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The Quarterly Journal of Mathematics
, Volume Advance Article – May 23, 2018

21 pages

/lp/ou_press/equivariant-embeddings-of-rational-homology-balls-s3NNjcYagf

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
- ISSN
- 0033-5606
- eISSN
- 1464-3847
- D.O.I.
- 10.1093/qmath/hay016
- Publisher site
- See Article on Publisher Site

Abstract We generalize theorems of Khodorovskiy and Park–Park–Shin, and give new topological proofs of those theorems, using embedded surfaces in the 4-ball and branched double covers. These theorems exhibit smooth codimension-zero embeddings of certain rational homology balls bounded by lens spaces. 1. Introduction The rational blow-down operation was introduced by Fintushel and Stern in [6] and has been a useful tool in constructing small exotic 4-manifolds; see for example [18, 19, 21]. The basic setup is that one has two 4-manifolds C and B with diffeomorphic boundary Y, and with B a rational homology 4-ball. Given a closed smooth 4-manifold X containing C, the manifold Z=X⧹C∪YB is a rational blow-down of X. For certain favourable examples of C and B, this operation preserves many properties of the smooth structure of X, including in particular non-vanishing of Seiberg–Witten invariants. The most important examples of triples (C,B,Y) for this purpose are as follows. Let p>q be coprime natural numbers, and let Yp,q be the lens space L(p2,pq−1). Let Cp,q be the negative-definite plumbed 4-manifold bounded by Yp,q. The lens space Yp,q is known to bound a rational ball Bp,q. One is then interested to know if a 4-manifold Z contains such a submanifold B so that it may be the result of a rational blow-down. Khodorovskiy [11] used Kirby calculus to show that Bp,1 embeds smoothly in a regular neighbourhood V−p−1 of any embedded sphere with self-intersection −(p+1), for p>1. She also showed that for odd p, Bp,1 embeds smoothly in a regular neighbourhood of an embedded sphere with self-intersection −4, and hence in CP¯2. Theorem 1.1 (Khodorovskiy [11]) For each p>1, the rational ball Bp,1embeds smoothly in V−p−1. Theorem 1.2 (Khodorovskiy [11]) For each odd p>1, the rational ball Bp,1embeds smoothly in V−4. For each even p>1, Bp,1embeds smoothly in B2,1#CP¯2. Park–Park–Shin used methods from the minimal model program for complex algebraic 3-folds to generalize Theorem 1.1. For each p,q, they described a linear graph which is roughly speaking half of the negative-definite plumbing graph associated to Cp,q, and denoted by Zp,q the plumbing of disk bundles over spheres according to this graph. This Zp,q is called the δ-half linear chain associated to the pair (p,q). Theorem 1.3 (Park–Park–Shin [17]) For any p>q≥1, the rational ball Bp,qembeds smoothly in the δ-half linear chain Zp,q. This recovers Theorem 1.1 since Zp,1 is diffeomorphic to the regular neighbourhood of a sphere of self-intersection −(p+1). Note that if p>2 then Theorem 1.3 gives two different embeddings, since Bp,q≅Bp,p−q but Zp,q≇Zp,p−q. The purpose of this paper is to give relatively simple topological proofs for the Khodorovskiy and Park–Park–Shin results, which also lead to some new embeddings. Our method is to view Yp,q as the double cover of S3 branched along the two-bridge knot Kp,q=S(p2,pq−1). The plumbing Cp,q is the double cover of B4 branched along the black surface associated to an alternating diagram of Kp,q, and the rational ball Bp,q is the double cover of B4 with branch locus Δp,q which is a slice disk (if p is odd) or a disk and a Möbius band (if p is even). Using an induction argument inspired by those of Lisca [14, 15], we show that Zp,q is the double cover of the 4-ball branched along a surface Fp,qδ and that there is a smooth embedding of pairs (B0.54,Δp,q)↪(B14,Fp,qδ), where Br4 is the ball of radius r. We say that Δp,q is a sublevel surface of Fp,qδ⊂B4. Taking branched double covers then gives the required embedding of the rational ball Bp,q. The proof also yields a clearer understanding of why the δ-half linear plumbing shows up in Theorem 1.3: each Δp,q is obtained by placing a band exactly half-way along the two-bridge diagram. Figure 1 illustrates this for (p,q)=(7,5). Figure 1. View largeDownload slide The δ-half embedding. The top diagram represents the slice disk Δ7,5. There are 5 crossings on either side of the pink band. The bottom diagrams exhibit the slice disk as a sublevel surface of the δ-half surface F7,5δ. The equivalence of the bottom two diagrams is explained in Section 4. Figure 1. View largeDownload slide The δ-half embedding. The top diagram represents the slice disk Δ7,5. There are 5 crossings on either side of the pink band. The bottom diagrams exhibit the slice disk as a sublevel surface of the δ-half surface F7,5δ. The equivalence of the bottom two diagrams is explained in Section 4. We give a proof along similar lines for Theorem 1.2. The method may also be used to find further embeddings of these rational balls. We give the following examples. Theorem 1.4 For each p>1, the slice surface Δp2,p−1is a sublevel surface of the boundary sum Δp,1♮P−, where P−is an unknotted Möbius band. Taking double branched covers yields a smooth embedding Bp2,p−1↪Bp,1#CP¯2. Theorem 1.5 For each n≥0, the slice surface ΔF(2n+2),F(2n)is a sublevel surface of the unknotted Möbius band P−, where F(n)is the nth Fibonacci number. Taking double branched covers yields a smooth embedding BF(2n+2),F(2n)↪CP¯2. It is interesting to compare Theorem 1.5 with the results of [4, 8], from which it follows that BF(2n+1),F(2n−1) embeds smoothly in CP2 for each natural number n. In the final section of the paper, we follow [11, 17] and show that all of the embeddings listed above are simple. Essentially this means they are not useful for constructing interesting 4-manifolds. Our justification for this paper, then, is that it provides useful worked examples of a simple and natural way to realize 4-manifold embeddings. It also seems interesting that the result of [17], found using birational morphisms of 3-dimensional complex algebraic varieties, turns out to have an explanation on the level of two-bridge knot diagrams. The paper contains further results of independent interest, including a careful proof that various descriptions of Bp,q—as a symplectic filling of the tight contact structure on Yp,q coming from the universal cover, as the Milnor fibre of a cyclic quotient singularity, or as the double cover of a slice disk described by Casson and Harer—are the same up to diffeomorphism. We also provide a new recursive description of the plumbing graphs associated to Wahl singularities, which is related to the Stern–Brocot tree structure on the rational numbers. 2. Rational numbers and rooted binary trees In this section, we describe two trees which arise in the study of Wahl singularities and rational balls bounded by lens spaces; one of them is in fact the well-known Stern–Brocot tree, used for finding efficient rational approximations for real numbers [1, 2, 7, 20]. The motivating fact is the following: cyclic quotient singularities of type 1p2(1,pq−1) correspond to sequences of integers [c1,…,ck]. The set of all such sequences is obtained from the one-term sequence [4] by the recursive rule described in [24] [c1,…,ck]↦[c1+1,…,ck,2]or[2,c1,…,ck+1]. We will see that there is an alternative recursive rule which may be used: we obtain all sequences from [2+2] via [a1,…,ak−1,ak+bl,bl−1,…,b1]↦[a1,…,ak−1,(ak+1)+2,bl,bl−1,…,b1]or[a1,…,ak−1,ak,2+(bl+1),bl−1,…,b1]. These two recursions give rise to different labellings of a tree by pairs (p,q), which we now describe. An (infinite, complete, rooted) binary tree is a tree with a single root and such that each node has two children. We will consider labellings of the nodes by pairs (p,q) of coprime natural numbers with p>q. For example we take the tree W1, which we call the inverse Stern–Brocot tree. The root is labelled (2,1) and subsequent nodes are labelled according to the following recursive rule: The first three rows of the tree, and two further nodes, are shown below: It is easy to see that there is one node for each coprime pair (p,q) with p>q. The Stern–Brocot tree, here denoted W2, has a similar description, but at each point the pair (p,q) from W1 is replaced with (p,q−1modp). Thus the root is again labelled (2,1) and the recursive rule is where in each case (m,n−1) is short hand for (m,n−1modm). The first few rows are as follows: Remark 2.1 It is not a difficult exercise to show that this is the same as the recursion rule described in [1, 7]. For example, one may write lp+mq=1, with m=q−1modp and then use l and m to find expressions for various other inverses involved in the recursion rule. Remark 2.2 In fact the tree W2 is a subtree of the usual Stern–Brocot tree. The full tree has root labelled (1, 1), and is easily recovered from the tree described here. 3. Continued fractions and rational balls In this section, we set our conventions for two-bridge links and lens spaces. We also describe a family of slice surfaces Δp,q; we further verify that these are the same as those described by Casson and Harer in [3], and that the double cover of B4 branched along Δp,q is the same rational ball Bp,q referred to in each of [6, 11, 13, 17, 18, 23, 24]. We use Hirzebruch–Jung continued fractions, with the following notation: [a1,a2,…,ak]≔a1−1a2−⋱−1ak. Given a pair of coprime natural numbers p and q with p>q, let p/q=[a1,…,ak], with each ai≥2. We now describe a planar graph Γp,q with k+1 vertices v0,…,vk. For each 1≤i<k, this graph has a single edge joining vi to vi+1. For each i>0, there are edges between v0 and vi so that the valence of vi is ai. We draw the first set of edges in a line, and the second set of edges all on the same side of this line. An example is shown in Fig. 2. We then draw a link diagram for which Γp,q is the Tait graph; this is obtained from the unlink diagram which is the boundary of a tubular neighbourhood of the vertices by adding a right-handed crossing along each edge. See for example Fig. 2. We refer to this as the standard alternating diagram of the two-bridge link S(p,q). Note that according to this convention S(3,1) is the left-handed trefoil. We define the lens space L(p,q) to be the double cover of S3 branched along S(p,q). Figure 2. View largeDownload slide The planar graph Γ18,11 and the two-bridge link S(18,11). Note that 18/11=[2,3,4]. Figure 2. View largeDownload slide The planar graph Γ18,11 and the two-bridge link S(18,11). Note that 18/11=[2,3,4]. It will be convenient to recall how to pass between the continued fractions for p/q and for p/(p−q). This can be done using the Riemenschneider point rule, as described in [14, 17]. We can also use planar graphs: the planar dual of Γp,q is Γp,p−q, with the continued fraction coefficients read from left to right for both. An example is shown in Fig. 3. Figure 3. View largeDownload slide Continued fractions and planar graphs. This example shows that if p/q=[3n], then p/(p−q)=[2,3n−1,2]. We have drawn v0 ‘at infinity’ for convenience. Figure 3. View largeDownload slide Continued fractions and planar graphs. This example shows that if p/q=[3n], then p/(p−q)=[2,3n−1,2]. We have drawn v0 ‘at infinity’ for convenience. The following lemma about continued fractions is presumably known to experts (cf. [8, Lemma 8.5, 12, Remark 3.2]. Lemma 3.1 Let p>qbe coprime natural numbers. Suppose that pq=[a1,…,ak]and pp−q=[b1,…,bl].Then p2pq−1=[a1,…,ak−1,ak+bl,bl−1,…,b1],and p2pq+1=[a1,…,ak−1,ak,2,bl,bl−1,…,b1]. Proof We use structural induction via the inverse Stern–Brocot tree, or in other words based on the fact that each ordered coprime pair of natural numbers may be obtained from (2,1) via a finite sequence of the following steps: (p,q)↦(p+q,q) and (p,q)↦(2p−q,p). The statement holds for the base case and the inductive step follows easily from [14, Lemma 9.1] together with the following identities: p+qq=1+pq=[a1+1,…,ak],p+qp=2−1p/(p−q)=[2,b1,…,bl],2p−qp=2−1p/q=[2,a1,…,ak],2p−qp−q=1+pp−q=[b1+1,…,bl]. (3.1) □ We recall one more well-known fact about continued fractions. Note this gives the well-known isotopy S(p,q)=S(p,q−1modp). Lemma 3.2 Let p>qbe coprime natural numbers. If pq=[a1,…,ak],then pq−1modp=[ak,…,a1]. Proof This can be proved by induction on k. For details, see for example [9].□ We pause to justify the statement made at the beginning of Section 2. Lemma 3.3 Let S1denote the set of strings obtainable from [4]by iteration of [c1,…,ck]↦[c1+1,…,ck,2]or[2,c1,…,ck+1].Let S2denote the set of strings obtainable from [4]=[2+2]via [a1,…,ak−1,ak+bl,bl−1,…,b1]↦[a1,…,ak−1,(ak+1)+2,bl,bl−1,…,b1]or[a1,…,ak−1,ak,2+(bl+1),bl−1,…,b1].Then S1=S2is the set of Hirzebruch–Jung continued fraction expansions of {p2pq−1:p>q>0,(p,q)=1}. Similarly the set of continued fraction expansions of {p2pq+1:p>q>0,(p,q)=1}is obtainable from [2,2,2]either via [c1,…,ck]↦[c1+1,…,ck,2]or[2,c1,…,ck+1]or via [a1,…,ak−1,ak,2,bl,bl−1,…,b1]↦[a1,…,ak−1,(ak+1),2,2,bl,bl−1,…,b1]or[a1,…,ak−1,ak,2,2,(bl+1),bl−1,…,b1]. Proof The proof is by induction, using the fact that each coprime pair (p,q) with p>q>0 appears exactly once as a node label in each of the trees W1 and W2 described in Section 2. For the base case of the first statement, the root of both trees is labelled (2,1) and the continued fraction expansion of 22/(2·1−1) is [4]. The inductive step follows from Lemmas 3.1 and 3.2 using (3.1) and is left as an exercise for the reader. The second statement is similar.□ We now describe a family of slice surfaces Δp,q bounded by the links Kp,q≔S(p2,pq−1). The first such is Δ2,1, shown in Fig. 4; applying the band move shown in pink converts the diagram to one of the two-component unlink. Figure 4. View largeDownload slide The ribbon surface Δ2,1. Figure 4. View largeDownload slide The ribbon surface Δ2,1. There are two ways to recursively build the family Δp,q. Starting with the left diagram in Fig. 4, we apply the recursive rule indicated in Fig. 5. Alternatively we may start with the right diagram in Fig. 4 and apply the recursion from Fig. 6. Figure 5. View largeDownload slide Moving down the inverse Stern–Brocot tree. The box marked β contains a 3-braid. Figure 5. View largeDownload slide Moving down the inverse Stern–Brocot tree. The box marked β contains a 3-braid. Figure 6. View largeDownload slide Moving down the Stern–Brocot tree. Figure 6. View largeDownload slide Moving down the Stern–Brocot tree. In either case, we see that if the band move indicated in the top diagram converts the link to a two-component unlink, then the same is true for each of the diagrams below. Suppose that the link in the top diagram in either case is a two-bridge link corresponding to the continued fraction [a1,…,ak−1,ak+bl,bl−1,…,b1]. Then the two lower diagrams in Fig. 5 are the standard alternating diagrams of the two-bridge links corresponding to [a1+1,…,ak−1,ak+bl,bl−1,…,b1,2] and [2,a1,…,ak−1,ak+bl,bl−1,…,b1+1], respectively, and the two lower diagrams in Fig. 6 are the standard diagrams of the two-bridge links corresponding to [a1,…,ak−1,(ak+1)+2,bl,bl−1,…,b1] and [a1,…,ak−1,ak,2+(bl+1),bl−1,…,b1], respectively. Comparing with Lemma 3.1, we see that if the top diagram of Fig. 5 represents a band move converting Kp,q to the two-component unlink, then the bottom two diagrams represent such a band move for Kp′,q′, where (p′,q′)=(p+q,q) on the left and (2p−q,p) on the right; this corresponds to the recursive rule for the inverse Stern–Brocot tree W1 from Section 2. Recursively we obtain a ribbon surface Δp,q bounded by each Kp,q, which has two zero-handles and a single 1-handle. Similarly if the top diagram of Fig. 6 represents a band move converting Kp,q−1 to the two-component unlink, then the bottom two diagrams represent such a band move for Kp′,q′−1, where (p′,q′)=(p+q,q) on the left and (2p−q,p) on the right; this corresponds to the recursive rule for the Stern–Brocot tree W2 from Section 2. We again obtain a ribbon surface Δp,q′ bounded by each Kp,q, which has two zero-handles and a single 1-handle. We observe by induction (using either Fig. 5 or 6 for the inductive step) that the bands giving the two surfaces have their ends on the same component of Kp,q, and if both band moves are applied one after the other, then the second one is the standard band move converting the two-component unlink to the three-component unlink. This shows that the surface given by the pair of band moves is obtained from either Δp,q and Δp,q′ by adding a cancelling pair of critical points, and thus that the slice surfaces Δp,q and Δp,q′ are isotopic to each other. It is straightforward to draw the bands described above for a particular example. Begin by drawing the standard alternating diagram of Kp,q=S(p2,pq−1) as described above. Then as in Fig. 7, the band obtained from Fig. 5 goes horizontally across the bottom of the diagram, attached just inside the last crossing at each end. The band obtained from Fig. 6 is placed vertically, half way along the diagram, with the same number of crossings on either side. Note in particular that the number of crossings in each region to the right and left of the vertical band may be read off from the continued fraction expansions of p/q and p/(p−q) as in Lemma 3.1. To distinguish between these band moves, we will refer to them from now on as the horizontal band and the vertical band associated to Δp,q. Figure 7. View largeDownload slide The slice disk Δ7,5. Each label gives a count of crossings in the labelled region. Note how these correspond to the continued fraction coefficients of 7/5=[2,2,3] and 7/2=[4,2] (compare Lemmas 3.1 and 3.2, noting that pq+1=−(pq−1)−1modp2). Figure 7. View largeDownload slide The slice disk Δ7,5. Each label gives a count of crossings in the labelled region. Note how these correspond to the continued fraction coefficients of 7/5=[2,2,3] and 7/2=[4,2] (compare Lemmas 3.1 and 3.2, noting that pq+1=−(pq−1)−1modp2). The following lemma and proposition tell us that various constructions of rational balls bounded by L(p2,pq−1) considered in the literature in relation to rational blow-down are all the same up to diffeomorphism. This is known to experts, but nonetheless there is some confusion in the literature, so we sketch a proof. The author is grateful to Yankı Lekili and Paolo Lisca for helpful conversations on this point. Lemma 3.4 Let p>qbe coprime natural numbers with p/q=[a1,…,ak]andp/(p−q)=[b1,…,bl],where each aiand bjis at least 2. Then the double cover of B4branched along Δp,qis given by the relative Kirby diagram in Fig.8. Proof The part of the Kirby diagram shown in black, with bracketed framing indices, is a Kirby diagram for the double cover of the 4-ball branched along one of the chessboard surfaces of the unlink diagram which results from the vertical band move. The branched double cover of the link cobordism given by inverting the vertical band move is the cobordism given by attaching a 2-handle along the (−1)-framed red curve.□ Proposition 3.5 For each coprime pair of natural numbers p>q, the slice surface Δp,qbounded by Kp,q=S(p2,pq−1)is isotopic to the slice surface described by Casson and Harer in [3]. Furthermore the double cover of the 4-ball branched along Δp,qis diffeomorphic to the Milnor fibre Bp,qof the cyclic quotient singularity of type 1p2(1,pq−1). Proof We first consider the slice surface given by Casson and Harer [3]. In the notation of that paper, we take c=−1, x=1/0, y=−q/p, and z=−(p−q)/p. Then the third diagram down on [3, page 32], with the crossing in the band shown there changed in order for the band move to yield the unlink, may be seen to be isotopic to that shown in Fig. 9. The reader may verify, with reference to Lemma 3.1, that this agrees with the description of Δp,q given above, with the vertical band. The last statement of the proposition follows, as in [13, Lemma 3.1], from Lisca’s classification of symplectic fillings of the tight contact structure ξ¯st on a lens space coming from that on its universal cover S3 [16]. Using the method of proof of [16, Corollary 1.2], one may show that it follows from [16, Theorem 1.1] that there is a unique rational ball symplectic filling of (L(p2,pq−1),ξ¯st) up to diffeomorphism. This is the manifold Wp2,pq−1(a1,…,ak,1,bl,…,b1) which is shown in Fig. 8. We have seen in Lemma 3.4 that this is diffeomorphic to the double cover of B4 branched along Δp,q. The last statement of the proposition now follows, since the Milnor fibre of the cyclic quotient singularity of type 1p2(1,pq−1) is a rational ball symplectic filling of (L(p2,pq−1),ξ¯st) [13].□ Figure 8. View largeDownload slide A Kirby diagram for Wp2,pq−1(a1,…,ak,1,bl,…,b1). This represents a single 2-handle attached to S1×B3. Figure 8. View largeDownload slide A Kirby diagram for Wp2,pq−1(a1,…,ak,1,bl,…,b1). This represents a single 2-handle attached to S1×B3. Figure 9. View largeDownload slide The slice surface of Casson and Harer. For the rational tangle notation used in this diagram, see [3, Description 2]. Figure 9. View largeDownload slide The slice surface of Casson and Harer. For the rational tangle notation used in this diagram, see [3, Description 2]. 4. Embeddings via double branched covers In this section, we provide proofs for the theorems stated in the introduction. The proofs will involve manipulations of ‘knot with bands’ diagrams representing properly embedded surfaces in B4. These are diagrams consisting of a knot or link K together with a set of bands attached, such that the band moves convert K to an unlink U. As usual, we interpret this as describing a movie for a surface embedding in B4 to which the radial distance function restricts to give a Morse function, with a minimum for each component of U and a saddle for each band. Maxima of such a surface would result in K being replaced by a union of K and an unlink, but these will not occur in the embeddings we consider. The knot with bands diagram does not specify the order in which the saddles occur during the movie, but changing this order does not change the isotopy class of the embedding. By a slight abuse of notation, we will occasionally use the same letter to refer to an embedded surface in B4 or to a knot with bands diagram representing that surface. Following [22] we note that the resulting embedded surface is also unchanged up to isotopy by any sequence of band slides or band swims. These moves are shown in Fig. 10. Both moves may be interpreted as an isotopy of the blue band coming from the top of the diagram after applying the band move indicated by the pink band; one is then free to slide and swim any band over any other, using the freedom to change the order of the saddle points. It is shown in [10, 22] that these moves together with introduction and removal of cancelling pairs gives a complete calculus for 2-knots, but we will not need this here. Figure 10. View largeDownload slide Band slide and band swim. Figure 10. View largeDownload slide Band slide and band swim. Given two surfaces F1 and F2 in B4, we say that F1 is a sublevel surface of F2 if there is a smooth embedding of pairs (B0.54,F1)↪(B14,F2), where Br4 is the ball of radius r. This is equivalent to existence of a movie for F2 whose final scenes consist of a movie for F1. This in turn may be realized as a knot with bands diagram for F2 which yields a knot with bands for F1 after applying a subset of the band moves (and possibly also removing an unlink corresponding to some minima). One way to produce properly embedded surfaces in the 4-ball is to take an embedded surface with no closed components in S3 and push its interior inside the 4-ball. To obtain a knot with bands diagram of this, we choose a handle decomposition of the surface with 0- and 1-handles and attach bands dual to the 1-handles. To put this another way, we choose a set of properly embedded arcs in the surface that cut it up into a union of disks; neighbourhoods of these arcs give bands. The examples of the negative Möbius band P− and the twisted annulus F−4 are shown in Fig. 11. Another example is shown in the bottom two diagrams of Fig. 12; a different knot with bands representation of the same surface is shown in the bottom right diagram of Fig. 1. Proof of Theorems 1.1 and 1.3 Let p>q be a coprime pair of natural numbers. We will describe a surface Fp,qδ whose branched double cover is the δ-half linear chain Zp,q, and which has Δp,q as a sublevel surface. Theorem 1.3, and hence also Theorem 1.1, then follows on taking branched double covers. We first describe the δ-half linear chain Zp,q. As usual we take p/q=[a1,…,ak] with each ai≥2. Then Zp,q is the linear plumbing of disk bundles over S2 with weights [a1,…,ak−1,ak+1]. This is the double cover of B4 branched along the pushed-in black surface of the standard alternating diagram of the corresponding two-bridge knot, which is S(p+q−1modp,q−1modp). This in turn is obtained from the standard diagram for S(p2,pq−1) by drawing a vertical line through the diagram with two more crossings on the left than on the right, and capping off the portion of the diagram to the left of this line. We denote this surface by Fp,qδ. An example is shown in Fig. 12, which also indicates our convention for which is the black chessboard surface. We next describe a knot with bands representation of a further surface Fp,q′. Start with the standard alternating diagram of Kp,q=S(p2,pq−1). Add the pink vertical band to obtain the slice surface Δp,q, as in the first diagram in Fig. 1. Then going to the right from the centre of the diagram, replace all but the first of the crossings corresponding to edges of Γp,q incident to v0 with blue bands, as in the second diagram of Fig. 1. Call the resulting surface Fp,q′. It is clear that this has the same boundary link as Fp,qδ. We claim that in fact these two embedded surfaces in B4 are isotopic. We prove this claim using structural induction on the Stern–Brocot tree W2. The proof of the base case is shown in Fig. 13. For the inductive step, we consider Fig. 6. Let (p,q) correspond to the top diagram in Fig. 6, and let (p′,q′) and (p″,q″) correspond to the diagrams on the lower left and the lower right, respectively. By induction, Fp,q′ represents the same surface as Fp,qδ. In particular, the blue bands in the bottom right of the given diagram of Fp,q′ may be moved by a sequence of band slides to give the blue bands in the top left of the given diagram of Fp,qδ. In the case of (p″,q″), we begin with a band slide as shown on the left side of Fig. 14. In both cases, we then move any blue bands inherited from the (p,q) diagram, using the fact that all three diagrams in Fig. 6 become isotopic after applying the pink band move. Finally we isotope the pink band as shown on the right side of Fig. 14. We have now established that the knot with bands Fp,q′ represents the surface Fp,qδ in the 4-ball. Applying all the blue band moves first, we see the slice surface Δp,q as a sublevel surface, as required.□ Proof of Theorem 1.2 Let p≥3. This proof is based on the diagram in Fig. 15. The reader may verify that performing the red band move yields the two-component unlink, while performing the blue band move converts the link to K2,1. We will see that this diagram may be interpreted as the middle frame of a movie exhibiting Δp,1 as a sublevel set of F−4, if p is odd, or of the boundary sum Δ2,1♮P− if p is even. Figure 16 shows a diagram consisting of the link K2,1=S(4,1) together with two bands; after performing the indicated isotopies and the blue band move, this is seen to contain Δp,1 as a sublevel surface. It remains to simplify this diagram. The first diagram in Fig. 16 shows two nested bands attached near the ends of a twist region. There are p crossings between the ends of one of them, and p+2 between the ends of the other. Using a band swim of the form , we can move the blue band inside the red one, so that now one of the pair has p crossings between its ends and the other has p−2 crossings. Iterate until we finish up with one of the two diagrams shown in Fig. 17; thus we have realized Δp,1 as a sublevel surface of F−4 if p is odd and of Δ2,1♮P− is p is even.□ Proof of Theorem 1.4 This is very similar to the proof of Theorem 1.2 so we omit some details. Figure 18 shows a diagram consisting of the link Kp2,p−1 together with two bands; performing the (+1)-labelled red band move yields the two-component unlink, while performing the blue band move converts the link to Kp,1. The knot with bands that we need is given by performing the blue band move and drawing in its inverse, that is the band which undoes the previous band move. This gives two (+1)-labelled bands which are nested in a similar way to those encountered in the proof of Theorem 1.2. Performing p band swims as in that proof results in the diagram of Fig. 19 representing the boundary sum of Δp,1 and P−.□ Proof of Theorem 1.5 The usual recursive definition of the Fibonacci numbers easily implies F(2n+2)=3F(2n)−F(2n−2), which by an easy induction gives F(2n+2)F(2n)=[3n]. Now using Fig. 3, we have F(2n+2)F(2n+2)−F(2n)=[2,3n−1,2], and then Lemma 3.1 yields F(2n+2)2F(2n+2)F(2n)−1=[3n−1,5,3n−1,2]. Consider now the surface embedded in B4 depicted in Fig. 20. We see that the (−1)-labelled blue band move converts this into a diagram of ΔF(2n+2),F(2n), which is thus a sublevel surface. Also observe that the boundary link is an unknot: to see this perform a sequence of 2n isotopies of the form , followed by a Reidemeister II and a Reidemeister I move. (The first of these isotopies involves the four crossings to the immediate left of the red band; in general these 2n isotopies can be located by noting that they always involve the single non-alternating crossing, which in the initial diagram is the one to the left of the blue band.) The proof is then completed by the sequence of handleslides shown in Fig. 21.□ Remark 4.1 Applying almost the same proof for odd Fibonacci numbers realizes ΔF(2n+1),F(2n−1) as a sublevel surface of P+, and hence a smooth embedding of BF(2n+1),F(2n−1) in CP2. It is well known that in fact an embedding of BF(2n+1),F(2n−1) in CP2 arises from singularity theory since (1,F(2n−1),F(2n+1)) is a Markov triple, cf. [1, 4, 8]. Figure 11. View largeDownload slide The surfaces P− and F−4. The branched double covers are CP2¯ and V−4, respectively. Figure 11. View largeDownload slide The surfaces P− and F−4. The branched double covers are CP2¯ and V−4, respectively. Figure 12. View largeDownload slide The surface F7,5δ. The top diagram shows S(49,34), which is then cut vertically one crossing to the right of the centre line yielding S(10,7). The diagram on the bottom left shows the black surface for S(10,7). Pushing the interior into the 4-ball yields F7,5δ, which is shown on the right. Figure 12. View largeDownload slide The surface F7,5δ. The top diagram shows S(49,34), which is then cut vertically one crossing to the right of the centre line yielding S(10,7). The diagram on the bottom left shows the black surface for S(10,7). Pushing the interior into the 4-ball yields F7,5δ, which is shown on the right. Figure 13. View largeDownload slide The proof of Theorem 1.3 for p=2, q=1. The second diagram is obtained from the first by sliding the blue band over the pink one, and the third is isotopic to the second. Figure 13. View largeDownload slide The proof of Theorem 1.3 for p=2, q=1. The second diagram is obtained from the first by sliding the blue band over the pink one, and the third is isotopic to the second. Figure 14. View largeDownload slide A band slide and an isotopy. Figure 14. View largeDownload slide A band slide and an isotopy. Figure 15. View largeDownload slide The middle frame of a movie. Here we use arcs to represent bands; the label beside each arc is the signed number of crossings in the band. We may think of the (+1)-labelled red band move as decreasing the radial distance function and the blue one as increasing it. Figure 15. View largeDownload slide The middle frame of a movie. Here we use arcs to represent bands; the label beside each arc is the signed number of crossings in the band. We may think of the (+1)-labelled red band move as decreasing the radial distance function and the blue one as increasing it. Figure 16. View largeDownload slide A surface bounded by K2,1. We see that this contains Δp,1 as a sublevel surface. Figure 16. View largeDownload slide A surface bounded by K2,1. We see that this contains Δp,1 as a sublevel surface. Figure 17. View largeDownload slide Two surfaces bounded by K2,1. The surface on the left is F−4, and the one on the right is the boundary sum of Δ2,1 and P− (cf. Figs 4 and 11). Figure 17. View largeDownload slide Two surfaces bounded by K2,1. The surface on the left is F−4, and the one on the right is the boundary sum of Δ2,1 and P− (cf. Figs 4 and 11). Figure 18. View largeDownload slide The middle frame of another movie. Figure 18. View largeDownload slide The middle frame of another movie. Figure 19. View largeDownload slide The surface Δp,1♮P−. Figure 19. View largeDownload slide The surface Δp,1♮P−. Figure 20. View largeDownload slide A surface bounded by the unknot. Figure 20. View largeDownload slide A surface bounded by the unknot. Figure 21. View largeDownload slide Simplifying a surface bounded by the unknot. Figure 21. View largeDownload slide Simplifying a surface bounded by the unknot. 5. Simple embeddings In [11], Khodorovskiy called an embedding Bp,q↪Zsimple if the corresponding rational blow-up X=Z⧹B∪Yp,qCp,q may be obtained from Z by a sequence of ordinary blow-ups, or in other words if X=Z#kCP2¯. She pointed out in that paper that the embeddings in Theorem 1.1, and those in Theorem 1.2 with odd p, are simple, and Park–Park–Shin showed that the embeddings described in Theorem 1.3 are also simple. One could extend the notion of simple to embeddings of the form Bp,q↪Bp′,q′#CP2¯ by saying that such an embedding is simple if the resulting rational blow-up of Bp,q has the same effect as rationally blowing up Bp′,q′, together with a sequence of ordinary blow-ups. With this terminology the proof of [11, Corollary 5.1] applies to show that the embeddings in Theorem 1.2 are all simple. Given a sublevel surface Δp,q of a properly embedded surface F in the 4-ball, the equivariant rational blow-up of Δp,q is the surface obtained by replacing Δp,q by the pushed-in black surface of its boundary link. Examples are shown in Figs 22–25. We say a sublevel surface Δp,q of a properly embedded surface F in the 4-ball is simple if the equivariant rational blow-up yields F♮kP− for some k∈N, or if F=Δp′,q′ and the equivariant rational blow-up yields Fp′,q′♮kP−, where Fp′,q′ is the pushed-in black surface of Kp′,q′. Proposition 5.1 All of the sublevel surfaces Δp,q↪Fexhibited in the previous section are simple. Proof The proofs are fairly straightforward and it suffices to illustrate with examples. For the case of Theorems 1.1 and 1.3, see Fig. 22. For Theorem 1.2, see Fig. 23. For Theorem 1.4, see Fig. 24. Finally for Theorem 1.5, see Fig. 25.□ Remark 5.2 It is very interesting to note that Finashin [5] has given examples of the use of equivariant rational blow-down in the construction of exotic smooth manifolds homeomorphic to CP2#5CP2¯. Figure 22. View largeDownload slide Simplicity of the sublevel surface Δ8,3↪F8,3δ. The equivariant rational blow-up is achieved by replacing the band for Δ8,3 with those for the black surface of K8,3. To go from the second diagram to the third, slide the blue bands to the left, and then note that a P− boundary summand can be moved anywhere in the diagram; to move it past another band, slide the other band over the trivial (+1)-labelled band twice. Figure 22. View largeDownload slide Simplicity of the sublevel surface Δ8,3↪F8,3δ. The equivariant rational blow-up is achieved by replacing the band for Δ8,3 with those for the black surface of K8,3. To go from the second diagram to the third, slide the blue bands to the left, and then note that a P− boundary summand can be moved anywhere in the diagram; to move it past another band, slide the other band over the trivial (+1)-labelled band twice. Figure 23. View largeDownload slide Simplicity of the sublevel surface Δ4,1↪Δ2,1♮P−. Figure 23. View largeDownload slide Simplicity of the sublevel surface Δ4,1↪Δ2,1♮P−. Figure 24. View largeDownload slide Simplicity of the sublevel surface Δ9,1↪Δ3,1♮P−. Figure 24. View largeDownload slide Simplicity of the sublevel surface Δ9,1↪Δ3,1♮P−. Figure 25. View largeDownload slide Simplicity of the sublevel surface Δ8,3↪P−. The sequence of slides and isotopies from the third diagram to the sixth give an inductive step to go from n to n−1. Figure 25. View largeDownload slide Simplicity of the sublevel surface Δ8,3↪P−. The sequence of slides and isotopies from the third diagram to the sixth give an inductive step to go from n to n−1. Acknowledgements The author thanks Paolo Aceto, Marco Golla, Yankı Lekili, Paolo Lisca and Jongil Park for helpful conversations. We also thank the referee for helpful suggestions to improve the exposition. References 1 M. Aigner , Markov’s theorem and 100 years of the uniqueness conjecture , Springer , Cham , 2013 . Google Scholar CrossRef Search ADS 2 A. Brocot , Calcul des rouages par approximation, nouvelle méthode, Revue Chonométrique 3 ( 1861 ), 186–194. 3 A. J. Casson and J. L. Harer , Some homology lens spaces which bound rational homology balls , Pacific J. Math. 96 ( 1981 ), 23 – 36 . Google Scholar CrossRef Search ADS 4 J. D. Evans and I. Smith , Markov numbers and Lagrangian cell complexes in the complex projective plane , Geom. Topol. 22 ( 2018 ), 1143 – 1180 . Google Scholar CrossRef Search ADS 5 S. 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The Quarterly Journal of Mathematics – Oxford University Press

**Published: ** May 23, 2018

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