Efficient geodesics and an effective algorithm for distance in the complex of curves
aa r X i v : . [ m a t h . G T ] M a y EFFICIENT GEODESICS AND AN EFFECTIVE ALGORITHM FORDISTANCE IN THE COMPLEX OF CURVES
JOAN BIRMAN, DAN MARGALIT, AND WILLIAM MENASCOA
BSTRACT . We give an algorithm for determining the distance between twovertices of the complex of curves. While there already exist such algorithms, forexample by Leasure, Shackleton, and Webb, our approach is new, simple, andmore effective for all distances accessible by computer. Our method gives a newpreferred finite set of geodesics between any two vertices of the complex, calledefficient geodesics, which are different from the tight geodesics introduced byMasur and Minsky. F IGURE
1. Vertices of C ( S ) with distance 4 and intersection number12; this is the smallest possible intersection for vertices with distance 4
1. I
NTRODUCTION
The complex of curves C ( S ) for a compact surface S is the simplicial complexwhose vertices correspond to isotopy classes of essential simple closed curves in S and whose edges connect vertices with disjoint representatives. We can endow the0-skeleton of C ( S ) with a metric by defining the distance between two vertices tobe the minimal number of edges in any edge path between the two vertices.The geometry of C ( S ) —especially the large-scale geometry—has been a topicof intense study over the past two decades, as there are deep applications to thetheories of 3-manifolds, mapping class groups, and Teichm¨uller space; see, e.g.,[14]. The seminal result, due to Masur and Minsky in 1996, states that C ( S ) is d -hyperbolic [13]. Recently, several simple proofs of this fact have been found, andit has been shown that d can be chosen independently of S ; see [2, 6, 7, 10, 15].In 2002, Leasure [11, § C ( S ) , and since then other algorithms have been devised by Shack-leton [17], Webb [19], and Watanabe [18]. About his algorithm, Leasure says: We do not mention this in the belief that anyone will ever imple-ment it. The novelty is that finding the exact distance between twocurves in the curve complex should be so awkward.
The first author gratefully acknowledges partial support from the Simons Foundation, under Col-laborative Research Award
One goal of this paper is to give an algorithm for distance—the efficient geodesicalgorithm—that actually can be implemented, at least for small distances. Thethird author and Glenn, Morrell, and Morse [9] have in fact already developed animplementation of our algorithm, called Metric in the Curve Complex [8]. Theirprogram is assembling a data bank of examples as we write.
Known examples.
Let S g denote a closed, connected, orientable surface of genus g and let i min ( g , d ) denote the minimal intersection number for vertices of C ( S g ) with distance d . The Metric in the Curve Complex program has been used to showthat:(1) i min ( , ) =
12 and(2) i min ( , ) ≤ i min ( , ) —was dis-covered using the program. See Section 2 for a discussion of this example and aproof using the methods of this paper that the distance is actually 4.We are only aware of one other explicit picture in the literature of a pair ofvertices of C ( S ) that have distance four, namely, the example of Hempel thatappears in the notes of Saul Schleimer [16, Figure 2] (see [9, Example 1.6] for aproof that the distance is 4). This example has geometric intersection number 25.Using the bounded geodesic image theorem [12, Theorem 3.1] of Masur andMinsky (as quantified by Webb [21]) it is possible to explicitly construct examplesof vertices with any given distance; see [17, Section 6]. We do not know how tokeep the intersection numbers close to the minimum with this method, but Aougaband Taylor did in fact use this method to give examples of vertices of arbitrarydistance whose intersection numbers are close to the minimum in an asymptoticsense; see their paper [3] for the precise statement. Local infinitude.
One reason why computations with the complex of curves are sodifficult is that it is locally infinite and moreover there are infinitely many geodesics(i.e. shortest paths) between most pairs of vertices. Masur and Minsky [12] ad-dressed this issue by finding a preferred set of geodesics, called tight geodesics, andproving that between any two vertices there are finitely many tight geodesics; seeSection 2.2 for the definition. Our first goal is to give a new class of geodesics thatstill has finitely many elements connecting any two vertices but is more amenableto certain computations.
Efficient geodesics.
Our approach to geodesics in C ( S ) is defined in terms of in-tersections with arcs. First, suppose that g is an arc in S and a is a simple closedcurve in S . We say that g and a are in minimal position if a is disjoint from theendpoints of g and the number of points of intersection of a with g is smallest overall simple closed curves that are homotopic to a through homotopies that do notpass through the endpoints of g .Let v , . . . , v n be a geodesic of length at least three in C ( S ) , and let a , a , and a n be representatives of v , v , and v n that are pairwise in minimal position (thisconfiguration is unique up to isotopy of S ). A reference arc for the triple a , a , a n FFICIENT GEODESICS AND AN EFFECTIVE ALGORITHM FOR DISTANCE 3 is an arc g that is in minimal position with a and whose interior is disjoint from a ∪ a n ; such arcs were considered by Leasure [11, Definition 3.2.1].We say that the oriented geodesic v , . . . , v n is initially efficient if | a ∩ g | ≤ n − g (this is independent of the choices of a , a , and a n by the uniqueness statement above). Finally, we say that v = v , . . . , v n = w is efficient if the oriented geodesic v k , . . . , v n is initially efficient for each 0 ≤ k ≤ n − v n , v n − , v n − , v n − is also initially efficient.We emphasize that to test the initial efficiency of v k , . . . , v n we should look atreference arcs for the triple v k , v k + , and v n and we allow n − k − v k + with any such reference arc. Existence of efficient geodesics.
Our main result is that efficient geodesics alwaysexist, and that there are finitely many between any two vertices.
Theorem 1.1.
Let g ≥ . If v and w are vertices of C ( S g ) with d ( v , w ) ≥ , thenthere exists an efficient geodesic from v to w. What is more, there is an explicitlycomputable list of at most n g − vertices v that can appear as the first vertex on an initially efficient geodesicv = v , v , . . . , v n = w . In particular, there are finitely many efficient geodesics from v to w.
We emphasize that our theorem is only for closed surfaces; see the discussionon page 18 about surfaces with boundary for an explanation. We also mention thatthis theorem is stronger than Theorem 1.1 in the first version of this paper [4]; seeProposition 3.7 and the accompanying discussion.
Finitely many reference arcs.
While a priori there are infinitely many referencearcs that need to be checked in the definition of initial efficiency there are in factfinitely many. Indeed, let a , a , and a n be representatives of v , v , and w that haveminimal intersection pairwise. Since d ( v , w ) ≥ a and a n fill S ,which means that they together decompose S into a collection of polygons. We canendow each such polygon with a Euclidean metric and replace each segment of a in each polygon with a straight line segment.There are finitely many non-rectangular polygons in the decomposition sinceeach 2 k -gon contributes − ( k − ) / c ( S ) . And each reference arc in a rect-angular region is parallel to one in a non-rectangular region. Thus in order tocheck initial efficiency, it is enough to consider reference arcs that lie in a non-rectangular polygonal region. Furthermore, it is enough to consider reference arcsthat are straight line segments connecting the midpoints of the a -edges of a poly-gon. Indeed, such an arc is necessarily in minimal position with a and any otherreference arc can be extended to such a reference arc.In the special case that the reference arc connects the midpoints of a -edgesthat are consecutive in a polygon, the reference arc is parallel to the a n -edge in JOAN BIRMAN, DAN MARGALIT, AND WILLIAM MENASCO between. In this case points of a ∩ g are in bijection with points of a ∩ a n ,and so the definition of initial efficiency can be translated into a statement aboutintersections of a with a n ; see Proposition 3.7 below. Finitude of efficient geodesics.
The main point of Theorem 1.1 is the existencestatement; the finiteness statement can be dispensed with immediately. Indeed, forany geodesic v , . . . , v n let a , a , and a n be representatives of v , v , and v n thathave minimal intersection pairwise. As above, a and a n decompose S g into acollection of polygons.If we cut S g along a we obtain a surface S ′ g with two boundary componentson which a n becomes a collection of arcs. The a n -arcs cut S ′ g into a collection ofeven-sided polygons. We can choose reference arcs in S ′ g that are disjoint from eachother, that have interiors disjoint from the a n -arcs, and that cut S ′ g into hexagons.Such a collection is obtained by taking one reference arc parallel to each parallelfamily of arcs of a n and then taking additional reference arcs cutting across anyremaining polygons with more than six sides.An Euler characteristic count shows that any such collection of reference arcshas 6 g − a is disjoint from a the curve a is determinedup to homotopy by the number of intersections it has with each reference arc. Bythe definition of initial efficiency, each of these intersection numbers is between 0and n −
1. This gives the bound stated in Theorem 1.1.
Discussion of the proof.
Our method for proving Theorem 1.1 is detailed in Sec-tion 3. Briefly, the idea is to show that if some geodesic v = v , . . . , v n = w is notinitially efficient then we can modify v , . . . , v n − by surgery in order to reduce theintersection of v with v and v n . The basic surgeries we use in our proof are notnew. The crucial point—and our new idea—is that it is usually not possible to re-duce intersection by modifying a single vertex; rather, it is often the case that wecan reduce intersection by modifying a sequence of vertices all at the same time. v n v v v ′ v v v ′ v ′ F IGURE Left: two arcs of v and a simplifying surgery; Right: arcsof v and v and a simplifying surgery Here is what we mean by this. Suppose we have a geodesic v , ..., v n . Say thereis a v - v n polygon with two parallel arcs of v as in the first picture in the left-handside of Figure 2. Then we can perform a surgery along the dotted reference arc as inthe figure in order to find a vertex v ′ that is simpler in that it has fewer intersectionswith v and v n . The vertex v ′ can replace v in the geodesic since the surgery didnot create any intersections with v or v .Now suppose we have four parallel arcs of v , v , v , and v (in order) as inthe right-hand side of Figure 2. We cannot surger v as in the previous paragraph FFICIENT GEODESICS AND AN EFFECTIVE ALGORITHM FOR DISTANCE 5 because this would create an intersection with v —an arc of v is in the way. How-ever, we can perform surgery simultaneously on v and v along the dotted arc asin the figure. This gives two new vertices v ′ and v ′ and again we can replace v and v with these new, simpler vertices.Our basic strategy is to show that whenever we have an inefficient geodesic wecan find a similar surgery in order to reduce intersection with v and v n . If thereference arc only sees v and v then the surgeries in the previous two paragraphsapply. The problem is that when there are more vertices v i involved, there are moreand more complicated surgeries needed, and the combinatorics get to be unwieldy;look ahead to Figures 11 and 13 for examples of more complicated surgeries.To deal with this problem, we introduce a new tool, the dot graph. This is agraphical representation of the sequence of vertices v i seen along a reference arc;there is a dot at the point ( k , i ) in the plane if the k th vertex along the arc is v i (see Figure 8 below). The existence of a simplifying surgery is translated into theexistence of certain two-dimensional shapes in the dot graph (see Figure 9 below).In this way, the unwieldy combinatorial problem becomes a manageable geometricone. Efficiency versus tightness.
We already mentioned that there are finitely many tightgeodesics between two vertices of C ( S g ) and so Theorem 1.1 gives a second finiteclass of geodesics connecting two vertices of C ( S g ) . The next proposition showsthat the class of efficient geodesics is genuinely new. Proposition 1.2.
Let g ≥ . In C ( S g ) there are geodesics of length three that are... (1) efficient and tight, (2) tight but not efficient, and (3) efficient but not tight. We do not know if between any two vertices there always exists a geodesic thatis efficient and tight.Proposition 1.2 is proved by explicit construction; see Section 2.2. The mostsubtle point is the third one, as it is in general not easy to prove that a given geodesicis not contained in any tight multigeodesic.While the examples of geodesics in Proposition 1.2 all have length three, weexpect that the result holds for all distances at least three. It is also worth notingthat our constructions are all delicate: it is not obvious how to modify our examplesin order to obtain infinite families of examples.
The efficient geodesic algorithm.
We now explain how Theorem 1.1 can be usedin order to give an algorithm for distance in C ( S g ) , which we call the efficientgeodesic algorithm . It is straightforward to determine if the distance between twovertices is 0, 1, or 2. So assume that for some k ≥ C ( S g ) have distance 0 , . . . , k . We would like to givean algorithm for determining if the distance between two vertices is k + v and w be two vertices of C ( S g ) . By induction we can check if d ( v , w ) ≤ k . If not, then as in Theorem 1.1 we can explicitly list all possible vertices v on an efficient geodesic v = v , . . . , v k + = w . If d ( v , w ) = k for some choice of JOAN BIRMAN, DAN MARGALIT, AND WILLIAM MENASCO v , then d ( v , w ) = k +
1; otherwise it follows from Theorem 1.1 (the existence ofefficient geodesics) that d ( v , w ) = k + Corollary 1.3.
The efficient geodesic algorithm computes distance in C ( S g ) . The special case of the efficient geodesic algorithm when the distance is fourwas explained to us by John Hempel and served as inspiration for the cases oflarger distance.
Comparison with previously known algorithms.
Our efficient geodesic algorithmis in the same spirit as the algorithms of Leasure, Shackleton, and Watanabe forcomputing distance in C ( S g ) . All three show that there is a function F of threevariables so that for any two vertices v and w of C ( S g ) with d ( v , w ) = n there is ageodesic v = v , . . . , v n = w with i ( v , w ) bounded above by F ( g , n , i ( v , w )) . Thisgives an algorithm in the same way as our efficient geodesic algorithm, since thereis an explictly computable list of v with i ( v , v ) = i ( v , w ) ≤ F ( g , n , i ( v , w )) .While the theorems of Leasure, Shackleton, and Watanabe apply to surfaces thatare not closed, we restrict here to the case of closed surfaces for simplicity.Our approach also gives such a function F . By only considering reference arcsthat are parallel to arcs of a n \ a (where a and a n are minimally-intersectingrepresentatives of v and v n ), we deduce that for any initially efficient geodesic v = v , . . . , v n = w we have i ( v , v n ) ≤ ( n − ) i ( v , w ) (this uses a slight strengtheningof a special case of Theorem 1.1; see Proposition 3.7 below). So we can take F BMM ( g , n , i ( v , w )) = ( n − ) i ( v , w ) . However, this bound does not use the full strength of initial efficiency as it doesnot give information as to how these points of intersection are distributed along a n nor does it take into account reference arcs that are not parallel to a n .Leasure’s function is F L ( g , n , i ( v , w )) = ( ( g − ) + ) n i ( v , w ) . We can illustrate the improvement of our methods over Leasure’s with the examplein C ( S ) from Figure 1. To prove the distance is 4, we can suppose for contradic-tion that it is 3. According to Leasure, if v is the first vertex we meet on a length3 geodesic from v to w , then we can choose v so that it satisfies i ( v , w ) ≤ ( ( g − ) + ) i ( v , w ) = · = , , . By contrast, any v on an efficient geodesic of length 3 satisfies i ( v , w ) ≤
12 and,what is more, we know there is at most one intersection of v along each edgeof the polygonal decomposition of S determined by v and w (cf. Proposition 3.7below). Because of these strong restrictions, the computation can be carried out byhand, and in fact we apply the algorithm by hand to this example in Section 2.Shackleton’s function depends only on i ( v , w ) and g , but not d ( v , w ) . As ex-plained by Watanabe [18], Shackleton’s function is F S ( g , n , i ( v , w )) = i ( v , w )( ( g − ) ) i ( v , w ) . FFICIENT GEODESICS AND AN EFFECTIVE ALGORITHM FOR DISTANCE 7
Watanabe recently improved on Shackleton’s result by replacing the exponen-tial function with a linear one. His work, like Webb’s, uses the theory of tightgeodesics. Specifically, Watanabe’s function is: F W ( g , n , i ( v , w )) = R g i ( v , w ) where R g = ( g − ) · ( M + ) ( g − ) g − and M is the minimal possible constantin the bounded geodesic image theorem. Since R g is independent of n , it followsthat when n is large compared to g Watanabe’s bounds give a better algorithm fordistance than the efficient geodesic algorithm. However, the smallest known upperbound for M is 102 (see [22]), and so even for g =
2, we have R g = · , , > , , . Thus, even for g = g : F W ′ ( g ) = ( g − ) (cid:0) ( g − ) − g + (cid:1) g − , which for g = v that need to be tested instead of the quantity i ( v , v n ) . In Webb’salgorithm, this number is bounded above by:2 ( g + ) min { n − , } ( g − − ) (here we are really counting the number of candidate simplices s along a multi-geodesic from v to w ); see the appendix of this paper for an explanation. On theother hand, our Theorem 1.1 states that the number of candidate vertices v alongan efficient geodesic v , . . . , v n is bounded above by n g − . Our bound is smallerthan Webb’s when min { n − , } = n −
2. In the case that min { n − , } = ( g )( ) and we find that our bound issmaller than Webb’s for all distances less than 2 ( ) , which is approximately 10 .We conclude that among all known algorithms for distance in C ( S ) our methodsare by far the most effective for all distances accessible by modern computers. Acknowledgments.
We would like to thank Ken Bromberg, Chris Leininger, YairMinsky, Kasra Rafi, and Yoshuke Watanabe for helpful conversations. We areespecially grateful to John Hempel for sharing with us his algorithm, to RichardWebb for sharing many ideas and details of his work, and to Tarik Aougab for manyinsightful comments, especially on the problem of constructing geodesics that arenot tight. Finally, we would like to thank Paul Glenn, Kayla Morrell, and MatthewMorse for supplying numerous examples generated by their program Metric in theCurve Complex.
JOAN BIRMAN, DAN MARGALIT, AND WILLIAM MENASCO
2. E
XAMPLES
In this section we do two things. First we illustrate the efficient geodesic algo-rithm by applying it to the example from Figure 1. Then we prove Proposition 1.2by giving explicit examples for each of the three statements. All of the exampleswill be presented in terms of the branched double cover of S g over the sphere,which we now explain. The branched double cover.
Let X g + denote a sphere with 2 g + S g . Thepreimage of a simple arc in X g + connecting two marked points is a nonsepa-rating simple closed curve in S g , and the preimage of a simple closed curve thatsurrounds 2 k + S g that cutsoff a subsurface of genus k .Minimally intersecting curves and arcs in X g + lift to minimally intersectingcurves and arcs in S g . This follows from the work of the first author and Hildenon the symmetric mapping class group [5]; see also the paper by Winarski [23].Also, if two minimally intersecting curves or arcs fill X g + —meaning that thecomplementary components are all disks with at most one marked point each—then the preimages fill S g since the preimage of a disk with at most one markedpoint is a disk.2.1. An example of the efficient geodesic algorithm.
Consider the two arcs d and e in X shown in the left-hand side of Figure 3 (we depict X g + by drawing2 g + g + v and w denote the corresponding verticesof S , the two-fold branched cover over X . We would like to show that the distancebetween v and w in C ( S ) is 4 (it so happens that v and w are the same as thevertices of C ( S ) shown in Figure 1, but we will not need this). The distancebetween v and w in C ( S ) can be computed with the computer program Metric inthe Curve Complex, but here we explain how to apply our algorithm by hand. d e de + e − F IGURE Left: the arcs d and e in X corresponding to the curvesshown in Figure 1; Right: the disk D obtained by cutting along e First, we will show that d ( v , w ) ≤
4. To do this, we observe that the horizontalline segment connecting the second and third marked points in the left-hand sideof Figure 3 corresponds to a vertex u in C ( S ) with i ( u , v ) = i ( u , w ) =
1. It followsthat d ( u , v ) = d ( u , w ) = d ( v , w ) ≤ FFICIENT GEODESICS AND AN EFFECTIVE ALGORITHM FOR DISTANCE 9
If we cut X along e , we obtain a disk D , the shaded disk in the right-hand sideof Figure 3. The boundary of D consists of two copies of e , say, e + and e − , andin the figure points of e + and e − are identified in X exactly when they lie on thesame vertical line. The arc d becomes a collection of arcs in D as shown in thefigure. Since the arcs of d cut D into a disjoint union of disks with at most onemarked point each, it follows that d and e fill S and so d ( v , w ) ≥ d ( v , w ) ≥
4. As-sume that d ( v , w ) were equal to three. By Theorem 1.1 there is a path v , v , v , w sothat the number of intersections of v with each arc of w \ v is at most two (considera reference arc parallel to the arc of w \ v ). Proposition 3.7 below gives an improve-ment: there is a choice of v so that the intersection with each arc of w \ v is at mostone point. Also, since v , v , v , w is a path, this choice of v satisfies d ( v , w ) ≤ v and w do not fill S .A special feature of the genus two case is that every vertex of C ( S ) is obtainedas the preimage of a curve or arc in X (this again follows from the work of the firstauthor with Hilden). In this way, any v as in the previous paragraph correspondsto an arc or curve b in D that intersects each arc of d in at most one point. There areonly six such candidates for b , namely the six straight line segments connectingmarked points in the interior of D . It is straightforward to check that the arc in X corresponding to each fills with d . Therefore there is no v as in the previousparagraph and we have d ( v , w ) = Efficiency versus tightness.
We will now prove Proposition 1.2—that thereare geodesics in C ( S g ) that are efficient and tight, geodesics that are efficient butnot tight, and geodesics that are tight but not efficient. First we recall the definitionof a tight geodesic. Tight geodesics. A tight multigeodesic is a sequence of simplices s , . . . , s n in C ( S ) where(1) s and s n are vertices,(2) the distance between v i and v j is | j − i | whenever i = j and v i and v j arevertices of s i and s j , respectively, and(3) for each 1 ≤ i ≤ n − s i can be represented as the union ofthe essential components of the boundary of a regular neighborhood in S of minimally-intersecting representatives of s i − and s i + .This definition is due to Masur and Minsky. We will refer to any sequence ofvertices v , . . . , v n with v i ∈ s i as a tight geodesic . Proof of Proposition 1.2.
We begin with the first statement, there there are geodesicsin C ( S g ) that are both efficient and tight. Consider the arcs d , d , d , and d in X shown in Figure 4. As above, each arc d i represents a vertex v i of C ( S ) . We have d ( v , v ) = d and d fill X .To see that the geodesic v , v , v , v is efficient we first note that d intersectsonly two regions of X determined by d and d . One of these regions is a bigon Masur and Minsky used the term “tight geodesic,” instead of “tight multigeodesic,” language weprefer to avoid because the object in question is not a geodesic. with one marked point; the preimage of this is a rectangular region in S and so itcan be ignored. The other region is a disk with no marked points and its preimageis a pair of disks in S . The preimage of d passes through each of these disks in S once, whence the initial efficiency of v , v , v , v . There is an obvious symme-try of X reversing the geodesic and so v , v , v , v is initially efficient. Hence v , v , v , v is indeed efficient. d d ′ d ′′ d d ′ d ′′ d d F IGURE
4. Arcs giving a geodesic in C ( S ) that is both efficient and tight Let v ′ i and v ′′ i denote the vertices of C ( S ) corresponding to the arcs d ′ i and d ′′ i .The simplices s = { v , v ′ , v ′′ } and s = { v , v ′ , v ′′ } give a tight multigeodesic v , s , s , v with v ∈ s and v ∈ s , certifying that v , v , v , v is a tight geo-desic. (To verify this, note that the preimage in S g of a disk with two marked pointsin X g + is an annulus.)For any odd g > g a slightmodification is needed; for instance to obtain an analogous example for S fromFigure 4, we move the left-hand endpoints of d and d together and we move theright-hand endpoints together as well, giving a collection of arcs in X . d d d F IGURE Left:
Arcs in X giving a tight geodesic in C ( S ) ; Middle:
The outer 10-gon in X cut along d and d shown with arcs of d ; Right:
The preimage of the 10-gon in S shown with the preimage of d We now give examples of geodesics that are tight but not efficient. Consider thearcs d , d , and d in shown in the left-hand side of Figure 5. Let d be the boundaryof a regular neighborhood of d ∪ d ; this d is a curve surrounding three markedpoints. Let v , v , v , v be the corresponding path in C ( S ) . We have d ( v , v ) ≥ d and d fill X . By definition v , v , v , v is tight at v (meaning that thethird part of the definition of a tight multigeodesic is satisfied for i =
1) and it isstraightforward to check that it is tight at v ; so more than being contained in a tight FFICIENT GEODESICS AND AN EFFECTIVE ALGORITHM FOR DISTANCE 11 multigeodesic, the given geodesic is itself a tight multigeodesic (in other words, v , v , v , v is a tight multigeodesic with a single associated tight geodesic).We will now show that the oriented geodesic v , v , v , v is not initially efficient.If we cut X along d and d there is a single region that is not a bigon with onemarked point, namely, the region containing the (umarked!) point at infinity. Thereare five arcs of d in this disk as shown in the middle picture of Figure 5 (the exactconfiguration relative to the marked point is important here). The preimage of this10-gon in S is a 20-gon, and the arcs of the preimage of d are arranged as inthe right-hand side of Figure 5. It is easy to find a reference arc in this polygonthat intersects the preimage of d in more than two points. Thus v , v , v , v isnot initially efficient; of course this implies that v , v , v , v is not efficient. Thegeneralization to higher genus should be clear. d d d d d F IGURE
6. Arcs giving an efficient geodesic in C ( S ) that is not tight Finally we give examples of geodesics that are efficient but not tight. Considerthe arcs d , d , d , and d shown in Figure 6 (the arcs d , d , and d are shown inthe top picture of the figure and the arcs d , d , and d are shown at the bottom).Again, each d i represents a vertex v i of C ( S ) and again d ( v , v ) = d and d fill X .To see that the oriented geodesic v , v , v , v is initially efficient we notice that d lies in a single region of X determined by d and d and in that region it connectstwo marked points, one of which lies on d . It follows that the preimage of d in S is a single nonseparating simple closed curve and if we cut S along the preimagesof d and d then this nonseparating curve becomes a single diagonal in a singlepolygonal region of the cut-open surface. From this it follows that v , v , v , v isinitially efficient. A similar argument shows that the oriented geodesic v , v , v , v is initially ef-ficient. Indeed, the intersection of the arc d with each region of X determined by d and d is a single arc. It follows that the preimage of d in S intersects eachpolygonal region of S in one or two arcs (depending on whether the correspond-ing arc in X terminates at a marked point not contained in d ∪ d ). As such, anyreference arc in S for the preimages of d and d can intersect the preimage of d in at most two points. The efficiency of v , v , v , v follows.We will now show that v , v , v , v is not tight, in other words that v , v , v , v isnot contained in any tight multigeodesic. Suppose s , s , s , s were a tight multi-geodesic containing v , v , v , v . First of all, by definition we would have s = v and s = v . Second, since (representatives of) v and v fill the complement of(a representative of) v we must have that s = v . Now we notice that v doesnot lie in a regular neighborhood of the union of representatives of s = v and s = v (since we can find an arc in X that intersects d without intersecting d or d ). Therefore, for any choice of simplex s containing v we will still have theproperty that s does not lie in a regular neighborhood of the union of representa-tives of s and s ; in particular, for any choice of s containing v , the sequence s , s , s , s is not tight at s . Hence v , v , v , v is not tight, as desired. Againthe generalization to higher genus is clear. (cid:3)
3. E
XISTENCE OF EFFICIENT PATHS
In this section we prove the main result of this paper, Theorem 1.1. The mainpoint is to prove the existence of initially efficient geodesics (Proposition 3.2), andthis will occupy most of the section. At the end we give the additional inductiveargument for the existence of efficient geodesics (Theorem 1.1). Let g ≥ Setup: a reducibility criterion.
Our first goal is to recast the problem offinding initially efficient paths in terms of sequences of numbers; see Proposi-tion 3.1 below.
Standard representatives and intersection sequences.
Let v and w be vertices of C ( S g ) with d ( v , w ) ≥
3. Let v = v , . . . , v n = w be an arbitrary path from v to w .We can choose representatives a i of the v i with the following properties:(1) each a i is in minimal position with both a and a n ,(2) each intersection a i ∩ a i + is empty, and(3) all triple intersections of the form a i ∩ a j ∩ a k are empty.To do this, we take the a i to be geodesics with respect to some hyperbolic metricon S g and then perform small isotopies to remove triple intersections. We say thatsuch a collection of representatives for the v i is standard . Note that we do not insistthat a i and a j are in minimal position when 0 < i , j < n and | i − j | > g be a reference arc for the standard set of representatives a , . . . , a n , bywhich we mean that:(1) g has its interior disjoint from a ∪ a n ,(2) g has endpoints disjoint from a , . . . , a n − , FFICIENT GEODESICS AND AN EFFECTIVE ALGORITHM FOR DISTANCE 13 (3) all triple intersections a i ∩ a j ∩ g are trivial for i = j , and(4) g is in minimal position with each of a , . . . , a n − .A reference arc for a , . . . , a n is automatically a reference arc for the triple a , a , a n as in the introduction, but not the other way around. We will need to deal with thisdiscrepancy in the proof of Proposition 3.2 below.Denote the cardinality of g ∩ ( a ∪ · · · ∪ a n − ) by N . Traversing g in the di-rection of some chosen orientation, we record the sequence of natural numbers s = ( j , j , . . . , j N ) ∈ { , . . . , n − } N so that the i th intersection point of g with a ∪ · · · ∪ a n − lies in a j i . We refer to s as the intersection sequence of the a i along g . Complexity of paths and reducible sequences.
We define the complexity of an ori-ented path v , . . . , v n in C ( S ) to be n − (cid:229) k = ( i ( v , v k ) + i ( v k , v n )) . We say that a sequence s of natural numbers is reducible under the followingcircumstances: whenever s arises as an intersection sequence for a (standard setof representatives for) path v , . . . , v n in C ( S g ) there is another path v ′ , . . . , v ′ n with v ′ = v and v ′ n = v n and with smaller complexity. With this terminology in hand, theexistence of initially efficient paths is a consequence of the following proposition. Proposition 3.1.
Suppose s is a sequence of elements of { , . . . , n − } . If s hasmore than n − entries equal to 1, then s is reducible. We can deduce the existence of initially efficient geodesics easily from Proposi-tion 3.1.
Proposition 3.2.
Let g ≥ . If v and w are vertices of C ( S g ) with d ( v , w ) ≥ , thenthere exists an initially efficient geodesic from v to w.Proof of Proposition 3.2 assuming Proposition 3.1. Let v and w be vertices of C ( S g ) with d ( v , w ) ≥
3. Since the complexity of any path from v to w is a natural number,there is a geodesic of minimal complexity. We will show that any geodesic from v to w that has minimal complexity must be initially efficient.To this end, we consider an arbitrary geodesic v = v , . . . , v n = w and we as-sume that it is not initially efficient. In other words there is a set of representa-tives a , a , a n for v , v , v n that are in minimal position and a reference arc g for a , a , a n with | a ∩ g | > n − a , a , a n to a set of standard representatives a , . . . , a n for the whole geodesic v , . . . , v n . What is more, we may assume that g is a refer-ence arc for this full set of representatives a , . . . , a n .Indeed, if g is not in minimal position with some a i with 2 ≤ i ≤ n − g and a i cobound an embedded bigon; if we choose an innermost such bigon (withrespect to g ) and push the corresponding a i across, then we can eliminate the bigonwithout creating any new points of intersection between g with any a j or between any two a j . (Alternatively, as in the introduction, we can assume that each a i with1 ≤ i ≤ n − a and a n and we can take g to be any straight line segment; this procedure always yields a g that is in minimal position with each a i ).Since we did not change a , the new intersection sequence of a , . . . , a n with g still has more than n − s isreducible. This implies that v , . . . , v n does not have minimal complexity, and weare done. (cid:3) Notice that the approach established in Proposition 3.1 disregards all informa-tion about a path in C ( S g ) except its intersection sequences. For instance, we willnot need to concern ourselves with how the strands of the a i are connected outsideof a neighborhood of g .We will prove Proposition 3.1 in three stages. First, in Section 3.2 we describea normal form for sequences of natural numbers (Lemma 3.3 below) and also de-scribe an associated diagram for the normal form called the dot graph. Next inSection 3.3 we will show that if the dot graph exhibits certain geometric features—empty boxes and hexagons—then the sequence is reducible (Lemma 3.4). Finallyin Section 3.4 we will show that any sequence in normal form that does not sat-isfy Proposition 3.1 has a dot graph exhibiting either an empty box or an emptyhexagon, hence proving Proposition 3.1.3.2. Stage 1: Sawtooth form and the dot graph.
The main goal of this sectionis to give a normal form for sequences of natural numbers that interacts well withour notion of reducibility. We also describe a way to diagram sequences in normalform called the dot graph.
Sawtooth form.
We say that a sequence ( j , j , . . . , j k ) of natural numbers is in sawtooth form if j i < j i + = ⇒ j i + = j i + . An example of a sequence in sawtooth form is ( , , , , , , , , , , , , ) . If asequence of natural numbers is in sawtooth form, we may consider its ascendingsequences , which are the maximal subsequences of the form k , k + , . . . , k + m . Inthe previous example, the ascending sequences are ( , ) , ( , , ) , ( , ) , ( , ) ,and ( , , , ) . Lemma 3.3.
Let s be an intersection sequence. There exists an intersection se-quence t in sawtooth form so that t differs from s by a permutation of its entriesand so that s is reducible if and only if t is.Proof. Suppose s = ( j , . . . , j N ) is the intersection sequence for a set of standardrepresentatives a , . . . , a n along an arc g ⊆ a n \ a . The basic idea we will useis that if | j i − j i + | >
1, then we can modify a j i and a j i + to new curves a ′ j i and a ′ j i + so that the new curves still form a set of standard representatives for thesame path and so that the new intersection sequence along g differs from s by atransposition of the consecutive terms j i and j i + ; see Figure 7. We call this theresulting modification of s a commutation. FFICIENT GEODESICS AND AN EFFECTIVE ALGORITHM FOR DISTANCE 15 g a j i a j i + a ′ j i a ′ j i + F IGURE
7. A commutation
It suffices to show that if a sequence s is not in sawtooth form, then it is possibleto perform a finite sequence of commutations so that the resulting sequence t is insawtooth form. Indeed, the sequence t appears as an intersection sequence for aparticular path in C ( S g ) if and only if s does (the key point is that commutationsnever result in a nonempty intersection of the form a i ∩ a i + ).We say that s fails to be in sawtooth form at the index i if j i + > j i +
1. Let k = k ( s ) be the highest index at which s fails to be in sawtooth form, and saythat k is zero if s is in sawtooth form. Assuming k >
0, we will show that wecan modify s by a sequence of commutations so that the highest index where theresulting sequence fails to be in sawtooth form is strictly less than k .We decompose s into a sequence of subsequences of s , namely, ( s , s , s , s ) where s is the singleton ( j k ) and s is the longest subsequence of s starting fromthe ( k + ) st term so that each term is greater than j k +
1. The sequences s and s are thus determined, and one or both might be empty.By a series of commutations, we can modify s to the sequence s ′ = ( s , s , s , s ) . We claim that k ( s ′ ) < k ( s ) . Since the length of s is k −
1, it is enough to showthat the subsequence ( s , s , s ) is in sawtooth form.By the definition of k , we know that s is in sawtooth form. Next, the lastterm of s is greater than j k + s is j k , and sothese terms satisfy the definition of sawtooth form. We know s = ( j k ) and thefirst term of s , call it j , is at most j k +
1, and so these terms are also in sawtoothform. Finally, the subsequence s is in sawtooth form by the definition of k . Thiscompletes the proof. (cid:3) Dot graphs.
It will be useful to draw the graph in R ≥ of a given sequence ofnatural numbers, where the sequence is regarded as a function { , . . . , N } → N .The points of the graph of a sequence s will be called dots . We decorate the graphby connecting the dots that lie on a given line of slope 1; these line segments willbe called ascending segments . The resulting decorated graph will be called the dotgraph of s and will be denoted G ( s ) ; see Figure 8. F IGURE
8. Example of dot graph of a sequence in sawtooth form
Stage 2: Dot graph polygons and surgery.
The goal of this section is to de-scribe certain geometric shapes than can arise in a dot graph, and then to prove thatif the dot graph G ( s ) admits one of these shapes then the sequence s is reducible(Lemma 3.4). Dot graph polygons.
We say that a polygon in the plane is a dot graph polygon if(1) the edges all have slope 0 or 1,(2) the edges of slope 0 have nonzero length, and(3) the vertices all have integer coordinates.The edges of slope 1 in a dot graph polygon are called ascending edges and theedges of slope 0 are called horizontal edges .Let s be a sequence of natural numbers in sawtooth form. A dot graph polygonis a s -polygon if:(1) the vertices are dots of G ( s ) and(2) the ascending edges are contained in ascending segments of G ( s ) . F IGURE
9. A box, a hexagon of type 1, and a hexagon of type 2; the red(darker) dots are required to be endpoints of ascending segments, whilethe blue (lighter) dots may or may not be endpoints A box in G ( s ) is a s -quadrilateral P with the following two properties:(1) the leftmost ascending edge contains the highest point of some ascendingsegment of G ( s ) and(2) the rightmost ascending edge contains the lowest point of some ascendingsegment of G ( s ) . FFICIENT GEODESICS AND AN EFFECTIVE ALGORITHM FOR DISTANCE 17
We will also need to deal with hexagons. Up to translation and changing theedge lengths, there are four types of dot graph hexagons; two have an acute exteriorangle, and we will not need to consider these. Notice that a dot graph hexagon nec-essarily has a leftmost ascending edge, a rightmost ascending edge, and a middleascending edge. This holds even for degenerate hexagons since horizontal edgesare required to have nonzero length.A hexagon of type 1 in G ( s ) is a s -hexagon where:(1) no exterior angle is acute,(2) the middle ascending edge is an entire ascending segment of G ( s ) , and(3) the minimum of the middle ascending edge equals the minimum of theleftmost ascending edge,(4) the leftmost ascending edge contains the highest point of an ascendingsegment of G ( s ) .Similarly, a hexagon of type 2 in G ( s ) is a s -hexagon that satisfies the first twoconditions above and the following third and fourth conditions:(3 ′ ) the maximum of the middle ascending edge equals the maximum of therightmost ascending edge,(4 ′ ) the rightmost ascending edge contains the lowest point of an ascendingsegment of G ( s ) .See Figure 9 for pictures of boxes and hexagons of types 1 and 2.The following lemma is the main goal of this section. We say that a horizontaledge of a s -polygon is pierced if its interior intersects G ( s ) . Also, we say that a s -polygon is empty if it there are no points of G ( s ) in its interior. Lemma 3.4.
Suppose that s is a sequence of natural numbers in sawtooth formand that G ( s ) has an empty, unpierced box or an empty, unpierced hexagon of type1 or 2. Then s is reducible. Before we prove Lemma 3.4, we need to introduce another topological tool,surgery on curves.
Surgery.
Let a be a simple closed curve in a surface and let g be an oriented arcso that a and g are in minimal position. We can form a new curve a ′ from a by performing surgery along g as follows. We first remove from a small openneighborhoods of two points of a ∩ g that are consecutive along g . What remainsof a is a pair of arcs; we can connect the endpoints of either arc by another arc d that lies in a small neighborhood of g in order to create the new simple closedcurve a ′ (the other arc of a is discarded); see Figure 10.We draw a neighborhood of g in the plane so that g is a horizontal arc orientedto the right. We say that a ′ is obtained from a by ++ , + − , − + , or −− surgeryalong g ; the first symbol is + or − depending on whether the first endpoint of d (as measured by the orientation of g ) lies above g or below, and similarly for thesecond symbol.In general, for a given pair of intersection points of a curve a with g , exactlytwo of the four possible surgeries result in a simple closed curve. If we orient a , then the two intersection points of a with g can either agree or disagree. If theyagree, then the + − and − + surgeries, the odd surgeries , result in a simple closedcurve, and if they disagree, the ++ and −− surgeries, the even surgeries , result ina simple closed curve. a ag ++ −− + − − + F IGURE
10. The four types of surgery on a curve along an arc
These surgeries will of course only be of use to us if the curve a ′ is an essentialsimple closed curve in S . One variant of the well-known bigon criterion is thata curve a and an arc g are in minimal position if and only if every closed curveformed from a and g as above is essential. Indeed, the proof in the case where a and g are both curves (see [1, Proposition 3.10]) can be adapted to this case. Thusour a ′ is essential. Surfaces with boundary.
In order to show that our surgered curves are essential,we used a version of the bigon criterion. This bigon criterion is exactly what failsin the case of surfaces with boundary. For instance, suppose that the surface S hasat least two boundary components and consider a simple closed curve a that cutsoff a pair of pants in S . If g is an arc that intersects a in two points then both ofthe curves obtained by surgering a along g are homotopic to components of theboundary of S , neither of which represents a vertex of C ( S ) .We now use the surgeries described above to prove that a dot graph with anempty, unpierced box or an empty, unpierced hexagon of type 1 or 2 correspondsto a sequence that is reducible. Proof of Lemma 3.4.
Suppose that s appears as an intersection sequence for a ref-erence arc g for a set of standard representatives a , . . . , a n for a path v , . . . , v n in C ( S g ) . We need to replace the a i with new curves a ′ i so that the resulting pathfrom v to v n has smaller complexity. We treat the three cases in turn, accordingto whether G ( s ) has an empty, unpierced box or an empty, unpierced hexagon oftype 1 or 2.Suppose G ( s ) has an empty, unpierced box P . By the definitions of sawtoothform and empty boxes there are no ascending edges of G ( s ) in the vertical stripbetween the two ascending edges of P , that is, the dots of P correspond to a con-secutive sequence of intersections along g : a k , . . . , a k + m , a k , . . . , a k + m where 1 ≤ k ≤ k + m ≤ n − FFICIENT GEODESICS AND AN EFFECTIVE ALGORITHM FOR DISTANCE 19 ′ ′ ′ − + ++ + − F IGURE
11. An example of a set of surgeries as in the box case of Lemma 3.4
First, for i / ∈ { k , . . . , k + m } we set a ′ i = a i . We then define a ′ k , . . . , a ′ k + m induc-tively: for i = k , . . . , k + m , the curve a ′ i is obtained by performing surgery along g between the two points of a i ∩ g corresponding to dots of P and the surgeries arechosen so that they form a path in the directed graph in Figure 12 (of course foreach i we must choose one of the two surgeries that results in a closed curve). +++ − − + −− F IGURE
12. The directed graph used in the proof of Lemma 3.4
The vertices of the graph in Figure 12 correspond to the four types of surgeries: ++ , + − , − + , and −− , and the rule is that the second sign of the origin of adirected edge is the opposite of the first sign of the terminus. Since every vertex hasone outgoing arrow pointing to an even surgery and one outgoing arrow pointingto an odd surgery, the desired sequence of surgeries exists; in fact it is completelydetermined by the choice of surgery on a k , and so there are exactly two possiblesequences. See Figure 11 for an example of this procedure; there we perform + − surgery on a , then ++ surgery on a , then − + surgery on a .For 0 ≤ i ≤ n , let v ′ i be the vertex of C ( S g ) represented by a ′ i . We need to checkthat the v ′ i certify the reducibility of s , namely that(1) v ′ = v and v ′ n = v n ,(2) each v ′ i is connected to v ′ i + by an edge in C ( S g ) , and (3) the complexity of v ′ , . . . , v ′ n is strictly smaller than that of v , . . . , v n .The first condition holds because 1 ≤ k ≤ k + m ≤ n −
1. The second conditionholds because each intersection a i ∩ a i + is empty and the surgeries do not createnew intersections. For the third condition, we claim that something stronger is true,namely, that i ( v , v ′ i ) + i ( v ′ i , v n ) ≤ i ( v , v i ) + i ( v i , v n ) for all i and that i ( v , v ′ k ) + i ( v ′ k , v n ) < i ( v , v k ) + i ( v k , v n ) . Indeed, if we consider the polygonal decomposition of S g determined by a ∪ a n we see that when we surger two strands of some a i along g we create no newintersections with a ∪ a n and we remove two intersections with a ∪ a n (we mightalso create a bigon, but this would only help our case). Since we performed at leastone surgery—on a k —our claim is proven. ′ ′ ′ ′ × ′ − + ++ + − × −− − + IGURE
13. An example of a set of surgeries as in the hexagon case ofLemma 3.4
The cases of empty, unpierced hexagons of types 1 and 2 are similar, but onenew idea is needed. These two cases are almost identical, and so we will only treatthe first case, that is, we suppose G ( s ) has an empty, unpierced hexagon P of type1. By the definition of sawtooth form and the definition of an empty, unpiercedhexagon of type 1, there are no ascending segments of G ( s ) in the vertical stripbetween the leftmost and middle ascending edges of P and any ascending segmentsof G ( s ) that lie in the vertical strip between the middle and rightmost ascendingsegments have their highest point strictly below the lower-right horizontal edge of P . It follows that the dots of P correspond to a sequence of intersections along g ofthe following form a k , . . . , a k + m , a k , . . . , a k + ℓ , a j , . . . , a j p , a k + ℓ , . . . , a k + m FFICIENT GEODESICS AND AN EFFECTIVE ALGORITHM FOR DISTANCE 21 where 1 ≤ k ≤ k + ℓ ≤ k + m ≤ n − p ≥
0, and each j i < a k + ℓ . See Figure 13 foran example where k = ℓ = m =
4, and p = a i with i / ∈ { k , . . . , k + m } we set a ′ i = a i . Each of the remaining a i corresponds to exactly two dots in P except for a k + ℓ , which corresponds tothree. Let a ′ k + ℓ be the curve obtained from a ′ k + ℓ via surgery along g between thefirst two (leftmost) points of a ′ k + ℓ ∩ g corresponding to dots of P and satisfyingthe following property: a ′ k + ℓ does not contain the arc of a k + ℓ containing the third(rightmost) point of a k + ℓ ∩ g corresponding to a dot of P . As always, there are twochoices of surgery given two consecutive points of a k + ℓ ∩ g ; one contains this thirdintersection point and one does not.We then define a ′ k + ℓ − , . . . , a ′ k inductively as before using the diagram above(notice the reversed order), and finally we define a ′ k + ℓ + , . . . , a ′ k + m inductively asbefore.By our choice of a ′ k + ℓ , we have that a ′ k + ℓ ∩ a ′ k + ℓ + = /0, as required; indeed,we eliminated the strand of a ′ k + ℓ that was in the way between the two strands of a k + ℓ + being surgered. Also, since each j i is strictly less than k + ℓ , the curves a ′ k + ℓ + , . . . , a ′ k + m satisfy the condition that a ′ i ∩ a ′ i + = /0. The other conditions inthe definition of a reducible sequence are easily verified as before. This completesthe proof of the lemma. (cid:3) Stage 3: Innermost polygons.
In this section we will put together Lem-mas 3.3 and 3.4 in order to prove Proposition 3.1. We begin with two lemmas.
Lemma 3.5.
If a dot graph G ( s ) contains a box P pierced in exactly one edge,then it contains an unpierced box.Proof. Denote the ascending edges of P by e and f . There is an ascending segment e ′ intersecting the interior of exactly one of the two horizontal edges of P ; wechoose e ′ to be rightmost if it intersects the bottom edge of P and leftmost if itintersects the top edge. Either way, we find a box P ′ pierced in at most one edgeand where one ascending edge is contained in e ′ and the other ascending edge iscontained in P . The box P ′ has horizontal edges strictly shorter than those of P .Therefore, we may repeat the process until it eventually terminates, at which pointwe find the desired unpierced box. (cid:3) Lemma 3.6.
Among all unpierced boxes and hexagons of type 1 and 2 in a dotgraph G ( s ) , an innermost unpierced box or hexagon of type 1 or 2 is empty.Proof. We treat the three cases separately. First suppose that P is an unpierced boxthat is not empty. We will show that P either contains another unpierced box oran unpierced hexagon of type 1. Let e be an ascending segment contained in theinterior of P . We choose e so that max ( e ) is maximal among all such ascendingsegments, and we further choose e to be rightmost among all ascending segmentswith maximum equal to max ( e ) .There is a unique (possibly degenerate) hexagon P ′ of type 1 with one edge equalto e , and the other two edges contained in the ascending edges of P ; see the left-hand side of Figure 14. If P ′ is unpierced, we are done, so assume that P ′ is pierced. F IGURE
14. Inside a box, inside a hexagon, inside a hexagon
By construction, the top horizontal edge of P ′ and the lower-right horizontal edgeof P ′ are unpierced. Suppose that the interior of the lower-left horizontal edge of P ′ were pierced. Let e ′ be the rightmost ascending segment of G ( s ) that piercesthis edge of P ′ . By the choice of e , we have that max ( e ′ ) ≤ max ( e ) , and so there isa box pierced in at most one edge whose ascending edges are contained in e ′ and e . By Lemma 3.5, there is an unpierced box contained in this pierced box, and so P is not innermost.The second case is where P is an unpierced hexagon of type 1. Again supposethat P is not empty. Let e be an ascending segment contained in the interior of P that has the largest maximum max ( e ) over all such segments and is rightmostamong all such ascending segments. Let m denote the middle ascending edge of P . It follows from the fact that s is in sawtooth form that there are no ascendingsegments of G ( s ) that lie inside P and to the right of m ; so e lies to the left of m .We now treat two subcases, depending on whether max ( e ) > max ( m ) or not.If max ( e ) > max ( m ) , there is a maximal hexagon P ′ of type 1 with ascendingedges contained in P ∪ e as in the middle picture of Figure 14. By the same ar-gument as in the previous case, P ′ is either unpierced or it contains an unpiercedbox.If max ( e ) ≤ max ( m ) , the argument is similar. There is a hexagon P ′ of type 2as shown in the right-hand side of Figure 14. The topmost edge of P ′ is unpiercedby the choice of e . The bottom edge of P ′ is unpierced since it is a horizontal edgefor P , which is unpierced. And if the third horizontal edge of P ′ were pierced, wecould find a box pierced in at most one edge, hence an unpierced box, as in theprevious cases. It follows that P ′ is unpierced and again P is not innermost.The third and final case is where P is an unpierced hexagon of type 2. This iscompletely analogous to the previous case; in fact, if we rotate the two picturesfrom the type 1 case by p we obtain the required pictures for the type 2 case. (cid:3) We can now use the two previous lemmas to prove Proposition 3.1.
Proof of Proposition 3.1.
Let s be a sequence of elements of { , . . . , n − } . ByLemma 3.3 we may assume that s is in sawtooth form without changing the num-ber of entries equal to 1; call this number k . Let e , . . . e k denote the ascendingsegments of G ( s ) with minimum equal to 1, ordered from left to right.If max ( e i + ) < max ( e i ) for all i , then since max ( e ) ≤ n − k ≤ n −
1. Therefore, it suffices to show that if max ( e i + ) ≥ max ( e i ) for some i then s is reducible. FFICIENT GEODESICS AND AN EFFECTIVE ALGORITHM FOR DISTANCE 23
Suppose then that max ( e i + ) ≥ max ( e i ) for some i . The first step is to show that G ( s ) has an unpierced box. Let e be the first ascending segment (from left to right)that appears after e i and has max ( e ) ≥ max ( e i ) . Because min ( e ) ≥ min ( e i ) = P with two edges contained in e i and e and two horizontal edges with heights min ( e ) and max ( e i ) . By the definition of e ,the interior of the upper horizontal edge of P is disjoint from G ( s ) , so P is piercedin at most one edge. By Lemma 3.5, P contains an unpierced box.Let P now be an innermost unpierced box or hexagon of type 1 or 2; such P exists because each s -polygon contains a finite number of dots of G ( s ) and apolygon contained inside another polygon contains a fewer number of dots. ByLemma 3.6, the polygon P is empty. By Lemma 3.4, s is reducible. (cid:3) From initially efficient geodesics to efficient geodesics.
At this point wehave established the existence of initially efficient geodesics (Proposition 3.2). Itremains to establish the existence of efficient geodesics (Theorem 1.1).
Total complexity.
For an oriented path q in C ( S g ) with vertices w , . . . , w n definethe complexity k ( q ) as before: k ( q ) = n − (cid:229) k = ( i ( w , w k ) + i ( w k , w n )) . Next, for an oriented path p with vertices v , . . . , v n , let p be the oriented path v n , . . . , v n − and let p k be the oriented path v n − k − , . . . , v n for 2 ≤ k ≤ n −
1. Wewill relabel the vertices of p k as w , . . . , w n k . The total complexity of a path p is theordered ( n − ) -tuple: ˆ k ( p ) = ( k ( p ) , . . . , k ( p n − )) . We order the set N n − —hence the set of total complexities—lexicographically. Proof of Theorem 1.1.
Let v and w be vertices of C ( S g ) with d ( v , w ) ≥
3. We claimthat any geodesic from v to w that has minimal total complexity must be efficient.Let p be an arbitrary geodesic v = v , . . . , v n = w and assume that p is not effi-cient. In other words, one of the corresponding paths p k with vertices w , . . . , w n k is not initially efficient. This is the same as saying that there is a set of represen-tatives b , b , b n k for w , w , w n k that are in minimal position and a reference arc g with | b ∩ g | > n k − b , b , b n k to a fullstandard set of representatives b , . . . , b n k for p k . And as in that proof there aresurgeries that reduce the complexity of p k . The curves obtained by these surgeriesnot only give a new path between the endpoints of p k , but they also give rise to anew path between v and w .The key observation here is that, by our choice of the order of the p i , the surg-eries used in modifying p k do not increase the complexity of any p i with i < k .Indeed, these surgeries do not increase the intersection between any of the curves b , . . . , b n k and all of the vertices of p used in the computation of k ( p i ) with i < k are already vertices of p k , namely, the vertices represented by b , . . . , b n k . Thetheorem follows. (cid:3) An improved algorithm in a special case.
We end this section by statingand proving the alternate version of the efficient geodesic algorithm that was usedin the example at the start of Section 2.1. This proposition is equivalent to the maintheorem (Theorem 1.1) of the first version of this paper [4].
Proposition 3.7.
Suppose v and w are vertices of C ( S g ) with d ( v , w ) ≥ . Let a and b be representatives of a and b that are in minimal position. Then there isa geodesic v = v , . . . , v n = w and a representative a of v so that the number ofintersections of a with each arc of b \ a is at most d ( v , w ) − .Proof. The proof is essentially the same as the proof of Theorem 1.1. The onlyadded observation is that, since g is a subset of b , every intersection sequence canbe taken to have entries in { , . . . , n − } instead of { , . . . , n − } . (cid:3) Note that in the special case that vertices v and w have representatives a and b that cut the surface into rectangles and hexagons only (e.g. the example of Sec-tion 2), then every reference arc is parallel to a reference arc as in Proposition 3.7,and so in this case there are geodesics that are extra efficient in the sense that theintersection of a representative of v with any reference arc is at most n − n −
1. A
PPENDIX
A. W
EBB ’ S ALGORITHM
In this appendix we give an exposition of Webb’s algorithm for computing dis-tance in C ( S ) . As with the efficient geodesic algorithm we will make the inductivehypothesis that for some n ≥ , . . . , n − n . First we introduce an auxiliarytool, the arc complex for a surface with boundary. Arc complex.
Let F be a compact surface with nonempty boundary. The arc com-plex A ( F ) is the simplicial complex with k -simplices corresponding to ( k + ) -tuples of homotopy classes of essential arcs in F with pairwise disjoint represen-tatives. Here, homotopies are allowed to move the endpoints of an arc along ¶ F ,and an arc is essential if it is not homotopic into ¶ F . The algorithm.
A maximal simplex of A ( F ) can be regarded as a triangulation ofthe surface obtained from F by collapsing each component of the boundary to apoint. If F is a compact, orientable surface of genus g with m boundary compo-nents, then the number of edges in any such triangulation is 6 g + m − v and w be two vertices of C ( S ) with d ( v , w ) ≥
3. As in the efficient geodesicalgorithm, it suffices by the induction hypothesis to list all candidates for vertices v on a tight geodesic v = v , . . . , v n = w . Since there are finitely many vertices ineach simplex of C ( S ) it further suffices to list all candidates for simplices s on atight multigeodesic v = s , . . . , s n = w .Suppose we have such a tight multigeodesic v = s , . . . , s n = w . We can chooserepresentatives a i of the s i so that a i ∩ a i + = /0 for all i and so that each a i liesin minimal position with a . If we cut S along a , we obtain a compact surface S ′ ,some of whose boundary components correspond to a . FFICIENT GEODESICS AND AN EFFECTIVE ALGORITHM FOR DISTANCE 25
For each i >
1, the representative a i gives a collection of disjoint arcs in S ′ and hence a simplex t i of A ( S ′ ) (some arcs of a i might be parallel and these getidentified in A ( S ′ ) ). For i ≥
3, the collection of arcs is filling, which means thatwhen we cut S ′ along these arcs we obtain a collection of disks and boundary-parallel annuli, and we say that the corresponding simplex of A ( S ′ ) is filling.Since there is a unique configuration for a n and a in minimal position, thereis a unique possibility for t n . As t n ∪ t n − is contained in a simplex of the arccomplex of S ′ and since t n is filling, there are finitely many possibilities for t n − (and we can explicitly list them). This is the key point: there are infinitely manyvertices of C ( S ) that correspond to any given simplex in the arc complex, but thereare finitely many choices for the simplex itself.Because t i is filling whenever i ≥
3, we can continue this process inductively,and explicitly list all possibilities for t . Now, by the definition of a tight multi-geodesic in C ( S ) , the simplex s is represented by the union of the essential com-ponents of the boundary of a regular neighborhood of a ∪ a . Equivalently, anysuch s is given by a regular neighborhood of the union of ¶ S ′ with a representa-tive of t . Hence there are finitely many (explicitly listable) possibilities for s , asdesired. A bound on the number of candidates.
In the introduction we stated that the num-ber of candidate simplices s produced by Webb’s algorithm when d ( v , w ) = n isbounded above by 2 ( g + ) min { n − , } ( g − − ) . We will now explain this bound; we are grateful to Richard Webb for supplying uswith the details.We can think of the sequence t n , . . . , t as a path in the filling multi-arc complex,that is, the simplicial complex whose vertices are simplices of A ( S ′ ) whose geo-metric realizations fill S ′ and whose edges correspond to simplices with geometricintersection number zero. Then we obtain t by extending this path by one moreedge and taking some nonempty subset of the simplex of A ( S ′ ) represented by theendpoint ˆ t of this extended path.Webb proved that the degree of an arbitrary vertex of this filling multi-arc com-plex is bounded above by 2 g + (this is for the case where we start with a closedsurface of genus g and cut along a single simple closed curve, as above); see hispaper [19]. Our extended path from t n to ˆ t n − has length n − ( g + )( n − ) for the number of possibilities for ˆ t . However,there is a version of the bounded geodesic image theorem which tells us that, be-cause the t i arise from a geodesic in C ( S ) , the actual distance in the filling multi-arccomplex between t n and ˆ t is bounded above by 21. This gives the first multipli-cand in the desired bound. The second multiplicand comes from the number ofways of choosing a nonempty sub-simplex t of ˆ t . The number of vertices of t is bounded above by 6 g −
6, and so there are 2 g − − t from ˆ t . R EFERENCES [1]
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