The number of surfaces of fixed genus in an alternating link complement
TTHE NUMBER OF SURFACES OF FIXED GENUS IN ANALTERNATING LINK COMPLEMENT
JOEL HASS, ABIGAIL THOMPSON, AND ANASTASIIA TSVIETKOVA
Abstract.
Let L be a prime alternating link with n crossings. We show that for eachfixed g , the number of genus g incompressible surfaces in the complement of L is boundedby an explicitly given polynomial in n . Previous bounds were exponential in n . Overview
Let L be a non-split prime alternating link with an n -crossing diagram and let M L denotethe complement of L in S . In this paper we give a bound on the number of isotopy classesof closed incompressible surfaces of genus g embedded in M L . For each choice of genus g , this bound is a polynomial in n . For example, we show that the number of genus twoincompressible surfaces in M L is at most 12 n . More generally, we show that the numberof genus g incompressible surfaces in the complement of an n -crossing alternating diagramis at most C g n g , where C g is a constant depending only on the genus.The surfaces we consider are closed and incompressible, but not necessarily disjoint.The number of disjoint incompressible surfaces in a manifold is much easier to bound,as originally observed by Kneser [7]. Kneser showed that the number of such surfaces isbounded by a linear function of the number of tetrahedra t required to triangulate themanifold. For link complements, the number t is itself a linear multiple of the number ofcrossings in a link diagram n .In non-hyperbolic manifolds there can be infinitely many distinct incompressible surfacesof a fixed genus, as for example in the 3-torus, which has infinitely many non-isotopicessential tori. In a hyperbolic manifold the number of such surfaces is always finite, as canbe seen by isotoping each surface to a least area representative and applying the Gauss-Bonnet Theorem and Schoen’s curvature estimates [11, 2]. This argument applies also to π -injective immersions, but is not constructive and gives no explicit bound on the numberof surfaces of a given genus.One approach to counting the number of embedded incompressible surfaces of genus g isthrough normal surface theory. Each incompressible surface can be isotoped to be normal,and can then be expressed as a sum of certain fundamental normal surfaces. However this Key words and phrases. alternating link, incompressible surface.Partially supported by NSF grant 1117663, BSF grant 2012188, and the Ambrose Monell Foundation.Partially supported by NSF grant DMS-1207765; Neil Chriss and Natasha Herron Chris Founders’ CircleMember, IAS 2015-2016.Partially supported by NSF grant DMS-1406588. a r X i v : . [ m a t h . G T ] J un JOEL HASS, ABIGAIL THOMPSON, AND ANASTASIIA TSVIETKOVA process leads to an exponential bound on the number of incompressible surfaces of genus g , either in terms of the number of tetrahedra in a triangulation t , or in terms of thecrossing number n . The main issue is that the number of fundamental surfaces of a givengenus, and even the number of vertex fundamental surfaces, can be exponential in t [3]. Asecond difficulty in applying normal surfaces is that an incompressible surface may not befundamental, so that one must also count all Haken sums of normal surfaces that can resultin a given genus [1].Bounds for the number of immersed π -injective surfaces in a closed hyperbolic 3-manifold,up to homotopy, have been obtained by Masters and by Kahn and Markovic. Mastersshowed that the number of surface subgroups in a closed hyperbolic 3-manifold M , up toconjugacy, is at most g Cg for some C [8]. Kahn and Markovic showed that this number isbounded below by ( cg ) g and above by ( Cg ) g for constants c, C depending on M [6]. Thesepapers consider the number of immersed surfaces in a fixed closed manifold as the genusgrows. In contrast, we obtain bounds on the number of embedded surfaces in terms of boththe genus of the surface and the complexity of the hyperbolic manifold. Our bounds applyto a class of link complements, as opposed to the closed manifolds previously considered.2. Standard position for a surface
We first review techniques of Menasco that place an incompressible surface in M L into astandard position with respect to a projection plane Q for L [9]. We begin by isotoping thelink L so that its diagram is alternating and reduced, and so that it lies in the projectionplane Q with the exception of two small arcs near each crossing, one of which drops below Q , and one of which rises above it. L then lies on a union of two overlapping 2-spheres in S , S + and S − , which agree with Q except along bubbles surrounding each crossing. At thebubbles, S + and S − go over the top and bottom hemispheres of each bubble, respectively.We denote by B + and B − the balls in S lying respectively above S + and below S − .We then compress along curves that are parallel to a meridian of the link. A meridianalcompression on F surgers a curve in F , parallel in M L to a meridian of L , and creates anew pair of meridianal punctures.A surface is meridianally incompressible if no meridianal surgery can be performed.Menasco showed in [9] how to isotope a meridianally incompressible surface into standardposition with respect to S + and S − . The properties of a surface in standard position, whichwe call F , are summarized in Lemma 2.1.We can assign to each curve C in F ∩ S + or F ∩ S − a word in the letters P and S , definedup to cyclic order, with P indicating a point where C crosses a strand of the link alonga puncture and S indicating that the curve passes through a bubble region adjacent to asaddle of F . Figures 1 (1), 1 (2) and 1 (3) depict a component of F ∩ S + giving an SSSS , SP P P P P and
P P P S curve on S + .We define a complexity | F | = p + s + c where p is the sum of the number of P ’s associatedto all curves in F ∩ S + , s is the sum of the number of S ’s associated to these words, and c is the sum of the number of curves in F ∩ S + plus the number of curves in F ∩ S − . If HE NUMBER OF SURFACES OF FIXED GENUS IN AN ALTERNATING LINK COMPLEMENT 3 (1) (2) (3)
Figure 1.
The link L and a curve from F ∩ S ± F minimizes this sum among standard position surfaces in its isotopy class, then F is thensaid to be in | F | minimizing standard position . Lemma 2.1.
Suppose an incompressible and meridianally incompressible surface F is in | F | minimizing standard position. Then the curves of F ∩ S + and F ∩ S − and the associatedwords in the letters P, S satisfy the following properties: (1)
Each curve of F ∩ S + bounds a disk in F ∩ B + , and similarly each curve of F ∩ S − bounds a disk in F ∩ B − . (2) No curve passes through the same saddle twice. (3)
An innermost curve of F ∩ S + or F ∩ S − does not go through two successive saddles. (4) An equal number of curves pass through each side of a saddle. (5)
No curve has two successive punctures on the same arc of L with no intermediatesaddles. (6) No curve passes through a saddle and then crosses an arc of L adjacent to the saddle. (7) No word has the form P i S j for j > . (8) Each word contains at least two P ’s. (9) Each word has length at least four.Proof.
Property (1) follows from the incompressibility of F and that fact that the complexity | F | = p + s + c is minimized. Figures 1 (1), 1 (2) and 1 (3) show curves that are ruled out by(2), (5), (6) respectively. Properties (2), (5), (6) are proven in Lemma 3.2 of [4]. Property(3) holds since successive saddles lie on opposite sides of a curve, and thus cannot exist foran innermost curve. Property (4) follows from the fact that each saddle of a surface resultsin one intersection curve on each side of a crossing. The proofs of (7) and (8) are given inLemma 2 of [9]. For (9), first note that any collection of closed curves on a plane intersectin an even number of points, so that w has an even number of letters. The words P S and SS are not possible by (7) and P P is ruled out by (5) and the fact that the link is prime. (cid:3)
Lemma 2.1 severely restricts the types of surfaces in a non-split prime alternating linkcomplement. For example, Property 8 implies that there are no closed incompressible andmeridianally incompressible surfaces in the complement of L . This was used by Menascoto show that there are no totally geodesic surfaces in such a link complement. JOEL HASS, ABIGAIL THOMPSON, AND ANASTASIIA TSVIETKOVA Counting genus two surfaces
We first bound the number of genus two surfaces in M L . Lemma 3.1 follows from thearguments in Menasco [9]. We give the argument for completeness. Lemma 3.1.
Suppose F is a closed incompressible genus two surface in M L . Let F be theresult of a maximal number of meridianal compressions on F . Then F is a four-puncturedsphere. Furthermore, when placed in | F | minimizing standard position relative to S + ∪ S − , F intersects S + in either a single P P P P curve or in two
P SP S curves.Proof.
Meridianally compress F to obtain the meridianally incompressible surface F , andplace F in | F | minimizing standard position. Since F is incompressible, F intersects S + in at least one curve. The word associated to every intersection curve has length at leastfour, and has at least two P ’s by Lemma 2.1 (8) and (9). By Lemma 2.1 (7), every wordis either of the form P SP S , or
P P P P , or has length at least six. Since F has genus two,there are four meridianal punctures in F , and we have the following cases.1) If F has zero or two punctures, then the intersection pattern with S + has only two P ’s. Then there can then be only a single intersection curve which must have the form P SP S . But a saddle through which this curve passes has a distinct curve passing throughits opposite side, so at least four P ’s appear in curves of F ∩ S + .2) If F is a 4-punctured sphere, then every curve in F ∩ S + has associated word of the form P P P P or P SP S , with at least two P ’s. Since L is alternating, an innermost curve cannothave two consecutive saddles not separated by punctures. Moreover, if it contains a saddle,then F ∩ S + contains a second innermost curve. Since just four P ’s are available, thereis either a single P P P P curve which is innermost on both sides or each of two innermostcurves must have associated word
P SP S and this accounts for all P ’s and therefore allcurves. (cid:3) We now count the number of such curves. Suppose F is a closed incompressible genustwo surface in M L , and F is obtained from F by a sequence of two meridianal compressionsand that F is in standard position relative to S + ∪ S − with | F | minimized. Lemma 3.2.
The number of isotopy classes of curve configurations of F ∩ S + with F arisingfrom meridianal compressions of a genus two incompressible surface in M L is less than n .Proof. There are 2 n arcs in an n -crossing link diagram. For a P P P P word, the last punctureis determined by the location of the other three. Indeed, suppose two distinct
P P P P curves, c and c , have exactly three punctures that coincide. Then one can form a closed curvegoing through the other two punctures, and corresponding to a P P word. But such a curvein a reduced alternating diagram of a prime link is trivial, and therefore c is isotopic to c . A cyclic reordering of the initial edge punctured gives the same curve, so we divide byfour to get the number of configurations. Hence the number of P P P P curves is less than(2 n )(2 n − n − / n − n + n , obtained by picking three successive edges of thediagram to cross.We now bound the number of P SP S curves. There are n crossings, each with two sidesthrough which a curve can pass and contribute an S to a word. Suppose two curves emerge HE NUMBER OF SURFACES OF FIXED GENUS IN AN ALTERNATING LINK COMPLEMENT 5 from the same side of a crossing going in the same direction, cross distinct punctures, andthen enter a second saddle on the same side of a second crossing. Then there is a closedcurve intersecting the diagram in two punctures only, formed by joining the two arcs afterthey leave the first saddle and again just before they enter the second. This loop intersectsthe diagram in a
P P word, and since the diagram is prime, this loop is trivial and intersectsthe same arc of the diagram twice. It follows that the two original arcs are isotopic and thata
P SP S curve is completely determined by a choice of two saddles. One
P SP S curve thendetermines the second, since it determines the location of the second pair of saddles. Thenumber of
P SP S curves is then at most (cid:18) n (cid:19) = 2 n − n . Adding to the previous count givesa bound on the number of configurations of 2 n − n + n + 2 n − n = 2 n − n < n . (cid:3) Theorem 3.3.
The number of closed incompressible genus two surfaces in M L is less than n .Proof. A genus two surface is obtained from a meridianally incompressible surface by tubingtogether pairs of boundary components. By Lemma 3.1, every closed incompressible surfacemeridianally compresses to a surface with four boundary punctures.We obtain a closed surface from a surface with meridian boundary curves by a tubing pro-cedure that joins punctures on the same component of L in pairs. These pairs of puncturescannot be interleaved as one traverses once around a component of the link, as otherwisethe tubed surface could not be embedded. If L is a knot, there are two ways to partitionthe 4 punctures into pairs with no interleaving, and each of these gives rise to three pairs oftubes. Figure 2 (1) shows a knot L and a four-punctured sphere in M L . Figures 2 (2)–2 (7)show the six possible tubings. This is the n = 2 case of the general formula for the numberof ways to form n tubes starting with 2 n curves, which equals (cid:18) nn (cid:19) [10].If L is a link with more than one component, then each component of L meets at most fourpunctures. If two components meet two punctures each, then there are two ways to undo ameridianal boundary compression and obtain a closed surface for each of the components,giving a total of four choices. If one component of L meets all four punctures, there are atmost six ways to add tubes to F as before.Lemma 3.2 then implies that the number of closed incompressible surfaces of genus twoin M L is less than 6(2 n ) = 12 n . (cid:3) Decomposing a surface into polygons
Let F ⊂ M L be an incompressible genus g surface and let F be a meridianally incom-pressible surface obtained from F . Place F in standard position relative to S + ∪ S − with | F | minimized. In this section, we bound the number of curves in F ∩ S + and we bound themaximum length of the word associated to each curve. The method is based on studying adecomposition of F into polygons given by its intersections with S + and S − . Lemma 4.1.
Suppose that F is an embedded genus g incompressible surface in M L and that F is obtained by meridianal compression on F along pairwise non-parallel curves. Then JOEL HASS, ABIGAIL THOMPSON, AND ANASTASIIA TSVIETKOVA (1) (2) (3) (4)(5) (6) (7)
Figure 2.
Six ways to add two tubes to a four-punctured sphere in M L the number of meridianal compressions is at most g − and the number of meridianalboundary curves of F is at most g − .Proof. The compressing curves in F cut F into a collection of complementary surfaces F i .Each F i has an even number of boundary curves, since meridianal compression results in asurface that meets L in an even number of punctures. No component F i an annulus, sincethe meridianal compressions occur along non-parallel curves. Thus the Euler characteristicof each F i is negative and even. A surface with negative even Euler characteristic canbe divided into four-punctured spheres, each with Euler characteristic -2. Since their Eulercharacteristics add to the Euler characteristic of F , equal to 2 − g , the number of resulting4-punctured spheres is at most (2 − g ) / ( −
2) = g −
1. Each 4-punctured sphere contributesat most four meridianal boundary curves to F , and it follows that the number of meridianalboundary curves after a maximal set of meridianal compressions, is at most 4 g − (cid:3) Corollary 4.2.
A closed genus g surface F yields at most g − curves in F ∩ S + .Proof. From Lemma 4.1 we can have at most 4 g − F . By Lemma 2.1 (8),each curve accounts for at least two P ’s, so there are at most 2 g − F ∩ S + or F ∩ S − . (cid:3) We now bound the maximum length of the word associated to each curve. By filling ineach puncture of F we obtain a new surface ¯ F with Euler characteristic χ ( ¯ F ) = χ ( F ) + p .Decompose ¯ F into polygons as follows. Outside the bubbles containing the saddles of ¯ F wetake the arcs of ¯ F ∩ Q to form part of a graph on ¯ F . Each saddle in ¯ F is a disk with fourarcs of ¯ F ∩ Q meeting its boundary, and we cone these to the center of the saddle, wherewe add a vertex. The resulting graph has valence four vertices at the center of the saddlesand cuts ¯ F into a collection of polygons. By Lemma 2.1 (1), each polygon is homeomorphic HE NUMBER OF SURFACES OF FIXED GENUS IN AN ALTERNATING LINK COMPLEMENT 7 to a disk which lies in B + or B − away from neighborhoods of its vertices that lie in thebubbles. Four polygons meet at each of the saddle disks of ¯ F .The Euler characteristic of ¯ F can be computed by summing the contribution of each diskregion. Lemma 4.3.
The Euler characteristic of ¯ F can be computed by adding the contribution ofeach disk region, with a disk region E contributing − s / to χ ( ¯ F ) , where s is the numberof S ’s in the word associated to the boundary of E .Proof. Enumerate all curves C i , i = 1 , ..., r , in F ∩ S + and F ∩ S − . Suppose C i is theboundary of a polygon E i of ¯ F with interior in B + or B − . The Euler characteristic of ¯ F can be recovered by summing the contributions of each of these polygons E i , i = 1 , ..., r .The Euler characteristic χ ( ¯ F ) = v − e + f can be distributed so that +1 / s vertices and the same number of edges is s / − s / − s / . (cid:3) Lemma 4.4.
Suppose F is a closed genus g surface. Any curve in F ∩ S ± has an associatedword of length at most g − .Proof. Suppose w is the word associated to a curve C ⊂ ¯ F ∩ S + and C is the boundary ofa disk E in ¯ F ∩ B + . Let s be the number of saddles in w . By Lemma 2.1 (7), no word hasa single saddle, so we have the following cases:1) The word w consists solely of P (cid:48) s , i.e. s = 0. Then the curve C also bounds a diskin B − and F ∩ S + is a single curve, ¯ F is a sphere, and the length of w is at most 4 g − s ≥
2, and E contributes 1 − s / χ ( ¯ F ) .In Case (2) each curve contributes at most +1 / χ ( ¯ F ). Since there are at most 2 g − F ∩ S + , any subset of the words in F ∩ S + bounds polygons that together contributeat most g − χ ( ¯ F ). The same applies to words in F ∩ S − . Thus the positive contributionto χ ( ¯ F ) from any subset of complementary polygonal disks is bounded above by 2 g − F has at most g − χ ( ¯ F ) ≤ g − − g ≤ χ ( F ) + p = χ ( ¯ F ) ≤ g −
2. No combination of complementary polygonaldisks contributes more than 2 g − χ ( ¯ F ), and all complementary polygonal disks togethersum to χ ( ¯ F ). Thus no single polygonal disk can contribute less than χ ( ¯ F ) − (2 g − ≥ − g .A word of length greater than 20 g −
16 has at least 16 g −
12 saddles, and therefore contributesless than 4 − g to χ ( ¯ F ), by Lemma 4.3. We conclude that the length of any word is atmost 20 g − (cid:3) Bounding the number of surfaces
Fix, g ≥ N g ( L ) denote the number of closed incompressible genus g surfacesin S − L , up to isotopy. The following theorem bounds N g ( L ) by a polynomial function JOEL HASS, ABIGAIL THOMPSON, AND ANASTASIIA TSVIETKOVA of the crossing number of L . The argument gives explicit values for the constants in thebound. Theorem 5.1.
Suppose a non-split reduced prime alternating link diagram L has n cross-ings. Then N g ( L ) ≤ C g n g ,where C g is a constant that depends only on g .Proof. We choose one surface F in every isotopy class, and perform a maximal set ofmeridianal surgeries on F to obtain a surface F that is in | F | minimizing standard position.There are at most 2 g − F ∩ S + by Lemma 4.2, and each word has length atmost 20 g −
16 by Lemma 4.4. We give a bound on the number of non-isotopic meridianallyincompressible surfaces by counting all possible ways that these words can occur in a fixedlink diagram.There are n crossings, and every crossing gives 2 choices for the location of an S adjacentto that crossing. There are 2 n between-crossing edges in the link diagram, giving 2 n choicesfor where to locate a puncture. So for each curve of length at most 20 g −
16, there are atmost (4 n ) g − ways to choose successive saddles and punctures. For 2 g − F ∩ S + , the total number of choices is bounded by (4 n ) (20 g − g − ≤ (4 n ) g .To bound the number of closed surfaces, we consider all ways to tube together up to4 g − (cid:18) g − g − (cid:19) by Mossessian in [10]. Thus the number of closedincompressible genus g surfaces is at most (cid:18) g − g − (cid:19) (4 n ) g = (cid:18) g − g − (cid:19) g n g . We let C g = (cid:18) g − g − (cid:19) g and the Theorem follows. (cid:3) We can obtain the same bounds in most closed manifolds obtained by surgery on L Corollary 5.2.
Let L be a reduced alternating hyperbolic link with n crossing. The numberof genus g surfaces in the closed manifold obtained by ( p, q ) -surgery on M L is at most C g n g , with finitely many exceptional surgeries. The number of exceptional surgeriesdepends only on g .Proof. The only new closed incompressible genus g surfaces after ( p, q )-surgery are onesresulting from genus g incompressible surfaces with slope ( p, q ) in M L . Theorem 4.1 in [5]provides an upper bound for the number of distinct slopes ( p, q ) that bound incompressiblesurfaces, as a function of the genus g . The bound is given by a function N ( g ) given in[5]. For manifolds obtained by surgeries other than this finite collection, the incompressiblesurfaces are incompressible in the complement of L . (cid:3) Remark . The argument in Theorem 5.1 also gives an upper bound for the number ofnon-isotopic embedded essential meridianal surfaces in S − L . For such surfaces we don’tneed to consider all ways to tube meridians, so the bound is smaller for each genus. HE NUMBER OF SURFACES OF FIXED GENUS IN AN ALTERNATING LINK COMPLEMENT 9
Remark . If we fix a link L , and let the genus vary, Theorem 5.1 together with theexplicit expression for C g in the proof provide an exponential upper bound for the numberof embedded surfaces of arbitrary genus in S − L . In particular, the number grows atmost as C g + gL for a constant C L that can be written explicitly in terms of the number ofcrossings of L . While Masters and Kahn and Markovic results provide a better growth forimmersed surfaces in a closed hyperbolic 3-manifold, this is the first estimate of a similarnature for surfaces in any cusped manifold (in our case, for embedded surfaces in a primealternating link complement). References [1] W. Haken,
Theorie der Normalfl¨achen: Ein Isotopiekriterium f¨ur den Kreisknoten’ , Acta Math. , 105(1961) 245–375.[2] J. Hass,
Acylindrical surfaces in 3-manifolds , Michigan Math. J. 42 (1995), no. 2, 357–365.[3] J. Hass, J. C. Lagarias and N. Pippenger,
The computational complexity of Knot and Link problems ,Journal of the ACM, 46 (1999) 185–211.[4] J. Hass, W. Menasco,
Topologically rigid non-Haken 3-manifolds , J. Austral. Math. Soc. Ser. A 55(1993), no. 1, 60–71.[5] J. Hass, H. Rubinstein, S. Wang,
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Counting essential surfaces in a closed hyperbolic three-manifold , Geom. Topol.16 (2012), 601–624.[7] H. Kneser, “Geschlossene Fl¨achen in dreidimensionalen Mannigfaltigkeiten”,
Jahresbericht Math.Verein. , 28 (1929) 248–260.[8] J. D. Masters,
Counting immersed surfaces in hyperbolic 3-manifolds , Algebr. Geom. Topol. 5 (2005),835–864.[9] W. W. Menasco,
Closed incompressible surfaces in alternating knot and link complements , Topology (1984), 37–44.[10] G. Mossessian, Stabilizing Heegaard Splittings of High-Distance Knots , arXiv:1507.07231.[11] R. Schoen,
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103 of Ann. ofMath. Studies. Princeton University Press, 1983.[12] W. P. Thurston,
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Department of Mathematics, University of California, Davis California 95616 & School ofMathematics, Institute for Advanced Study, Princeton NJ 08540
E-mail address : [email protected] Department of Mathematics, University of California, Davis California 95616 & School ofMathematics, Institute for Advanced Study, Princeton NJ 08540
E-mail address : [email protected] Department of Mathematics, University of California, Davis California 95616
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