OOn Crossing Ball Structure in Knot and Link Complements
Wei Lin
Abstract
We develop a word mechanism applied in knot and link diagrams for the illustra-tion of a diagrammatic property. We also give a necessary condition for determiningincompressible and pairwise incompressible surfaces, that are embedded in knot or linkcomplements. Finally, we give a finiteness theorem and an upper bound on the Eulercharacteristic of such surfaces.
Let L ⊂ S be a link and π ( L ) ⊂ S ( ⊂ S ) be a regular link projection. Additionally, let F ⊂ S − L be an closed incompressible surface. In 1981 Menasco introduced his crossingball technology for classical link projections [7] that replaced π ( L ) in S with two 2-spheres, S ± , which had the salient features that L was embedded in S ∪ S − and S \ ( S ∪ S − )was a collection of open 3-balls— B ± that correspond to the boundaries S ± and a collectof crossing balls . (Please see § F into normal position with respect to S ± so that F ∩ S ± is a collection of simple closed curves (s.c.c.’s). When one imposes the assumption that π ( L ) is an alternating projection, the normal position of an essential surface is exceedinglywill behaved to the point where by direct observation one can definitively state whetherthe link is split, prime, cabled or a satellite. As such, any alternating knot can by directobservation be placed into one of William Thurston’s three categories—torus knot, satelliteknot or hyperbolic knot [10].One salient result from [7] is that any essential surface in a non-split alternating link exte-rior will contain a meridional curve of a link component and, thus, studying such essentialsurfaces can be reduced to studying essential surfaces with meridional boundary curves thatare meridianally incompressible or pairwise incompressible . The importance of studyingpairwise incompressible surfaces has been reflected in the work of numerous scholars. Toname a few, Bonahon and Seibenmann’s work on arborescent knots [3], Oertel’s work onstar links [9], Adams’ work on toroidally alternating links [1], Adams’ et. al work almost al-ternating links [2], Fa’s initial cataloging of incompressible pairwise incompressible patterns1 a r X i v : . [ m a t h . G T ] J a n F is in normal position, for an alternating projectionthe existence of such a meridional curve is manifested by an innermost s.c.c. on S ± of S ∩ S ± .An alternative indirect way of arguing comes from the proof of Lemma 2 of [7]. Again inbrief, for F in normal position one considers an arbitrary s.c.c. c ⊂ F ∩ S ± . Then whenone considers how c (cid:48) s imposes a “nesting behavior” on the subset of curves in F ∩ S ∓ thatintersect c , the assumption of an alternating projection again forces the existence of somes.c.c. from this subset manifesting a meridian.This latter technique for detecting the existence of a meridional curve is the main toolemployed in this note. Our concluding remark is that it seem to be an underutilizedtool to-date in works that have exported the crossing ball technology to study pairwiseincompressibility in other settings. (a) The bubble. (b) A local view of the crossing ball with a saddle.The solid arcs on the crossing ball are part of F ∩ S , while the dotted arcs on the crossing ball arepart of F ∩ S − . Figure 12 .2 Main results
To review, a surface F properly embedded in the link L complement in S is called pairwiseincompressible if for each disk D in S meeting the L transversely in one point, with D ∩ F = ∂D , there is a disk D (cid:48) ⊂ F ∪ L meeting L transversely in one point, with ∂D (cid:48) = ∂D . We consider the following problem: Given the condition that a surface isincompressible in the link complements, how to characterize such a surface that is alsopairwise incompressible?As before, let π ( L ) be a connected regular projection from a knot or a link L ⊂ S to asphere S . We place a ball at each crossing of π ( L ), which we refer to as a bubble B (SeeFigure 1a). At each crossing, both the overstrand and the understrand are in the ∂B . Wedefine S to be the sphere S where the equatorial disk in each bubble is replaced by theupper hemisphere of the bubble, and B to be the 3-ball bounded by S that does notcontain any bubble B . Similarly, we can define S − and B − when replacing equatorial diskin each bubble by lower hemisphere of the bubble. Since such operations can be performedon an arbitrary link diagram on S , we use this convention that all π ( L ) of this note areassumed to be endowed with the crossing ball structure.Let F ⊂ S − L be a separating sphere, a closed incompressible pairwise incompressiblesurface, or an incompressible pairwise incompressible surface with meridional boundaries(when L is a knot, if F is a sphere, it is reducible). F can be isotoped to intersect eachbubble in a set of saddles. We assume the surface F is connected and is chosen to minimizethe total number of saddles and curves in lexicographical order. We can replace F (isotopewhen it is incompressible pairwise incompressible) with F (cid:48) such that it is in normal position(details are shown in § F in a position such that for each C ⊂ F ∩ S ± , C bounds a disk in either B ± , C does not pass through a bubble twice, andall meridional boundaries of F , i.e. punctures are on the arcs of F ∩ S ∩ S − , away fromthe crossing balls.With the above-mentioned concepts, we now introduce the word mechanism. We considerthe nontrivial situation when C has nonempty intersections with saddles and give thefollowing definition: Definition 1.1 (cyclic word) For a loop C ⊂ F ∩ S ± we consider it the union of two typesof arcs. Type I A ⊂ C ∩ S ∩ S − ; and, Type II A ⊂ C \ C ∩ S ∩ S − . (Notice a arc oftype I is away from bubbles and an arc of type II is on the boundary of a bubble.). Arcsof type I whose ends encounter two overcrossings of L on same side of C are assigned alabel R . Arcs of type I that have intersections with L are assigned a label P i , where i isthe number of times it intersects with L . Arcs of type II are assigned a label S . A cyclicword ω ( C ) is a word obtained by recording in order the labels of the arcs of C .From a cyclic word ω ( C ) and the number of loops at each bubble C crosses, we willbe able to obtain a virtual word, ω v ( C ) (see Definition 3.5). We will show that each3igure 2: A cyclic word that can be read as SP RSRSP SSRSS . R is a label such that SRS represents the successive saddles lie on the same side of the simple closed curve, while SS means the successive saddles lie on different sides of the simple closed curve. P i meansthe surface is punctured i times at the type II arc. The order of P and R on the same arcdoes not matter.virtual words related to incompressible pairwise incompressible surfaces in normal positionsatisfies a condition defined in § ω -reducible . The diagrammatic property mentionedat the beginning of this note can be illustrated by the following two theorems: Theorem 1.1
Let L ⊂ S be a link, π ( L ) be its link diagram on S , F ⊂ S − L be aseparating sphere or an incompressible pairwise incompressible surface, that is closed orwith meridional boundaries in normal position, then ω v ( C ) is ω -reducible for each simpleclosed curve C ⊂ F ∩ S ± . If in the link complement, there exists a separating sphere or a closed incompressiblepairwise incompressible surface F ⊂ S − L , we also give a theorem to show that theconfigurations of such sufaces are finite, and we can give an upper bound on the Eulercharacteristic of such surfaces through the pullback graph of F , i.e. the surface F endowedwith a 4-valent graph structure and labels on some of the edges (see § refchapter final),where | R | stands for the total number of type I arcs of F ∩ S (or F ∩ S − ) marked with R , and n -gons correspond to disks of F ∩ B ± which intersect with n saddles: Theorem 1.2
Let L be a link, F be a separating sphere, or a closed incompressible pairwiseincompressible surface in S − L . Then: a) If | R | = 4 or , F is a sphere. If | R | = 8 , F is either a sphere or a torus.(b) The maximum vertex number of a region in the pullback graph is bounded by | R | − .(c) For fixed | R | , there are only finitely many such surfaces F (up to isotopy when F isnot a sphere).(d) The Euler characteristics χ ( F ) of F and the number of n -gons F n subject to the fol-lowing restriction : χ ( F ) = | R |− (cid:88) n =2 F n − | R |− (cid:88) n =2 n F n = | R |− (cid:88) n =2 F n − | S | ≤ | R | − | S | Suppose F ⊂ S − L is a surface whose boundary curves are all meridians of L which donot intersect the bubbles. To each component C of F ∩ S ± can be associated a cyclic word.Note that Definition 1.1 modifies the original representation in [7]. When L is alternating, SP i +1 S of [7] will be denoted as SP i +1 RS in this note, where R represents successivesaddles lie on the same side of C ; If the puncture number is even, the notation of this noteis SP i S , according to the alternatingness of link. Proposition 2.1
Let F ⊂ S − L be a separating sphere or a closed incompressible pair-wise incompressible surface, then F can be replaced by another surface F (cid:48) of the sametype(isotopic to F when it is closed incompressible pairwise incompressible) that is in thefollowing position: (1) No word ω ( C ) associated to F is empty. (2) No loop of S ∩ S ± meets a bubble in more than one arc. (3) Each loop of S ∩ S ± bounds a disk in B ± . See [7] Lemma 1 for proof. Notably it is independent of the alternatingness of L . We say F is in normal position of closed surface if it satisfies all above conditions. Conditions (1) and(3) are the results of the incompressibility of F and are independent of the alternatingessof L . Condition (2) is the result of both the pairwise incompressibility of F and the choiceof the isotopic class of F to minimize saddles number, which are both independent on thealternating property of π ( L ).In addition to (1), (2) and (3) conditions for closed surface in normal position, we claim F can be isotoped to satisfy additional three conditions if it has meridional boundaries:5 roposition 2.2 Let F ⊂ S − L be an incompressible and pairwise incompressible surfacewith meridional boundaries, then F can be isotoped so that, in additional to (1), (2) and(3), F satisfies the following conditions:(4) No loop of F ∩ S ± meets both a bubble and an arc of L ∩ S ∩ S − having an endpointon that bubble.(5) No loop of F ∩ S ± meets a component of L ∩ S ± more than once.(6) There does not exist two loops α ⊂ F ∩ S and β ⊂ F ∩ S − , with arcs a, b ⊂ α ∩ β such that the interiors of a, b are contained in adjacent components of S ∩ S − − L , and ∂a ∩ ∂b = ∅ . See [8] Lemma 3 for proof, as these properties are independent of the alternatingness of L . We say a surface with meridonal boundaries is in normal position when it satisfies all(1)-(6) of the above conditions. (a) (b) Figure 3Let F ⊂ S − L be a surface that is closed or with meridional boundaries. Suppose F satisfies condition (1), (3), (4), (5), (6) of normal position, and C ⊂ F ∩ S ± . We show a fewexamples which will explain the motive for some technical definitions in next section. Example 2.3
Fig. 3a shows the situation C intersects with saddles alternatingly. Assume F is punctured, no matter how we “connect” the saddles on the same side of C in F ∩ S through disk(s), on F ∩ S − there is always a curve going through both sides of a bubble,contradicting the pairwise incompressibility of F (see [1], the proof of Lemma 1).In Fig. 3b, F is closed and C no longer intersects saddles alternatingly. However, as one ries to connect the 1st arc end with the 2nd, 3rd, or 4th arc end, the resolving diagramrepresents the result of different “connection of saddles” through disk in F ∩ S , and theresolving surface always contains a meridian curve. We will describe such “connection of saddles” in the next section (see Definition 3.3). Anatural question raised here is: How far away the link diagram is form alternating can weput a closed incompressible pairwise incompressible surface in the link complement?
Example 2.4
Fig. 4 shows two examples when multiple saddles exist in the same bubble.In Fig. 4a, F ∩ S − , an arc goes through the same side of a bubble twice; While in Fig. 4bthe surface F is placed in normal position. (a) (b) Figure 4In order to illustrate how certain arrangement of saddles can lead to the difference betweenFig. 4a and Fig. 4b, we give Definition 3.5 to keep record of the appearance of multiplesaddles.
Definition 3.1 ( ω -reducible) We say a cyclic word is ω -reducible if it can be reduced to ∅ after we perform finitely many times of the following operations:(I) R i → ∅ ,(II) S i RS i → R ,(III) ( SR ) i S → R , 7IV) P i → ∅ ,where i stands for a positive integer, S i means i saddles in a sequence, and similarly for R i , P i . If a cyclic word is not ω -reducible, we call it ω -irreducible . Remark ω -reductions or reductions . An ω -reduction is amodification of the cyclic word. A cyclic word without P must be even length, a (I), (II),or (III) reduction does not change the parity of word length.We focus on closed surface and cyclic word with no puncture for convenience, the results ofthe case with punctures are going to be similar due to reduction (IV). Assume F ⊂ S − L is a sphere, or a closed incompressible pairwise incompressible surface in normal position, C ⊂ F ∩ S (Similarly we can give definitions for C ⊂ F ∩ S − ). We denote the set of allsaddles contained in the bubbles that C crosses as Λ C . With these assumptions, we givethe following technical definitions: Definition 3.2 (partial word) A partial word ω ( C (cid:48) ) is a record of an arc C (cid:48) ⊂ C , in order,the intersections of C (cid:48) with the bubbles in the same sense as the cyclic word, where C (cid:48) is aunion of type I or type II arc(s). We say a partial word of odd length is R- ω -reducible ifit can be R after we perform finitely many times of (I), (II), (III) reductions. Otherwise,we call it R- ω -irreducible . (a) F ∩ S (b) F ∩ S − Figure 5: Two S ’s in the figure are paired up through a connecting operation f , similarlyfor the three S ’s. S is not paired up with S , since the arcs connecting S and S in F ∩ S − is either intersecting C or intersecting S , ∈ Λ C . Definition 3.3 (paired up saddles) Suppose two saddles that C crosses both intersect with8n arc A ⊂ F ∩ S − , and A − { C ∪ Λ C } is connected, then we these saddles are assignedwith a connecting operation f λ . We can replace S ’s in the cyclic word ω ( C ) related to thesesaddles with S λ . If a saddle is assigned with two connecting operations, then we identifythe associated connecting operations, and we say all the saddles are paired up through f λ if all of them are assigned with f λ . See Fig. 5. Remark C intersects is associated to, if any, a uniqueconnecting operation. And an S λ still represents a saddle.With the above definitions, we can now show a lemma to describe the situation when eachbubble that a selected loop C crosses contains only one saddle: Lemma 3.1
Let L ⊂ S be a link with a connected diagram π ( L ) on S , F ⊂ S − L bea separating sphere or a closed incompressible pairwise incompressible surface in normalposition. If each saddle intersecting C ⊂ F ∩ S ± is paired up to some other saddle(s)intersecting C , then ω ( C ) is ω -reducible. (a) a R- ω -reducible partial word ω ( C (1) ) reads as RSRSRSSRS (b) existence of a meridian curve
Figure 6: Examples of two different situations where saddles of ω ( C (1) ) are paired up witheach other. Proof.
Suppose F ⊂ S − L is a closed pairwise incompressible surface in normal position.Without lost of generality, we assume C ⊂ F ∩ S is a simple closed curve such that ω ( C )9igure 7: A R- ω -reducible partial word ω ( C ( k ) ) reads as SSRSSSSRSSR (omitting thesubscripts). ω ( C ( k ) ) does not contain any S λ k ’s. S λ j , S λ j are two saddles contained in ω ( C ( k ) ) that are not paired up with any saddles contained in ω ( C ( k ) ), and they are notrecorded in the partial word between any two paired up saddles contained in ω ( C ( k ) ).is ω -irreducible, and that each of the saddles in ω ( C ) is paired up to other saddle(s) in ω ( C ). Then each saddle can be assigned to a unique connecting operation f λ i , 1 ≤ λ i ≤ n . ω ( C ) always consists of S i after we perform finitely many times of (I), (II), (III) reductionssince it is by assumption ω -irreducible.Consider an adjacent pair of S λ ’s associated to f λ . Denote the two partial words of thearcs separated by cutting the loop C at two of the edge points of type- II arcs intersectingthis pair of saddles as ω ( C (1) ) and ω ( C (1)(1) ), so that ω ( C (1) ) does not consist of any S λ ’s and ω ( C (1)(1) ) contains all the S λ ’s of ω ( C ). Note that the length of both ω ( C (1) )and ω ( C (1)(1) ) are odd. Therefore at least one of these partial words is R- ω -irreducible,otherwise the word ω ( C ) would be ω -reducible. We can assume ω ( C (1) ) is R- ω -irreducible,because otherwise we can pick a different adjacent pair of S λ ’s associated to f λ so that ω ( C (1) ) is R- ω -irreducible and it does not consist of any S λ ’s.Not all of the saddles of ω ( C ) are paired up with each other, otherwise either ω ( C )is R- ω -reducible, see Fig. 6a, or F is not in normal position and it contains a meridiancurve, see Fig. 6b. We claim that we can find a connecting operation f λ k and two pairedup saddles S λ k ’s, together with the two partial words in between of this pair of saddles, ω ( C ( k ) ) and ω ( C ( k )( k ) ), C = C ( k ) ∪ C ( k )( k ) , so that:( i ) ω ( C ( k ) ) consists of ω ( C ( k − ) when k >
1, and ω ( C ( k ) ) is R - ω -irreducible, and does notconsist of any S λ k ’s;( ii ) All partial words that’s contained in ω ( C ( k ) ), and in between each pair of adjacentpaired up saddles are R- ω -reducible.Because otherwise we can take an adjacent paired up saddles S λ k +1 ’s contained in ω ( C ( k ) ),10ssociated to some connecting operation f λ k +1 , such that the partial word ω ( C ( k +1) ) that’scontained in ω ( C ( k ) ), and in between this pair of saddles is R - ω -irreducible, and the finite-ness of word length will guarantee the existence of paired up saddles satisfying condition ( i )and ( ii ). In other words, we can always find the “inner” partial word ω ( C ( k +1) ) containedin ω ( C ( k ) ) until the “innermost” partial word satisfies condition ( i ) and ( ii ).Let { S λ j } be all the saddles contained in ω ( C k ) that are not paired up with any saddlescontained in ω ( C k ), and each saddle of { S λ j } is not recorded in the partial word betweenany two paired up saddles contained in ω ( C k ), see Fig. 7.Suppose the cardinality of { S λ j } is larger or equal to two. According to ( ii ), we can firstperform (II), (III) reductions on S ’s which correspond to saddles paired up within ω ( C k )and the R- ω -reducible partial words in between, then perform (II), (III) reduction(s) on thesaddles of { S λ j } and we are left with an odd number of R ’s, see Fig. 7, then we perform the(I) reduction and we are left with only one R , contradicting ω ( C k ) being R- ω -irreducible;Suppose the cardinality of { S λ j } is one. We are in the situation similar to Fig. 6b, con-tradicting the pairwise incompressibility of F .The rest of definitions in this section are given in order to convert the word problem ofmultiple-saddle cases to the word problem that each bubble a selected loop C crossescontains only one saddle, so that Lemma 3.1 would apply. The purpose is to obtain abookkeeping diagram, in which we obtain the virtual word ω v ( C ) by recording the virtualbubbles C crosses and the labels on type I arcs of C . These definitions are unrelated to theconcept of virtual crossing. First, to exclude the case that a diagram of F ∩ S ± represents adisconnected surface in the link complement, we give the following definition. Furthermore,it has a strong connection with the virtual word in Definition 3.5: Definition 3.4 ( C -onion) A C -onion is a set of saddles satisfying either of the followingconditions:(1) Suppose S α k intersects C , S α m either intersects C , or from S ’s side of view, liesbeneath a saddle intersecting C . If S α k and S α m both intersect an arc A km ⊂ F ∩ S − ,such that A km − { C ∪ S α k ∪ S α m } is connected, then S α k and S α m belong to the same C -onion.(2) Suppose S α m is in a C -onion, and from S ’s side of view, S α m (cid:48) lies beneath a saddleintersecting C . If S α m and S α m (cid:48) both intersect an arc A mm (cid:48) ⊂ F ∩ S − , such that A mm (cid:48) −{ C ∪ S α m ∪ S α m (cid:48) } is connected, then S α m and S α m (cid:48) belong to the same C -onion. See Fig.8 11 a) F ∩ S (b) F ∩ S − Figure 8: (a) and (b) are dual diagrams illustrating the same C -onion. One can checkby definition that the five shadowed saddles in (a) are the only saddles not belong to this C -onion. S α k intersects C , S α k and S α m both intersect an arc A km ⊂ F ∩ S − , such that A km − { C ∪ S α k ∪ S α m } is connected. Therefore S α k and S α m belong to the same C -onion; S α m is in a C -onion, S α m and S α m (cid:48) both intersect an arc A mm (cid:48) ⊂ F ∩ S − , such that A mm (cid:48) − { C ∪ S α m ∪ S α m (cid:48) } is connected. Therefore S α m and S α m (cid:48) belong to the same C -onion. Remark C belong to the same C -onion. C -onions are disjoint sets if multiple C -onions exist.The following proposition says it is easy to determine the saddles of a certain C -onionwhen F is connected: Proposition 3.2
Suppose F is a connected surface that is closed or with meridional bound-aries, and C ⊂ F ∩ S (or C ⊂ F ∩ S − ). Let B be a bubble C crosses, l S be a saddle in B intersecting C , where l means there are totally l saddles beneath it from S ’s side ofview (or from S − ’s side of view, respectively), then all those l saddles belong to the same C -onion with l S . roof. We assume C ⊂ F ∩ S . By the connectness of F , we can find arcs and saddleswhich satisfy the conditions given in the definition of C -onion, so that each saddle liesbeneath l S from S ’s side of view belong to the same C -onion with l S . For C ⊂ F ∩ S − theproof is similar. Definition 3.5 (virtual cyclic word)
Virtual bubbles of a C -onion are a bunch of bubblesintersecting C obtained by replacing a bubble C crosses, that contains 1 + l saddles in the C -onion ( l ≥ l bubbles. These virtual bubbles intersect C alternatinglyin the following way: see Fig. 9. Each single saddle contained in a virtual bubble is calleda virtual saddle . With all bubbles that C crosses replaced by virtual bubbles, we obtain a virtual cyclic word ω v ( C ), that is a record in order of virtual saddles intersecting C , in thesame sense as the record of cyclic word. (a) when l is odd (b) when l is even Figure 9: Virtual bubbles. C is oriented in the same direction as ω ( C ) or ω v ( C ) is recorded.The bubble in which l S lies contains a total of 1 + l saddles, all of which belong to the same C -onion.In other words, we can obtain virtual words from cyclic words in the following way: Definition 3.6 ( l -reduction) Assume F is a connected surface in normal position, C is acurve of F ∩ S (or F ∩ S − ), l S intersects C , and from S ’s side of view (or from S − ’s side ofview, respectively), there are totally l saddles underneath it. We can replace the associated S of l S in the cyclic word ω ( C ) with l S . We call the following operation performed on acyclic word ω ( C ) an l -reduction : 13V) l S → S l +1 . l is a non-negative integer. The resulting word is a virtual cyclic word ω v ( C ). (a) We replace a 1 + l -saddle-bubble with 2 l + 1virtual bubbles arranged alternatingly. The dot-ted ellipse record the related arc-ends in order. (b) Orient C , and direct the arc A kmhg to connectthe related arc-ends. Figure 10The following definition of virtual diagram is the construction of a bookkeeping diagram,so that it and its dual diagram would satisfy Proposition 3.3.
Definition 3.7 (virtual diagram) Suppose F ⊂ S is a closed surface satisfying (1) and(3) of normal position, C ⊂ F ∩ S . A positive virtual diagram of C , denoted as F v + C ∩ S (or respectively, if C − ⊂ F ∩ S − , a negative virtual diagram of C − , F v − C − ∩ S − ), is a diagramdevoid of link and surface structure, obtained by modifying (part of) the diagram F ∩ S in the following way:Suppose S α k,h and S α m,g both belong to a C -onion, and both intersect an arc A kmhg ⊂ F ∩ S − ,such that A kmhg − { C ∪ all saddles in the C -onion } is connected, then we connect the virtualsaddles manifested by the bubbles containing S α k,h and S α m,g as shown in Fig. 10. Weperform such operations on each C -onion of C to obtain F v + C ∩ S .14oreover, we can produce the dual diagram in the same way that is shown in Fig. 1b, i.e. F v + C ∩ S − , of a positive virtual diagram F v + C ∩ S .Similar to the cyclic word, we can define ω -reducible or ω -irreducible virtual words. Definition 3.8 ( ω -reducible virtual cyclic word) We say a virtual cyclic word ω v ( C ) is ω -reducible if it is ∅ after we perform finitely many times of (I), (II), (III), (IV) reductions,otherwise we call it ω -irreducible . (a) F ∩ S (b) F v + C ∩ S Figure 11: (a) shows a torus F embedded in a link complement from S ’s side of view. (b)is the positive virtual diagram of C , in which the 4 red slashes on C separate 2 arcs eachmeeting 3 virtual bubbles.The following proposition transfers the word problem of multiple-saddle case to the casewhere each bubble a selected loop C crosses contains only one saddle, so that Lemma 3.1would apply for the corresponding virtual word: Proposition 3.3
Suppose F ⊂ S is a surface satisfying condition (1), (3), (4), (5), (6)of normal position, C ⊂ F ∩ S . Let { C i − } be all loops of F ∩ S − which intersect saddlesthat belong to the C -onion(s), and let { C vi − } be all loops of F v + C ∩ S − . Then there is anatural correspondence between F v + C ∩ S − and F ∩ S − :(1) If in F v + C ∩ S − , there exists a loop going through a virtual bubble manifested from abubble B twice, or two virtual saddles manifested from B are paired up, then there exists aloop in the diagram of F ∩ S − that goes through B twice.
2) The diagrams ( { C i − } , S − ) and ( { C vi − } , S − ) are ambient isotopic.(3) Loops of { C i − } have an one-to-one correspondence with loops of { C vi − } .Proof. This is by construction of the virtual diagrams, see Definition 3.5, Definition 3.7.To illustrate (2), (3), see also Example 3.4.
Example 3.4
A torus F in normal position embedded in a link complement, where C ⊂ F ∩ S , see Fig. 11a. The virtual diagram of C , F v + C ∩ S , see Fig. 11b. See also Fig.12for the dual diagrams. (a) F ∩ S − (b) F v + C ∩ S − Figure 12: The dual diagrams of Figure 11
Proof.
According to Proposition 2.1 and Proposition 2.2, we can assume without loss ofgenerality F is a surface in normal position. And suppose ω v ( C ) is ω -irreducible for some C ⊂ F ∩ S , then there exists a loop in the dual virtual diagram F v − C ∩ S passing througha virtual bubble twice, according to Lemma 3.1. Then according to the Proposition 3.3(1), either there is a loop in F ∩ S − passing through the same side of a bubble twice,contradicting the minimality of saddle number; Or there is a loop in F ∩ S − passingthrough different sides of a bubble twice, contradicting the pairwise incompressibility of F .16 An Upper Bound On The Euler Characteristic (a) F ∩ S (b) The pullback graph of F . Figure 13: The edges marked in R are colored in red in the pullback graph of F . Disksabove S are marked with +, while disks below above S − are marked with − .Let F be a separating sphere, or a closed incompressible pairwise incompressible surface in S − L that is in normal position. Consider F with all intersection curves both in F ∩ S and F ∩ S − projected on it. All saddles correspond to quadrilaterals, which we collapse tovertices to obtain a 4-regular graph on F . F can be checkerboarded so that for any twoadjacent faces in this graph, one corresponds to a region which is contained strictly above S , and the other to a region contained strictly below S − , see Fig.13. A region which has n vertices ( n must be even) will be denoted as D n . An edge can be marked with R if itcorresponds to a type- I arc labeled with R in F ∩ S , or equivalently, F ∩ S − . We call thisgraph a pullback graph of F . Denote | R | as the total number of R ’s in the graph, | S | asthe vertex number (also the number of saddles of F ). Lemma 4.1
Let L ⊂ S be a link, F ⊂ S − L be a separating sphere or an incompressiblepairwise incompressible surface, that is closed or with meridional boundaries and in normalposition, then for each simple closed curve C ⊂ F ∩ S ± that intersects with saddles, ω ( C ) contains at least two R ’s.Proof. Assume ω ( C ) contains only P ’s and S ’s, then according to the construction of17irtual cyclic word (see Definition 3.5), ω v ( C ) contains only P ’s and S ’s, i.e. C crossesvirtual saddles in an alternating pattern, which makes any of its virtual cyclic word ω -irreducible, contradicting Theorem 1.1.This means in the link complement, if we put the surface of our interest F in normalposition, then in the diagrams of F ∩ S ± the arrangement of saddles intersecting eachsimple closed curve can never be alternating. The following lemma is also true if wereplace S with S − Lemma 4.2
Let D n be a disk component of F ∩ S intersecting n saddles, ∂D n divides S into two regions, then the region that is over m saddles ( m ≤ n ) intersecting ∂D n also contains loop(s) of F ∩ S admitting | R in | ≥ m type- I arcs which are labeled with R .Proof. We assume by convention the region that contains m saddles is the inner one.Suppose m = 1, the region still contains at least one loop, admitting 2 R ’s on the cor-responding edges in pullback graph of F by Lemma 4.1. Suppose m ≥
2, and if the m saddles are all paired up with each other, then the loop(s) contained in the region admits atleast m R ’s. If m of the saddles are not paired up, then the m − m saddles are paired upthrough loop(s) admitting at least m − m R ’s, while the other saddles are going to induceloop(s) intersecting the not paired up saddles. These loop(s) induce k R ’s and contain atleast m saddles on the different side of these m saddles, where k is a positive integerand k + m ≥ m . If m ≥
1, we can repeat this counting process on the inner region(s)bounded by the loop(s) containing m saddles. In the ( i +1)th step we obtain m i − m i +1 R ’sthrough paired up saddles, k i +1 R ’s through the not paired up saddles, and m i +2 saddlesin the inner region(s) obtained through the ( i + 1)th step, where k i +1 + m i +2 ≥ m i +1 . Aswe repeat this process, after finitely many steps we will obtain second innermost region(s)that are over m s saddle(s) in total, and innermost loop(s) admitting at least m s R ’s. Intotal, we can obtain more or equal to m R ’s. Proof.
To prove (a), according to Lemma 4.1, when | R | = 4 or 6, we only need to checkthe situation in which F ∩ S contains two or three loops. When | R | = 8 the maximal loopnumber becomes 4. In each of the cases we can calculate the Euler characteristics of F toshow it is a sphere, see Fig.14, except for when | R | = 8 there are only two configurationsof torus shown in Fig. 15.To prove (b), we assume on the pullback graph, D N is an N -gon so that N is the maximumvertex number of the n -gons. Note that on F ∩ S the corresponding loop of ∂D N divides18igure 14: The configurations of sphere when | R | = 4, 6, or 8.19igure 15: The configurations of torus when | R | = 8. S into two regions, and this loop intersects with a total of N saddles. Then accordingto Lemma 4.2, these two regions contain loops admitting at least N type- I arcs which arelabeled with R . Moreover, ∂D N admits at least 2 R ’s. Therefore N ≤ | R | − | R | . Then by (b) the maximum vertices number of a region in thepullback graph is bounded by | R | −
2. Therefore there are only finitely many types of thepullback graph, each type corresponds to finitely many (isotopic classes) of F ⊂ S − L due to the limited amount of crossing numbers in π ( L ), and the finiteness of type- I arcswhich are labeled with R .To prove (d), we calculate the χ ( F ) through the pullback graph. χ ( F ) is equal to the vertexnumber V subtract the edge number E , and plus the region number F in the pullback graph.Thus, χ ( F ) = V − E + F = (cid:80) ∞ n =2 n F n − (cid:80) ∞ n =2 n F n + (cid:80) ∞ n =2 F n = (cid:80) ∞ n =2 F n − (cid:80) ∞ n =2 n F n . Here (cid:80) ∞ n =2 F n equals the region number F , and by Lemma 4.1 F ≤ | R | . (cid:80) ∞ n =2 n F n quals thevertex number V , which is equal to | S | . By (b) the maximum possible value of n is | R | − χ ( F ) = (cid:80) | R |− n =2 F n − (cid:80) | R |− n =2 n F n = (cid:80) | R |− n =2 F n − | S | ≤ | R | − | S | . Acknowledgement
I would like to thank William Menasco for his guidance and many helpful discussions.20 eferences [1] C. Adams, Toroidally alternating knots and links,
Topology
33, no. 2,(1994): 353-369[2] C.C.Adams, J. F.Brock, J.Bugbee, T.D.Comar, K.A.Faigin, A.M.Huston, A.M.Josephand D.Pesikoff, Almost alternating links,
Topology and its Applications ∼ Topology and its Applications , 80 (1997), no. 3, pp. 239–249.[5] J.Hass, A.Thompson and A.Tsvietkova, The Number of Surfaces of Fixed Genus inan Alternating Link Complement,
International Mathematics Research Notices , 2017,August 2015.[6] M.T.Lozanoand, J.H.Przytycki, Incompressible surfaces in the exterior of a closed 3-braid I, surfaces with horizontal boundary components, Math. Proc. Camb. Phil. Soc.98 (1985), 275–299.[7] W.Menasco, Closed Incompressible Surfaces In Alternating Knot And Link Comple-ments.
Topology
33, no.2(1984): 353-369[8] W.Menasco, Determining Incompressibility Of Surfaces In Alternating Knot And LinkComplements.
Pacific Journal Of Mathematics
Vol 117, No 2. 1985[9] U.Oertel, Closed incompressible surfaces in complements of star links,
Pacific Journalof Mathematics , 111 no. 1 (1984), pp. 209—230.[10] W.Thurston, On the Geometry of Topology of 3-manifolds.