Strategy for constructing examples.
We can reverse engineer this constructionwith an eye towards finding filling pairs of distance greater than 3. Suppose we are interestedin finding such a filling pair ( α, β ) with α non-separating. Any associated ˆ S will be aconnected DW H with two boundary curves. Initially, let us fill ˆ S with a maximal collection, A , of properly embedded essential arcs that are pairwise non-parallel. We specify 2 g + 1arcs to be arcs associated to the 1-handles. Thus, when we split the DW H along these2 g + 1 arcs, we obtain the underlying 2-disc. Figure 2-left illustrates such a configurationfor g = 2.To obtain the collection of ω -arcs that will be used to produce a curve β , we will assignweights to each arc of A such that there is a reasonable expectation that the Distance ≥ A , we will place thatnumber of parallel copies of the arc in the DW H . To this end, we consider the dual graph, G ( A ), to A in ˆ S . (Figure 2-right illustrates such a dual graph for the left configuration.)In graph-theoretic terms, we consider elementary circuits —edge paths that form simpleloops—in G ( A ). Each elementary circuit, γ ⊂ G ( A ), represents a possible vertex in Γ ofthe Distance ≥ γ ) be the sum of the weights of all the arcs in A that a circuit γ intersects. Since our speculative β curve will be the union of arcs parallel to those in A and ( γ, β ) should be a filling pair for any circuit γ , we require that Σ( γ ) be greater thanor equal to the minimal intersection number for a filling pair, i.e. 4 when g = 2 and 2 g − g > { γ , · · · , γ m } ⊂ G ( A ) is the complete set of elementary circuits,for each circuit we get an inequality of the form Σ( γ i ) ≥ g − g >
2) or ≥ g = 2). This gives us m inequalities that make up an integer linear program (ILP). We addto this the equality that states the sum of the weights of ++ arcs equals the sum of the PAUL GLENN, WILLIAM W. MENASCO, KAYLA MORRELL, AND MATTHEW MORSE w w w w w w w w w w w w Figure 2.
The left illustration is a genus one surface with two boundary curves—coded red and blue. C is a maximal collection of 6 weighted arcs. The weights, w , w , w , w , w , w , are non-negative integers. The green graph in the right illus-tration is G ( C ), the dual graph. Each edge of G ( C ) intersects exactly one arc of C once. weights of −− arcs. These combined equations are the constraints for the object equation, P , the sum of all the weights which we wish to minimize.Figure 2-right illustrates this correspondence between a G ( A ) and a ILP when g = 2. Inparticular, for this dual graph, there are 6 elementary circuits which yield 6 weight equations.We get a seventh equation coming from having the weights of the ++ and −− arcs beingequal. All this yields the following constraints for minimizing P = w + w + w + w + w + w : w + w + w + w ≥ w + w + w + w ≥ w + w ≥ w + w ≥ w + w + w + w ≥ w + w + w + w ≥ w = w w , w , w , w , w , w ≥ P is 8. Thus, we have a theoretical minimal intersection number for a fillingpair of distance 4 on S . This optimal value happens to be uniquely realized by the weightsolution [ w , w , w , w , w , w ] = [2 , , , , , ω -arcsinto the DW H of Figure 2-left, we then have 8 possible ways to identify ∂ + to ∂ − . Asdiscussed in § w curve. For ω -arcs corresponding to theweight solution [2 , , , , ,
0] there happen to be four identifications that yield a ( α, β )filling pair. Finally, by employing MICC’s distance functionality (described in § C ( S ).3. Proofs of main results.
Although the case counting is extensive, the discussion in § ICC: A TOOL FOR COMPUTING SHORT DISTANCES IN THE CURVE COMPLEX 9
Proof of Theorem 1.6.
First, we need to generate all possible genus 2
DW H dia-grams with weighted ω -arcs so that we may then generate the corresponding ILP’s. Initiallywe divide this generating process into two cases corresponding to whether α is a separatingor non-separating curve. If α is separating, we have exactly one possible DW H diagramwith weighted ω -arcs. α splits S into two genus one surfaces with boundary, and eachgenus one surface has three weighted ω -arcs. (See Figure 3.) We leave it to the reader to w w w w w w Figure 3. generate the unique ILP in this situation.For the ILP corresponding to Figure 3, we determined that the theoretical minimalintersection number for a filling pair ( α, β ) of distance 4 on S is 12; and, intersection12 is uniquely realized when all the weights equal 2. IBM’s software package CPLEXOptimization Studio (9) was utilized in solving this and all other ILP’s in this paper.Next, employing MICC’s permutation functionality, we determined that there are sixpossible identifications of ∂ + and ∂ − which yield a single β curve. However, MICC’s distancefunctionality determined that all of these filling pairs of intersection 12 were distance 3. Sowe turn to the non-separating case for α . w w w w w w w w w w w w Figure 4.
When α splits S into a connected genus one surface with two boundary components,we first need to generate all possible DW H diagrams along with all possible completecollections of weighted ω -arcs. Such a DW H will have three handles with each handle having the feature that it intersects one or both boundary curves. Since we must have atleast one handle intersecting both boundary curves, the possibilities are: all three handlesintersect both boundaries (as in Figure 2); two handles intersect both boundaries (as inFigure 4)); or, only one handle intersects both boundary curves. In the latter case, it isstraight forward to see that, due to the requirement that the number of ++ and −− arcsare equal, there will be boundary parallel ω -arcs. Since this would mean that | α ∩ β | is notminimal, we conclude that only the first two possibilities occur.Having settled on the DW H diagram of either Figure 2 or 4, we consider other choicesfor a maximal collection of weighted ω -arcs. For example, Figure 4-left and 4-right illus-trates two different choices (w.r.t. the DW H structure) in the case where we have just twohandles intersecting both boundaries. Note that the difference between the two collectionsis a different choice for the arcs associated with the weights w , w , w . However, by in-terchanging the roles of the w and w —in Figure 4-right, we view w as the co-core of ahandle instead of w —we obtain Figure 4-left (after a relabeling weights). Moreover, thereis a similar re-choosing of the co-cores of handles in Figure 4-left that will yield the weightedarc collection of Figure 2. Finally, for the alternate choice of the w , w , w ω -arcs in Figure2, one can again re-choose the handle co-cores to produce the collection of Figure 2. (Weleave the details to the reader.) Thus, we need only consider the collection of weighted arcsin Figure 2 and its associated ILP (1). As mentioned at the end of §
2, the optimal value (9 , , Figure 5. of ILP(1) is 8 and [2 , , , , ,
0] is the unique weight solution. Again, this was determinedby utilizing the software package CPLEX (9). Next, we employed MICC to determine thatany of the four associated filling pairs are only distance 3. Continuing, we utilized CPLEXto find all weight solutions for values P = 9 , ,
11. CPLEX found five weight solutions forvalue 9: [2 , , , , , , , , , , , , , , , , , , , , , , , , , P = 9 filling pairs are distance 3. Similarly, ICC: A TOOL FOR COMPUTING SHORT DISTANCES IN THE CURVE COMPLEX 11 for values P = 10 and P = 11 there are 55 and 79, respectively, weight solutions. Again,MICC found only distance 3 filling pairs.However, using CPLEX to find all weight solutions to ILP (1) for value P = 12 wefound 150 solutions, 9 of which had associated filling pairs that are of distance 4. To listthese solutions: [ w , w , w , w , w , w ] ∈ { [2 , , , , , , [2 , , , , , , [4 , , , , , , [4 , , , , , , [4 , , , , , , [2 , , , , , , [4 , , , , , , [0 , , , , , , [0 , , , , , } .(The power notation is to dedicate a multiplicity of distance 4 filling pairs.) Figure 5 is arealization of [4 , , , , , ≥ ≥
4. To establish distance 4, we must producea length 4 path between α and β . We refer the reader to the green arc in Figure 5 thatcrosses the ω -arc labeled 2 and has endpoints in the 5 segments of ∂ + and ∂ − . Thus, thisgreen arc can be understood as a closed curve that intersects α once and β once. Thisclosed curve, α , will represent a vertex v of a length 4 path { v = v , v , v , v , v = w } .We can obtain a curve, α , representing v by taking a regular neighborhood of α ∪ α andletting α correspond to its unique boundary curve. (Here, α = α and α = β .) Similarly, v is represented by a curve coming from the boundary curve of a regular neighborhood of α ∪ α . The fact that these neighborhoods are each topologically a torus-minus-disc makesall of these curves essential. Thus, we have a length 4 path. γ γ γ α γ γ Figure 6.
Proof of Theorem 1.8.
We will prove the
DW H diagram of Figure 6 is a genus 3distance 4 example of a filling pair have intersection 29. As described in § ∂ + & ∂ − and reading off the labels of the ω -arcs as theyare crossed. The labels of the segments of the boundary curves indicate the identificationmap that forms the curve α . As before, the black ω -arcs join together to form β . Forconvenience, we give the MICC input below: Input top identifications: 1,11,3,27,8,15,7,24,0,10,2,12,4,21,19,17,24,14,6,23,28,9,16,25,13,5,20,18,16Input bottom identifications: 0,10,2,26,7,14,6,23,28,9,1,11,3,22,20,18,25,13,5,22,27,8,15,26,12,4,21,19,17
In our genus 2 example, we saw that there were 6 curves representing vertices in theelementary circuit set Γ. In fact, MICC utilizes the set Γ ′ in its calculation. For the fillingpair of Figure 6, the set Γ ′ has 28 vertices. We can specific a curve representing a vertexof Γ ′ by a sequence of boundary segment labels. For example, the sequence [0, 22, 5,17, 24, 3, 20, 8] corresponds to a curve in this figure that intersects in cyclic order theboundary segments in this list. The set of green arcs in the figure correspond to this labelsequence and should be to understood as a curve, γ , representing ¯ γ ∈ Γ ′ . By inspectionobserve that ( α, γ ) is a filling pair. Similarly, the brown arcs of the figure correspond tothe label sequence [3, 24, 17, 7, 19, 26, 13] . The reader can also check by inspectionthat this curve, γ , representing ¯ γ ∈ Γ ′ , is also filling when paired with α .Having had MICC determine that all of the curves representing vertices in Γ ′ fill whenpaired with α , it remains for us to find a length 4 path so as to establish d ( v, w ) = 4.The magenta arc of Figure 6 corresponds to a curve, α , representing a v vertex of such apath. Notice α intersects β only twice. Thus, as a pair they cannot be filling. Taking anappropriate boundary curve of a regular neighborhood of α ∪ β will yield a a . Similarly,since | α ∩ α | = 1 we know d ( v, v ) = 2 and we can construct an α as we did for our genus2 example to give us a representative of v . (cid:3) Remark . As previously mentioned, the argument establishing (the genus 2) Theorem1.6 can be thought of as a proof of concept calculation. For higher genus the calculationis exactly the same, although more extensive due to the fact that there are more
DW H representations and, thus, a Γ-calculation with accompanying ILP equations for each
DW H .However, if we willing to settle for an estimate the calculation can be limited to just one
DW H representation and we can use MICC to “discover” some rough bounds. This isessentially how the example of Figure 6 was found. We choose a somewhat symmetricgenus 3
DW H and placed an initial set of disjoint proper essential arcs so that any simpleclosed curve drawn in the
DW H intersected these arcs at least 5(= [2 · −
1] = [2 · g − perm command discussed in § curves command discussed in § DW H and the possible curves in Γ.4.
Information for the user of MICC.
In this section we discuss the input format for MICC and the commands for analyzingand manipulating that input. The input is two sequences of numbers which correspondsto a “ladder” representation of curves in the surface. The commands are genus , distance , curves , matrix , faces and perm (for permutation). We will illustrate each of these featuresby further developing the example in Figure 5.4.1. MICC input.
Given a
DW H presentation of filling pair of curves ( α, β ) on a surface S , as previously discussed in §
2, we have specified the gluing of ∂ + and ∂ − by cyclicallylabeling their segments. Assuming we used k labels for the segments of ∂ ± , we now label ICC: A TOOL FOR COMPUTING SHORT DISTANCES IN THE CURVE COMPLEX 13 the ω -arcs of β , 0 through k −
1. Looking at Figure 5, we have a labeling using 0 through 11.For the purpose of extracting the MICC input from this example, the labeling assignmentcould have been done in a random fashion. However, for aesthetic reasons we have takencare to label the ω -arcs in the cyclic order they occur in β .With the ω -arcs of β in our DW H representation labeled as described above, we cannow extract the MICC input for our filling pair. Starting at segment 0 in ∂ − , we traversethis boundary component in the positive direction and record the labels of the ω -arcs as wepass over their endpoints. This sequence of labels will be the input for MICC’s “ Input topidentifications ”. Repeating this process on ∂ + we get the sequence of labels that will bethe input for MICC’s “ Input bottom identifications ”. Together these two sequencesare the basis of a ladder representation , L v ( w ), of a filling pair ( α, β ).A ladder representation is readily understood by again considering our example inFigure 5. Starting at the 0 segments of ∂ ± and reading off the two label sequences, ourinput for MICC would be: Input top identifications: 1,6,11,4,3,2,7,0,5,9,8,7Input bottom identifications: 0,5,10,3,2,1,6,11,4,10,9,8
The reader should readily grasp the ladder metaphor by considering the representation inFigure 7-left of a regular neighborhood of ( ∂ + ∪ ∼ ∂ − ) ∪ { all ω arcs } coming from Figure 5. Inthis illustration, our two boundary components which have been glued together to form α isrepresented by the horizontal segment that has its ends identified. (Our convention forcesthis identification to always occur in the middle of the 0-segment of ∂ ± .) Each verticalsegment above or below this horizontal α is half of an ω -arc. From left-to-right there are 12labels above these vertical ω -halves correspond to MICC’s Input top identifications .Similarly, the 12 labels below these vertical ω -halves correspond to MICC’s Input bottomidentifications . It will be useful to the reader to have an understanding of how MICC α β
Figure 7. utilizes a ladder representation. Specifically, for a ladder L v ( w ) we have, from left-to-right,0 through k − α with β . For the i th intersection we will associateto it a 1 × L v ( w )] i = [ v − ( i ) , w + ( i ) , v + ( i ) , w − ( i )]. Figure 8 illustrates thescheme for determining the values of the v ±′ s and w ±′ s. In particular, v − ( i ) = − ( i − v + ( i ) = i + 1; and w + ( i ) and w − ( i ) correspond to the adjacent (ladder) vertices along β with the parity determined by whether β is pointing down ( w + ( i ) < w − ( i ) >
0) or αβ v − w + v + w − Figure 8.
The illustration depicts how to generate [ L v ( w )] =[ v − (0) , w + (0) , v + (0) , w − (0)] = [ − , − , , up ( w + ( i ) > w − ( i ) <
0) at the i th intersection. Stacking the row vectors [ L v ( w )] i inorder, we produce a k × k is the intersection number of α and β . Figure7-right is the associated characteristic matrix , [ L v ( w )], for the ladder in Figure 7-left. Itshould be readily apparent to the reader that the information in [ L v ( w )] is sufficient toreproduce the ladder L v ( w ), and L v ( w ) is sufficient to reproduce a DW H representation.Thus, up to any permutation of labels and changes of orientation of our curves, L v ( w ) isdependent only on the classes v and w .Given the prompt, What would you like to calculate? , a reply of matrix willproduce [ L v ( w )].4.2. Genus command.
We now make concrete the filling calculation. We will use L v ( w )and its characteristic matrix [ L v ( w )] to compute the minimal genus of the surface that α and β fill. For a surface of genus g , with a minimally intersecting filling 4-valent graph, α ∪ β , we have g = − ( | V | − | E | + | F | ) + 1, where V = α ∩ β is the set of vertices and E is the set of edges of the graph α ∪ β . Also, F = S \ ( α ∪ β ) is a set of 2 n -gon disc regions.Then | E | = 2 | V | . Thus, determining g requires a count of the number of 2 n -gon regions of F . Listing the components of F using [ L v ( w )] is achieved by specifying their edge-pathboundaries in α ∪ β . When traversed such a edge-path will be alternating —an edge e α ∈ α followed by an edge e β ∈ β followed by e α ∈ α and so on, will be a cyclically ordered set { e α , e β , · · · , e αn , e βn } . Thus, e αj (respectively, e βj ) starts at vertex having integer label ∂ s e αj (respectively, ∂ s e βj ) and terminates at vertex having integer label ∂ t e αj (respectively, ∂ t e βj );and, ∂ t e βj − = ∂ s e αj and ∂ t e αj = ∂ s e βj , all modulo n .The entries of the [ L v ( w )] i ′ s gives us that ∂ t e αj ∈ { v + ( ∂ s e αj ) , | v − ( ∂ s e αj ) |} = { ( ∂ s e αj − , ( ∂ s e αj + 1) } . Additionally, we have ∂ s e βj ∈ {| w + ( ∂ t e βj ) | , | w − ( ∂ t e βj ) |} and ∂ t e βj ∈ {| w + ( ∂ s e βj ) | , | w − ( ∂ s e βj ) |} . Now fixing a 2 n -gon region and, by convention, traversing its edge path boundary so as toalways keep the region to our left, we have the following scheme for finding the terminusendpoint for the next e α or e β edge.T1– If ∂ t e αj = v + ( ∂ s e αj ) then ∂ t e βj = | w + ( ∂ t e αj ) | (with ∂ s e βj = ∂ t e αj ).T2– If ∂ t e αj = | v − ( ∂ s e αj ) | then ∂ t e βj = | w − ( ∂ t e αj ) | (with ∂ s e βj = ∂ t e αj ). ICC: A TOOL FOR COMPUTING SHORT DISTANCES IN THE CURVE COMPLEX 15
T3– If ∂ s e βj = | w + ( ∂ t e βj ) | and ∂ t e βj = | w − ( ∂ s e βj ) | then ∂ t e αj +1 = v + ( ∂ t e βj ) (with ∂ s e αj +1 = ∂ t e βj ).T4– If ∂ s e βj = | w − ( ∂ t e βj ) | and ∂ t e βj = | w + ( ∂ s e βj ) | then ∂ t e αj +1 = | v − ( ∂ t e βj ) | (with ∂ s e αj +1 = ∂ t e βj ).T5– If ∂ s e βj = | w + ( ∂ t e βj ) | and ∂ t e βj = | w + ( ∂ s e βj ) | then ∂ t e αj +1 = v + ( ∂ t e βj ) (with ∂ s e αj +1 = ∂ t e βj ).T6– If ∂ s e βj = | w − ( ∂ t e βj ) | and ∂ t e βj = | w − ( ∂ s e βj ) | then ∂ t e αj +1 = v − ( ∂ t e βj ) (with ∂ s e αj +1 = ∂ t e βj ).We illustrate this scheme using our [ L v ( w )] in Figure 7-right. Starting at vertex 2we can traverse the edge between 2(= ∂ s e α = v + (1)) and 1(= ∂ t e α = | v − (2) | ). Thus,we have the assumption of T2 for e α . This gives us ∂ t e β = 8(= | w − (1) | = ∂ s e α ) with1 = ∂ s e β = ∂ t e α . Since 1 = | w + (8) | , we have the T3 assumption and 9 = ∂ t e α (= v + (8)).But this gives us the T1 assumption for e α . Thus, we have that ∂ t e β = | w + (9) | which is 10.Since | w − (10) | = 9 we again have the T3 assumption which implies ∂ t e α = | v − (10) | = 9.Now, we again have the T2 assumption for e α which implies ∂ t e β = | w − (9) | = 2, back wherewe started. (Refer to the region in Figure 5 contain (9 , , | F | and, thus, are ableto compute genus.4.3. Faces command.
Having traversed all the boundaries of the 2 n -gon regions of S \ ( α ∪ β ), MICC records this calculation as a vector. Specifically, let F n ⊂ F be the numberof 2 n -gon regions for n ∈ { , , , · · · } . Then associated with the graph α ∪ β ⊂ S g wehave the vector [ F , F , F , · · · ]. The output of MICC is actually formatted as { F , F , F , · · · } . This vector solution to Euler characteristic equation of Lemma 4.1 of (7).The prompting inquiry and command appears as: What would you like to calculate?faces . Additionally, it lists each boundary edge in a truncated matter—it lists only the e αi ′ s. For our extended example of Figure 8, we would get: Vectorsolution : { , } (0 , , , , , , , , , , , , , , Curves command.
An alternate reply to the prompt,
What would you like tocalculate? , is curves . MICC applies the Theorem 1.3 test by listing all of the curvesrepresenting vertices of Γ ′ and computing the genus of their graphs when paired with α .MICC finds all curves representing elements of Γ ′ by applying a classical depth-first search(18) for elementary circuits of the graph G ( C ′ ), the dual graph to the proper arcs of α in S \ β . The MICC output is a cyclic sequence of e α edges. That is, a curve, γ , representinga vertex ¯ γ ∈ Γ ′ and a region f ∈ F , γ ∩ f will be a collection of proper arcs having theirendpoints on e α edges of the boundary of f . As we traverse γ , we will travel between regionsby passing through e α ′ s. Thus, ¯ γ can be characterized by giving a cyclic listing of these e α ′ s. Continuing with our extended example of Figure 8, the output response to curves would yield: Path [ , , , , , , , ] Curvegenus : 2
Path [ , , , ] Curvegenus : 2
Path [ , , , , , ] Curvegenus : 2
Path [ , , , , , ] Curvegenus : 2
Path [ , , , , , ] Curvegenus : 2
Path [ , , , , , , , ] Curvegenus : 2(3)4.5.
Distance command.
To determine d ( v, w ) one replies to the prompt, What wouldyou like to calculate? , by typing distance . If the genus of any of the pairs ( α, γ ) isless than g , then MICC will respond with Distance: 3 . If the genus of all such pairs is g , then MICC will respond with Distance: 4+ . All possible γ representing elements of Γ ′ are determined as described above.4.6. Perm command.
Finally, MICC has an experimental functionality. As most topol-ogist who have attempted to construct filling pairs on surfaces know, it is difficult to do sowhile avoiding the production of multi-curves. For example, starting with a
DW H represen-tation of the surface, after placing down some collection of ω -arcs finding an identificationof ∂ + and ∂ − so as to have β be a single curve is tedious at best. Fortunately, MICCautomates this process. Given any ladder top/bottom identification, it will first determinewhether β is a single curve or a multi-curve. If it is a multi-curve, then it will produce theinquiring prompt Would you like to shear this multi-curve? . With the reply yes , itwill search through all possible ∂ + / ∂ − identifications for those that yield a single β curveand print them out along with their distance. If MICC has been given a ladder identificationthat corresponds to a single β curve, one can still find all other ∂ + / ∂ − identifications thatyield a single curve. When given the prompt What would you like to calculate? justreply with perm (for permutation).
ICC: A TOOL FOR COMPUTING SHORT DISTANCES IN THE CURVE COMPLEX 17
For our extended example, the output would be:
Curve 1 Distance : [2 , , , , , , , , , , , , , , , , , , , , , , Curve 2 Distance : [5 , , , , , , , , , , , , , , , , , , , , , , Curve 3 Distance : +[1 , , , , , , , , , , , , , , , , , , , , , , Concluding remarks.
MICC was originally created as a tool that would help in the search for distance 4filling pairs on surface of genus greater than 2. We have since realized that it can be set toother uses. For example, it was recently used to find geodesic triangles where any pair ofvertices correspond to filling pairs in minimal position. As mentioned previously, we hoperesearchers will find additional uses.Remarking on the complexity of the algorithms employed in MICC, the most compu-tationally expensive task involves finding all cycles in the graph G ( A ′ ) of § G ( A ′ ), and thus on thecomplexity of the program itself.There are many ways in which we would like to improve the current version of MICC.Currently, MICC is only a partial implementation of the algorithm presented in (4). We planto extend the functionality of MICC to encompass the full scope of the Efficient GeodesicAlgorithm of (4). Yet this partial implementation is also manifestation of the complexitybarrier involved with the current exponential running time of the graph search. An improvedalgorithm will allow for more intricate curves to be studied, and parallelization of the MICCwould help future users to fully utilize their multicore computers in their research. Acknowledgements
The first, third and fourth authors are grateful to Joan Birman, DanMargalit and the second author for sharing results of their joint work as it developed. Ourthanks goes to John Ringland, Joaquin Carbonara and the URGE to Compute program atthe University at Buffalo and Buffalo State College for supplying a nurturing environmentfor our work. This work was supported in part by NSF CSUMS grants 0802994 and 0802964to the University at Buffalo and Buffalo State College. Finally, we thank the referees foralerting us to the recent results in the literature, for suggesting a strengthening of thestatement of Corollary 1.7 and numerous other expository improvements.6.
Addendum: All weight solutions of distance filling pairs in C ( S ) for , & intersections when one curve is non-separating. Below we list with multiplicity all solutions to ILP (1) for P -values 12, 13 and 14.For P = w + w + w + w + w + w = 12: [2 , , , , , , [2 , , , , , , [4 , , , , , , [4 , , , , , , [4 , , , , , , , , , , , , , , , , [0 , , , , , , [0 , , , , , For P = w + w + w + w + w + w = 13:[5 , , , , , , , , , , , [2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , [3 , , , , , , [4 , , , , , ,[4 , , , , , , , , , , , , , , , , , , , , , [2 , , , , , , , , , , , , , , , , , , , , , [3 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , [2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , [0 , , , , , , , , , , , , , , , , , , , , , , , , , , [4 , , , , , For P = w + w + w + w + w + w = 14:[4 , , , , , , [3 , , , , , , , , , , , [0 , , , , , , , , , , , [4 , , , , , , , , , , , [0 , , , , , , , , , , , , , , ,
2] [4 , , , , , , , , , , , , , , , , [3 , , , , , , , , , , , [4 , , , , , , [2 , , , , , , [2 , , , , , , , , , , , [2 , , , , , , , , , , , , , , , , [2 , , , , , , , , , , ,[2 , , , , , , [3 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , [4 , , , , , , , , , , , [1 , , , , , , , , , , ,[3 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,[0 , , , , , , [2 , , , , , , , , , , , , , , , , , , , , , , , , , ,[2 , , , , , , [4 , , , , , , , , , , , [2 , , , , , , [4 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,[4 , , , , , , [2 , , , , , , , , , , , [2 , , , , ,
0] [5 , , , , , , , , , , , , , , , , [4 , , , , , , , , , , , [2 , , , , , , , , , , , , , , , , , , , , , , , , , , [4 , , , , , , [4 , , , , ,
0] [2 , , , , , , , , , , , , , , , , , , , , , [2 , , , , , , [3 , , , , , , , , , , , [2 , , , , , ,[2 , , , , , , , , , , . Figure 9.
The weight solutions [2 , , , , ,
2] for P = 12 is particularly intriguing since it suggeststhere might be a high level of symmetry. Figure 9 is a 3-dimensional rendering of the onethe four associated distance 4 filling pair. The aesthetic of this rendering is so appealingthat it was placed at the begin of (4). We thank those authors for its use here. ICC: A TOOL FOR COMPUTING SHORT DISTANCES IN THE CURVE COMPLEX 19
MICC software package, software tutorial, and all known weight solutions yieldingfilling pairs having distance ≥ P ≤
25 with g = 2 (approximately 72,000 weightsolutions) are posted for download at micc.github.io. References [1] T. Aougab,
Uniform hyperbolicity of the graphs of curves , Geometry & Topology 17(2013) 2855-2875.[2] T. Aougab & S. Huang,
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Small intersection numbers in the curve graph , Bull. LondonMath. Soc. 46 (2014) 989-1002.[4] J. Birman, D. Margalit & W. Menasco,
Efficient geodesics and an effective algorithmfor distance in the complex of curves , arXiv:1408.4133.[5] B. Farb & D. Margalit,
A primer on mapping class groups , 2012 Princeton UniversityPress.[6] W. J. Harvey,
Boundary structure of the modular group , Riemann Surfaces and RelatedTopics: Proceedings of the 1978 Stony Brook Conference (I. Kra and B. Maskit eds.), Ann.Math. Stud. 97, Princeton, 1981.[7] J. Hempel, 3 -Manifolds as viewed from the curve complex , Topology, 40 (2001), 631-657.[8] J. Hempel, Personal email communication.[9] IBM ILOG CPLEX Optimization Studio.[10] A. Ido, Y. Jang & T. Kobayashi,
Heegaard splittings of distance exactly n , Algebraic& Geometric Topology 14 (2014) 1395-1411.[11] J. P. Leasure,
Geodesics in the complex of curves of a surface , PhD Thesis (2002), TheUniversity of Texas at Austin.[12] W. B. R. Lickorish,
A finite set of generator for the homeotopy group of a -manifold ,Proc. Cambridge Philos., 60:769-778[13] H. A. Masur & Y. N. Minsky, Geometry of the complex of curves. II. Hierarchicalstructure , Geom. Funct. Anal. 10 (2000), no. 4, 902-974.[14] H. A. Masur & Y. N. Minsky,
Geometry of the complex of curves. I. Hyperbolicity ,Invent. Math. 138 (1999), no. 1, 103-149.[15] P. Glenn, W. Menasco, K. Morrell & M. Morse,
Metric in the curve complex (MICC) ,software package available for download on the world wide web at https://micc.github.io,2014.[16] S. Schleimer,
Notes on the curve complex ∼ saulsch/math.html.[17] K. J. Shackleton, Tightness and computing distances in the curve complex , Geom.Dedicata 160 (2012), 243-259.[18] E. Shimon & G. Even,
Graph Algorithms , p. 46, Cambridge University Press 2012.[19] R. Webb,
Combinatorics of tight geodesics and stable lengths , arXiv:1305.3566.
Department of Mathematics, University at Buffalo—SUNY
E-mail address : [email protected] Department of Mathematics, University at Buffalo—SUNY
E-mail address : [email protected] Department of Mathematics, Buffalo State College—SUNY
E-mail address : [email protected] Department of Mathematics, University at Buffalo—SUNY
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