L-space surgeries on satellites by algebraic links
RRATIONAL L-SPACE SURGERIESON SATELLITES BY ALGEBRAIC LINKS
SARAH DEAN RASMUSSEN
Abstract.
Given an n -component link L in any 3-manifold M , the space L ⊂ ( Q ∪ {∞} ) n of rational surgery slopes yielding L-spaces is already fully characterized (in joint work bythe author [26]) when n = 1 and L is nontrivial. For n >
1, however, there are no previousresults for L as a rational subspace, and only limited results for integer surgeries L ∩ Z n on S . Herein, we provide the first nontrivial explicit descriptions of L for rational surgerieson multi-component links. Generalizing Hedden’s and Hom’s L-space result for cables, wecompute both L , and its topology, for all satellites by torus-links in S . For fractal-boundaried L resulting from satellites by algebraic links or iterated torus links, we develop arbitrarilyprecise approximation tools. We also extend the provisional validity of the L-space conjecturefor rational surgeries on a knot K ⊂ S to rational surgeries on such satellite-links of K . Theseresults exploit the author’s generalized Jankins-Neumann formula for graph manifolds [27]. Introduction
A connected, closed, oriented 3-manifold is called an
L-space if its reduced Heegaard Floerhomology vanishes. The present work focuses on the following relative notion of L-space.
Definition 1.1.
For a compact oriented 3-manifold Y with boundary a disjoint union of n tori, the L-space region L ( Y ) ⊂ (cid:81) ni =1 P ( H ( ∂ i Y ; Z )) ∼ = ( Q ∪ {∞} ) n , with complement NL ( Y ) ,is the space of (rational) Dehn-filling slopes of Y which yield L-space Dehn-fillings. Prior Results.
Until now, studies of multi-component L-space surgery slopes have been con-fined to integer surgeries on links in S . These primarily include numerical methods of Liu toplot individual points in L ∩ Z for 2-component links [21], Gorsky and Hom’s identification oftorus-link satellites with integer L-space surgery slopes in the positive orthant [9], and Gorskyand N´emethi’s work on integer torus-link surgeries [10] and on a partial characterization (com-plete for algebraic links) of which 2-component links have L ∩ Z bounded from below [11]. Present Motivation.
As subsets of ( Q ∪ {∞} ) n , L-space regions exhibit qualitative featuresinvisible to the set of integer L-space surgery slopes, such as nontrivial topological properties,fractal behaviors at the boundary of L , and symmetries such as the action of Λ in Theorem 1.2.Since non-L-space regions chart the silhouette of Heegaard Floer complexity as a functionof varying surgery slope, this creates a rich template to compare against the surgery regionssupporting any candidate geometric structure potentially responsible for nontrivial HF classes.Such comparisons for Seifert fibered spaces led to the L-space conjecture that non-L-spaces arecharacterized by the existence of left orders on fundamental groups and/or co-oriented taut fo-liations [4, 18]. Both L-spaces and L also constrain complex singularities: see Section 1.3.The author’s joint result with J. Rasmussen [26] characterizing nontrivial L for knot exteri-ors in 3-manifolds led both to our toroidal gluing theorem for L-spaces [26] and to the author’sindependent proof of the L-space conjecture for graph manifolds [27]. Our joint work on L combined with Hanselman and Watson’s studies of combinatorial properties of certain bor-dered Floer algebras [14] gave rise to a topological realization of bordered Floer homology forsingle-torus boundaries [13]. A multiple-boundary-component version of this should also exist. Key words and phrases.
Heegaard Floer, L-space, algebraic link, torus link, satellite, cable, Dehn surgery. a r X i v : . [ m a t h . G T ] D ec SARAH DEAN RASMUSSEN
Methods: Classification formula.
Despite reliance on an enhanced L-space gluing toolproved in Theorem 3.6, this paper was primarily made possible by the author’s classificationof graph manifolds admitting co-oriented taut foliations, with proof of the graph-manifoldL-space conjecture as by-product [27]. (This is not to be confused with the author’s jointwork with Hanselman et al [12].) This classification combines a new classification formula (Theorem 4.3), generalizing that of Jankins and Neumann for Seifert fibered spaces [17], witha structure theorem (Theorem 4.4) prescribing the interpretation of outputs of this formula.This classification tool also governs L-space regions for unions of graph manifolds withsingle-torus-boundary manifolds. In particular it gives a complete abstract characterizationof L for any graph-manifold-exterior satellite of any knot in any 3-manifold. The classifica-tion formula alternately composes a linear-fractional transformation φ P e ∗ , induced by a gluingmap φ e for each edge e , with a pair y v ± , for each vertex v , of extremizations of locally-finitecollections of piecewise-constant functions of slopes in a certain Seifert-data-compatible basis. Results.
Herein, we analyze the intricate behavior of solutions L to the classification formulaefor exteriors of such satellites. The bounded-chaotic behavior of these y v ± generically leadsto fractal-boundaried L , but we develop precise tools for local approximation and topologicalcharacterization. As sample applications of these tools, Theorems 1.6 and 1.7 construct globalinner approximations of L for satellites by algebraic links and iterated-torus-links, respectively.Moreover, for a satellite in S by an n -component torus link, the chaotic behavior of y v ± generically degenerates, and we provide an exact explicit description of L and its various possi-ble topologies, in Theorems 1.2 and 1.3, respectively. Lastly, in Theorem 1.4 and Corollary 1.5,we promote L-space conjecture results for knot surgeries to results for satellite surgeries.1.1. Torus-link satellites.
The T ( np, nq )-torus-link satellite K ( np,nq ) ⊂ M of a knot K ⊂ M in a 3-manifold M embeds the torus link T ( np, nq ) in the boundary of a neighborhood ν ( K ) of K ⊂ M . The exterior Y ( np,nq ) of K ( np,nq ) splices K ⊂ M to the multiplicity- q fiber of the Seifertfibered exterior of T ( np, nq ). This Seifert structure also prescribes 3 distinguished subsetsΛ , R , Z ⊂ (cid:81) ni =1 P ( H ( ∂ i Y ( np,nq ) )) of slopes. The lattice Λ acts on slopes by reparametrizationof Seifert data, and R \ Z catalogs reducible surgeries with no S × S connected summand. Theorem 1.2.
Suppose that K ⊂ S is a positive L-space knot of genus g ( K ) , and that n, p, q ∈ Z , with n, p > and gcd( p, q ) = 1 . Then the T ( np, nq ) torus-link satellite K ( np,nq ) ⊂ S of K has L-space surgery region given by the union of Λ -orbits L S = Λ · L ∗ S , where ( i ) If N := 2 g ( K ) − > qp and K ⊂ S is nontrivial, then L ∗ S := (cid:40) { ( ∞ , . . . , ∞ ) } p > (cid:83) ni =1 (cid:0) {∞} i − × [ N, + ∞ ] × {∞} n − i (cid:1) p = 1 . ( ii ) If g ( K ) − ≤ qp , and K ⊂ S is nontrivial, or if p, q > and K ⊂ S is the unknot(so that K ( np,nq ) = T ( np, nq ) ), then for N pq := pq − p − q + 2 g ( K ) p , we have L ∗ S := L − S ∪ ( R S \ Z S ) ∪ L + S ; R S \ Z S = (cid:96) ni =1 (cid:0) [ −∞ , pq (cid:105) ∪ (cid:104) pq, + ∞ ]) i − × { pq } × ([ −∞ , pq (cid:105) ∪ (cid:104) pq, + ∞ ]) n − i (cid:1) , L − S = [ −∞ , pq (cid:105) n \ [ −∞ , N pq (cid:105) n , L + S = (cid:104) pq, + ∞ ] n . Seven months after the current article’s appearance on the arXiv, Hanselman, Rasmussen, and Watson posteda revised version of [13] with a new L-space gluing theorem subsuming the current paper’s Theorem 3.6. The author conceived this foliation-classification project [27] shortly before her summons to collaborate withHanselman et al [12]. These two proofs of the graph-manifold L-space conjecture make contact with foliationsvia disparate mechanisms. The classification result itself is exclusive to the author’s independent work.
ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 3
Figure 1.
The L-space surgery region L S for the T (4 ,
46) satellite of the P ( − , ,
7) pretzel knot, with pq = 46 and N pq = 41. Here, Λ · L − S is dark grey,Λ ·L + S is light grey, R S is black except for Z S = { ( pq, pq ) } ⊂ NL S , and NL S is white except for the (dotted) conic B S = { α α = pq } ⊂ NL S of rationallongitudes of the satellite exterior. As usual, L S \ L ∗ S has radius 1 about R S . Remarks.
Positive L-space knots K ⊂ S have L S = [2 g ( K ) − , + ∞ ] [25]. Theorem 4.5 andits remark cover the remaining (redundant or less interesting) cases of negative L-space ornon-L-space knots K , and the fractal-boundaried case of K ( np,nq ) = T ( n,nq ) (for p = 1 and K the unknot). We use “ (cid:104) ” for open endpoints and implicitly intersect intervals with Q ∪ {∞} . Example:
Cables.
For n = 1, Λ is trivial and R \ Z = { pq } . Thus Theorem 1.2 yields( ii ) L S ( Y ( p,q ) ) = [ N pq , pq (cid:105) ∪ { pq } ∪ (cid:104) pq, + ∞ ] = [ N pq , + ∞ ] = [2 g ( K ( p,q ) ) − , + ∞ ]for K ⊂ S a nontrivial positive L-space knot with 2 g ( K ) − ≤ qp , and ( i ) L S ( Y ( p,q ) ) = {∞} for 2 g ( K ) − > qp , recovering well-known results of Hedden [15] and Hom [16] for cables. Topology of L ( Y ( np,nq ) ) . For any subset A ⊂ ( Q ∪ {∞} ) n (cid:44) → ( R ∪ {∞} ) n with complement A c := ( Q ∪ {∞} ) n \ A and real closure A ⊂ ( R ∪ {∞} ) n , we define A R := A \ A c ⊂ ( R ∪ {∞} ) n . Theorem 1.3.
Take
K, K ( np,nq ) ⊂ S , L , NL , N, and Λ as in Theorem 1.2, with q > , andlet B be the set of rational longitudes, as described in (4) , of the exterior Y ( np,nq ) of K ( np,nq ) . ( i.a ) If N > qp and p > , or if N > qp +1 and p = 1 , then L R deformation retracts onto Λ . ( i.b ) If N = qp +1 and n > , then rank H ( L R ) = (cid:0) n (cid:1) − and π ( L R ) (cid:39) ker( δ ) , for the map δ : F ( n ) → Λ sf , ( x ij ) e (cid:55)→ ( ε i − ε j ) e as in (117) , with F ( n ) := (cid:104) x ij (cid:105) i We briefly pause here to elaborate on the distinguished subsets R , Z , B , and Λ of sf -slopes. Reducible Slopes R and Z . Dehn filling along the smooth-fiber slope ∞ ∈ ( Q ∪ {∞} ) sf decomposes a Seifert fibered space as a connected sum, with one summand for each exceptionalfiber or boundary component. After this connected-sum decomposition has occurred, anyadditional ∞ -fillings create S × S summands. Thus, since ψ : ∞ (cid:55)→ pq + ∞ = pq , our reducible slopes R and their exceptional subset Z (see Definitions 2.4 and 2.5) satisfy R S ( Y ( np,nq ) ) = (cid:83) ni =1 { α ∈ ( Q ∪ {∞} ) nS | α i = pq } , Z S ( Y ( np,nq ) ) = (cid:83) i 0. In particular, for ∂M with n components, we have(3) Z = R ∩ B , and B R ∼ = T n − (cid:44) → (cid:96) ni =1 P ( H ( ∂ i M ; R )) (cid:39) ( R ∪ {∞} ) n ∼ = T n . For the exterior Y ( np,nq ) of K ( np,nq ) ⊂ S , Proposition 2.7 tells us B sf is the closure(4) B sf ( Y ( np,nq ) ) = (cid:110) y ∈ ( Q ∪ {∞} ) n sf (cid:12)(cid:12)(cid:12) pq + (cid:80) ni =1 y i = 0 (cid:111) of the linear subspace (cid:8) y ∈ Q n sf (cid:12)(cid:12) pq + (cid:80) ni =1 y i = 0 (cid:9) in ( Q ∪{∞} ) n sf . The change of slope-basis ψ transforms B sf ( Y ( np,nq ) ) into a degree- n hypersurface in ( Q ∪ {∞} ) nS . For example, the n = 2 case yields the conic B S ( Y (2 p, q ) ) = { α ∈ ( Q ∪{∞} ) nS (cid:12)(cid:12) α α = ( pq ) } , as in Figure 1. Symmetry by Λ. The lattice Λ sf ( Y ( np,nq ) ) ⊂ ( Q ∪{∞} ) n sf of Seifert-data reparametrizations,(5) Λ sf ( Y ( np,nq ) ) := { l ∈ Z n ⊂ ( Q ∪ {∞} ) n sf | (cid:80) ni =1 l i = 0 } , acts on sf -slopes by addition, y (cid:55)→ l + y , thereby determining an action of Λ on slopes in anybasis. In particular, in Theorem 1.2, ψ induces an action of Λ sf ( Y ( np,nq ) ) on ( Q ∪{∞} ) nS , via(6) l · α = ψ ( l + ψ − ( α )) , l ∈ Λ sf ( Y ( np,nq ) ) , α ∈ ( Q ∪ {∞} ) nS . The action of Λ induces homeomorphism on Dehn surgeries: S l · α ( K ( np,nq ) ) ∼ = S α ( K ( np,nq ) )for l ∈ Λ( Y ( np,nq ) ) and α ∈ ( Q ∪ {∞} ) nS . Thus Λ preserves R , Z , B , L , and NL as sets, andsince S L ∗ ( K ( np,nq ) ) ∼ = S L ( K ( np,nq ) ), L ∗ S completely catalogs the L-space surgeries on K ( np,nq ) . The L-space surgery slopes for K ( np,nq ) must retain their full Λ-orbits, but sf -slopes dothis automatically: the expression of L sf ( Y ( np,nq ) ) in Theorem 4.5 is naturally Λ-invariant.Moreover, L ∗ S ( Y ( np,nq ) ) is almost Λ-invariant. When L ∗ S = { ∞ } , L S \ L ∗ S is given by(7) (Λ S · { ∞ } ) \ { ∞ } = Λ S ∩ Q nS ⊂ ([ pq − , pq (cid:105) ∪ (cid:104) pq, pq + 1]) n . For any K ( np,nq ) in Theorem 1.2, the Λ-mismatch L S \L ∗ S lies inside a radius-1 neighborhood,(8) L S \ L ∗ S ⊂ (cid:83) ni =1 { α | α i ∈ [ pq − , pq (cid:105) ∪ (cid:104) pq, pq + 1] } =: U npq (1) , of R S = (cid:83) ni =1 { α | α i = pq } , and L S consists of a finite union of rectangles outside any positive-radius neighborhood of R S . Moreover, (8) implies the integer slopes in L S ( Y ( np,nq ) ) satisfy(9) L S ∩ ( Z ∪ {∞} ) n ⊂ L S ∗ ∪ S n (cid:16) { pq − , pq +1 }× ( Z ∪ {∞} ) n − (cid:17) , and it is a simple exercise to determine α ∈ ( L S \L S ∗ ) ∩ ( Z ∪{∞} ) n with α i ∈ { pq − , pq +1 } . ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 5 New features: Torus-link satellites vs One-strand Cables. L-space regions for satellitesby n > n = 1 cables.(a) For n > 1, the action of Λ becomes nontrivial, although as discussed before, this actiondoes not impact actual L-spaces resulting from surgery.(b) For n > 1, the codimension-1 subspace R S ⊂ ( Q ∪ {∞} ) nS acquires positive dimensionand the codimension-2 subspace Z S ⊂ ( Q ∪ {∞} ) nS becomes nonempty, although the set ofreducible L-space slopes R S \ Z S remains a disjoint union of hyperplanes ∼ = Q n − for all n .(c) For n = p = 1, both the L-space regions L S ( S \ ◦ ν ( K )) = [ N, ∞ ] = [ N q , ∞ ] = L S ( Y (1 ,q ) )and the spaces of resulting L-space surgeries S L ( K ) = S L ( K (1 ,q ) ) for K and K (1 ,q ) are identical,since the p = 1 cable affects framing without changing the knot. For n > 1, however, therelationship between S L ( K ) and S L ( K ( n,nq ) ) depends on the difference (2 g ( K ) − − qp :(=) S L ( K ( n,nq ) ) = S L ( K ) when 2 g ( K ) − > q ,( (cid:40) ) S L ( K ( n,nq ) ) (cid:40) S L ( K ) when 2 g ( K ) − q ,( (cid:41) ) S L ( K ( n,nq ) ) (cid:41) S L ( K ) when 2 g ( K ) − < q .(d) For n = 1, L ( Y ( p,q ) ) R is contractible and of dimension 0 or 1. For n > 1, however,Theorem 1.3 catalogs 6 distinct topologies that occur for L ( Y ( p,q ) ) R , including1. an infinite disjoint union of points — ( i.a ), p (cid:54) = 1;2. an infinite disjoint union of contractible 1-dimensional spaces — ( i.a ), p = 1;3. a connected 1-dimensional space with rank H = (cid:0) n (cid:1) − i.b );4. a contractible, 1-dimensional space with S closure — ( i.c ), 2 g ( K ) − qp +1, n = 2;5. a contractible n -dimensional space — ( i.c ), 2 g ( K ) − qp ;6. an n -dimensional space that deformation retracts onto T n − — ( ii ).1.2. The L-space Conjecture. The L-space conjectures, stated formally by Boyer-Gordon-Watson [4] and Juh´asz [18], posit the existence of left-invariant orders on fundamental groupsand of co-oriented taut foliations, respectively, for all prime, compact, oriented non-L-spaces.For Y a compact oriented 3-manifold with torus boundary, let F ( Y ) ⊂ P ( H ( ∂Y ; Z ))denote the space of slopes α ∈ P ( H ( ∂Y ; Z )) for which Y admits a co-oriented taut foliation(CTF) restricting to a product foliation of slope α on ∂Y . Along a similar vein, we define LO ( Y ) := { α ∈ P ( H ( ∂Y ; Z )) | π ( Y ( α )) is LO } , where LO stands for left-orderable. Theorem 1.4. Take K, K ( np,nq ) ⊂ S as in Theorem 1.2, with K nontrivial and p > . ( LO ) Suppose LO ( Y ) ⊃ N L ( Y ) , for Y := S \ ◦ ν ( K ) . ( lo .i ) If g ( K ) − > q +1 p , then LO ( Y ( np,nq ) ) = N L ( Y ( np,nq ) ) . ( lo .ii ) If g ( K ) − < qp , then LO ( Y ( np,nq ) ) ⊃N L ( Y ( np,nq ) ) \ Λ( Y ( np,nq ) ) · ([ −∞ , N pq (cid:105) n \ [ −∞ , N pq − p (cid:105) n ) . ( CTF ) Suppose F ( Y ) = N L ( Y ) . ( ctf .i ) If g ( K ) − > q +1 p , then F ( Y ( np,nq ) ) = N L ( Y ( np,nq ) ) \ R ( Y ( np,nq ) ) . ( ctf .ii ) If g ( K ) − < qp , then F ( Y ( np,nq ) ) ⊃ ( N L ( Y ( np,nq ) ) \ R ( Y ( np,nq ) )) \ Λ( Y ( np,nq ) ) · ([ −∞ , N pq (cid:105) n \ [ −∞ , N pq − p (cid:105) n ) . One might notice that our sharper results here lie in case ( i ) 2 g ( K ) − > qp of Theorem 1.2.While this case is the less interesting one from the standpoint of L-space production, it is themore nontrivial one from the standpoint of the L-space conjecture, since in this case everynon- S surgery on K ( np,nq ) has non-trivial reduced Heegaard Floer homology. SARAH DEAN RASMUSSEN For the 2 g ( K ) − < qp case, the difficulty with slopes α ∈ ([ −∞ , N pq (cid:105) n \ [ −∞ , N pq − p (cid:105) n ) isthat the existence of a CTF on Y ( np,nq ) ( α ) depends on the family of suspension foliations on ∂Y —necessarily of nontrivial holonomy—that arise from taking a CTF F of slope 2 g ( K ) − Y and restricting F to ∂Y . Such F can only be extended over the union Y ( np,nq ) if it matcheswith the boundary restriction of some CTF of sf -slope p ∗ − Nq ∗ q − Np on the Seifert fibered spaceglued to Y to form the satellite. A similar phenomenon occurs for LOs on the fundamentalgroup of Y ( np,nq ) ( α ). See Boyer and Clay [3] for more on this subtlety in gluing behavior.In Theorem 8.1 of Section 8, we prove a result analogous to the one above, but for satellitesby algebraic links or iterated torus-links. Instead of restating this theorem here, we state a Corollary 1.5. For K ⊂ S a positive L-space knot with exterior Y , suppose K Γ ⊂ S is analgebraic link satellite or iterated torus-link satellite of K ⊂ S , with g ( K ) − (cid:54) = q r +1 p r at theroot torus-link-satellite operation of Γ , such that K Γ ⊂ S has no L-space surgeries besides S . ( lo ) If LO ( Y ) = N L ( Y ) , then every non- S surgery on K Γ ⊂ S has LO fundamental group. ( ctf ) If F ( Y ) = N L ( Y ) , then every irreducible non- S surgery on K Γ ⊂ S admits a CTF. In [26], J. Rasmussen and the author conjectured that our L-space gluing theorem (see The-orem 3.5 below) also holds without the hypothesis of admitting more than one L-space Dehnfilling. Hanselman, Rasmussen, and Watson recently announced a proof of this conjecture in[13], implying that the above corollary also holds for any non-L-space knot K ⊂ S .1.3. Satellites by algebraic links. In the context of negative definite graph manifolds, thedistinction between L-space and non-L-space has consequences for algebraic geometry.N´emethi recently showed that the unique negative-definite graph manifold Link( X, ◦ ) bound-ing the germ of a normal complex surface singularity ( X, ◦ ) is an L-space if and only if ( X, ◦ ) isrational [22]. Due to results of the author in [27], we can promote this statement to a relativeversion: the subregion L nd ⊂ L ( Y Γ ) of negative-definite L-space Dehn filling slopes for a graphmanifold Y Γ parameterizes, up to equisingular deformation, the rational surface singularities( X, ◦ ) admitting “end curves” ( C, ◦ ) ⊂ ( X, ◦ ) (see [23]), such that Link( X \ C ) = Y Γ . If one such( X, ◦ ) is ( C , Y Γ is the exterior of an algebraic link, motivating the following study. Setup. Whereas a T ( np, nq )-satellite operation is specified by (an unknot complement in) theSeifert fibered exterior of T ( np, nq ) determined by the triple ( p, q, n ), a sequence of torus-link-satellite operations is specified by a rooted tree Γ determining the graph manifold exterior ofthe pattern link, where each vertex v ∈ Vert(Γ) specifies the Seifert fibered T v := T ( n v p v , n v q v )-exterior in S v , determined by the triple ( p v , q v , n v ), so that S v has 2 exceptional fibers λ v − and λ v of respective multiplicities p v and q v , and components of T v are regular fibers in S v .Since we direct the edges of Γ rootward , each vertex w has a unique outgoing edge e w , corre-sponding to the incompressible torus in whose neighborhood T w is embedded, or equivalently,to a gluing map φ e w splicing the multiplicity- q w fiber λ w ∈ S w to a Seifert fiber in S u , for u := v ( e w ), where we write v ( e ) to denote the vertex on which an edge e ∈ Edge(Γ) terminates.(A “splice” is a type of toroidal connected sum exchanging meridians with longitudes.)There are only two types of fiber in S u available for splicing: a regular fiber, which wethen regard as one of the n v components f uj ∈ S u of T v , or the multiplicity- p v fiber λ u − ∈ S u .When φ e w splices λ w ∈ S w to some j th component f uj ∈ S u of T u , we call φ e w a smooth splice ,set j ( e w ) := j ∈ { , . . . , n v } , and declare the JSJ component Y u at u to be the exterior of λ u (cid:113) T u in S u . When φ e w splices λ w ∈ S w to λ u − ∈ S u , we call φ e w an exceptional splice , set j ( e w ) = − 1, and define Y u to be the exterior of λ u − (cid:113) λ u (cid:113) T u in S u . Since this latter splicecould be redefined as a smooth one if p u = 1, we demand p u > ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 7 If we define J v ⊂ j ( E in ( v )) and its complement I v by(10) J v := j ( E in ( v )) ∩ { , . . . , n v } , I v := { , . . . , n v } \ J v , then I v catalogs the boundary components of Y v left unfilled, forming the exteriors of linkcomponents, so that the total satellite K Γ ⊂ S of K ⊂ S has (cid:80) v ∈ Vert(Γ) | I v | components. Thepattern link specified by Γ is then an algebraic link if and only if its graph manifold exterior isnegative definite, which, by straight-forward calculations as appear, for example, in Eisenbudand Neumann’s book [7], is equivalent to the condition that Γ is a tree, and that( i ) p v , q v , n v > v ∈ Vert(Γ) , ( ii ) ∆ e > e ∈ Edge(Γ) , ∆ e := (cid:40) p v ( e ) q v ( − e ) − p v ( − e ) q v ( e ) j ( e ) = − q v ( − e ) − p v ( e ) p v ( − e ) q v ( e ) j ( e ) (cid:54) = − . Conversely, given an isolated planar complex curve singularity ( ◦ , C ) (cid:44) → (0 , C ), one canobtain such a tree Γ from Newton-Puisseux expansions for the defining equations of C , oralternatively from the amputated splice diagram of the dual plumbing graph of (cid:101) X for a goodembedded resolution ( (cid:101) X, (cid:101) C ) → ( C , C ). Again, see [7] for details.If Y Γ denotes the exterior of the Γ-satellite K Γ ⊂ S of K ⊂ S , then for each JSJ compo-nent Y v of Y Γ , we again have reducible and exceptional subsets R v , Z v ⊂ (cid:81) i ∈ I v P ( H ( ∂ i Y v ; Z )),along with a lattice Λ v acting on (cid:81) i ∈ I v P ( H ( ∂ i Y v ; Z )) by addition of sf v -slopes. Theorem 1.6. Suppose Y Γ := S \ ◦ ν ( K Γ ) is the exterior of an algebraic-link satellite K Γ ⊂ S of a (possibly trivial) positive L-space knot K ⊂ S , and suppose the triple ( p r , q r , n r ) specifiesthe initial torus-link satellite operation, occuring at the root vertex r ∈ Vert(Γ) . ( i.a ) If K is nontrivial, q r p r < g ( K ) − , p r > , and − / ∈ j ( E in ( r )) , then L ( Y Γ ) = Λ Γ ; Y Γ ( y Γ ) = S for all y Γ ∈ L ( Y Γ ) . ( i.b ) If K is nontrivial, q r p r < g ( K ) − N , and p r = 1 , then L S ( Y Γ ) = Λ r · S | I r | ([ N, + ∞ ] ×{∞} | I r |− ) × (cid:89) e ∈ E in ( r ) Λ Γ v ( − e ) (cid:113) (cid:97) e ∈ E in ( r ) (cid:16) L S ( Y Γ v ( − e ) ) × Λ Γ \ Γ v ( − e ) (cid:17) , where Y Γ v ( − e ) denotes the exterior of the Γ v ( − e ) -satellite of K ⊂ S . ( ii ) If K is trivial, or if K is nontrivial with q r p r ≥ g ( K ) − and q r > g ( K ) − , then L sf Γ ( Y Γ ) ⊃ (cid:89) v ∈ Vert(Γ) (cid:0) L min − sf v ∪ R v \ Z v ∪ L min + sf v (cid:1) , where L min + sf v := (cid:40) y v ∈ Q | I v | (cid:12)(cid:12)(cid:12)(cid:12) (cid:80) i ∈ I v (cid:98) y vi (cid:99) ≥ (cid:41) , L min − sf v := (cid:40) y v ∈ Q | I v | (cid:12)(cid:12)(cid:12)(cid:12) (cid:80) i ∈ I v (cid:100) y vi (cid:101) ≤ m − v (cid:41) \ (cid:40) (cid:80) (cid:100) y vi (cid:101) = 0 , (cid:80) (cid:98) [ − y vi ]( q v − p v ) (cid:99) = 0 (cid:41) J v = ∅ ; j ( e v ) (cid:54) = − ∅ otherwise ,m − v := − (cid:88) e ∈ E in ( v ) (cid:40) j ( e ) (cid:54) = − (cid:108) p v ( − e ) p v ∆ e (cid:109) + 1 j ( e ) = − (cid:41) − J v (cid:54) = ∅ ; j ( e v ) (cid:54) = − (cid:108) p v ( ev ) p v ∆ ev (cid:109) + 1 j ( e v ) = − otherwise . SARAH DEAN RASMUSSEN This is not as horrible as it looks. Part ( i.a ) describes the case in which all L-space surgeriesyield S . For part ( i.b ), either we trivially refill all components of Y Γ except for one exteriorcomponent in the root, effectively replacing K Γ with K ; or, we trivially refill all exteriorcomponents but those of Y Γ v ( − e ) for some incoming edge e , replacing K Γ with some K Γ v ( − e ) .The notation Λ Γ v and the term “trivially refill” hide a subtlety, however. For both theabove theorem and for Theorem 1.7 for iterated torus-link satellites, we define Λ Γ v to mean(11) Λ Γ v = (cid:110) y Γ v (cid:12)(cid:12)(cid:12) Y Γ v ( y Γ v ) = S (cid:111) , Y Γ v := exterior of the Γ v -satellite of the unknot . While Λ Γ v ⊃ (cid:81) u ∈ Vert(Γ v ) Λ u , the two sets need not be equal. Similarly, if N > q r p r and j ( e (cid:48) ) = − e (cid:48) ∈ E in ( r ), then L ( Y Γ ) ⊃ L ( Y Γ v ( − e (cid:48) ) ) × Λ Γ \ Γ v ( − e (cid:48) ) , but the containmentcan be proper. Section 8.2 provides a more explicit characterization of L ( Y Γ ) in this case,along with a concrete description of Λ Γ in the case of iterated torus-link satellites.Part ( ii ) of Theorem 1.6 is analogous to Theorem 1.2.ii for torus-link satellites, but thissimilarity is masked by our transition from S -slopes to sf -slopes. For example, if v is a leafand its emanating edge e v does not correspond to an exceptional splice, then we have L min + S v = Λ v · (cid:104) p v q v , + ∞ ] | I v | , L min − S v = Λ v · (cid:16) [ −∞ , p v q v (cid:105) | I v | \ [ −∞ , p v q v − q v + p v (cid:105) | I v | (cid:17) . In all other cases, we still have L min + S v = Λ v · (cid:104) p v q v , + ∞ ] | I v | , but L min − S v now sits inside theunit-radius neighborhood of R v , with m − v providing a measure of how deeply inside it sits.In the set ψ − v (cid:16) [ −∞ , p v q v − q v + p v (cid:105) | I v | (cid:17) removed from L min − sf v at a smoothly-spliced leaf v (in part ( ii ) of the theorem), the notation [ · ] : Q → Z , [ x ] := x − (cid:98) x (cid:99) gives the fractionalpart of a rational number x , whereas the notation [ a ] b := a − (cid:106) a | b | (cid:107) | b | picks out the smallestnonnegative representative of a mod b for any integers a, b ∈ Z . For a suitable L-space regionapproximation when q r = 2( K ) − p r = 1, see line (194) and the associated remark.1.4. Iterated torus-link-satellites. For the case of iterated torus-link satellites, we onlyallow “smooth splice” edges, corresponding to the original type of torus-link satellite opera-tion. We also drop the algebraicity condition that ∆ e > 0, and while we keep all p v , n v > q v < q v (cid:54) = 0, for each v ∈ Vert(Γ). Theorem 1.7. Suppose Y Γ := S \ ◦ ν ( K Γ ) is the exterior of an iterated torus-link satellite K Γ ⊂ S of a (possibly trivial) positive L-space knot K ⊂ S , and suppose the triple ( p r , q r , n r ) specifies the initial torus-link satellite operation occuring at the root vertex r ∈ Vert(Γ) . ( i.a ) If K is nontrivial, q r p r < g ( K ) − , p r > , then L ( Y Γ ) = Λ Γ . ( i.b ) If K is nontrivial, q r p r < g ( K ) − N , and p r = 1 , then L S ( Y Γ ) = Λ r · S | I r | ([ N, + ∞ ] ×{∞} | I r |− ) × (cid:89) e ∈ E in ( r ) Λ Γ v ( − e ) (cid:113) (cid:97) e ∈ E in ( r ) (cid:16) L S ( Y Γ v ( − e ) ) × Λ Γ \ Γ v ( − e ) (cid:17) , where Y Γ v ( − e ) denotes the exterior of the Γ v ( − e ) -satellite of K ⊂ S . ( ii ) If K is trivial, or if K is nontrivial with q r p r ≥ g ( K ) − and q r > g ( K ) − , then L sf Γ ( Y Γ ) ⊃ (cid:89) v ∈ Vert(Γ) (cid:0) L min − sf v ∪ R v \ Z v ∪ L min + sf v (cid:1) , where ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 9 L min + sf v := (cid:40) y v ∈ Q | I v | (cid:12)(cid:12)(cid:12)(cid:12) (cid:80) i ∈ I v (cid:98) y vi (cid:99) ≥ m + v (cid:41) \ { (cid:80) (cid:98) y vi (cid:99) = (cid:80) (cid:98) [ y vi ][ p v ] q v (cid:99) = 0 } J v = ∅ ; p v q v > +1 { (cid:80) (cid:98) y vi (cid:99) = (cid:80) (cid:98) [ y vi ][ − p v ] q v (cid:99) = 0 } J v = ∅ ; q v < − ∅ otherwise , L min − sf v := (cid:40) y v ∈ Q | I v | (cid:12)(cid:12)(cid:12)(cid:12) (cid:80) i ∈ I v (cid:100) y vi (cid:101) ≤ m − v (cid:41) \ { (cid:80) (cid:100) y vi (cid:101) = (cid:80) (cid:98) [ − y vi ][ p v ] q v (cid:99) = 0 } J v = ∅ ; p v q v < − { (cid:80) (cid:100) y vi (cid:101) = (cid:80) (cid:98) [ − y vi ][ − p v ] q v (cid:99) = 0 } J v = ∅ ; q v > +1 ∅ otherwise , with m + v := − (cid:88) e ∈ E in ( v ) (cid:18)(cid:24) p v ( − e ) q v ( − e ) (cid:25) − (cid:19) + q v = − J v (cid:54) = ∅ ; q v < − or p v q v > +10 otherwise ,m − v := − (cid:88) e ∈ E in ( v ) (cid:18)(cid:22) p v ( − e ) q v ( − e ) (cid:23) + 1 (cid:19) − q v = +11 J v (cid:54) = ∅ ; q v > +1 or p v q v < − otherwise . Monotonicity. In both of the above theorems, (cid:81) v ∈ Vert(Γ) (cid:0) L min − sf v ∪ R v \ Z v ∪ L min + sf v (cid:1) isa component of what we call the monotone stratum L mono sf Γ ( Y Γ ) of L sf Γ ( Y Γ ), as discussed inSection 7.6. We say that y Γ ∈ L sf Γ ( Y Γ ) is monotone at v ∈ Vert(Γ) if(12) ∞ ∈ φ P e ∗ L ◦ sf v ( − e ) ( Y Γ v ( − e ) ( y Γ | Γ v ( − e ) )) ∀ e ∈ E in ( v ) and ∞ ∈ φ P e v ∗ L ◦ sf v ( Y Γ v ( y Γ | Γ v )) , or more prosaically (when the above interval interiors are nonempty) is monotone at v if y vj ( e )+ ≤ y vj ( e ) − ∀ e ∈ E in ( v ) , y v ( e v ) j ( e v )+ ≤ y v ( e v ) j ( e v ) − , as these are the respective endpoints of the above intervals. The monotone stratum L mono sf Γ ( Y Γ )of L sf Γ ( Y Γ ) is the set of slopes y Γ ∈ L sf Γ ( Y Γ ) such that y Γ is monotone at all v ∈ Vert(Γ).Specifying different collections of local monotonicity conditions allows one to decompose anL-space region into strata of disparate topologies. For example, for the (globally) monotonestratum, we have the following topological result, proved in Section 7.6. Theorem 1.8. Suppose that K Γ ⊂ S is an algebraic link satellite, specified by Γ , of a positiveL-space knot K ⊂ S , where either K is trivial, or K is nontrivial with q r p r > g ( K ) − . Let V ⊂ Vert(Γ) denote the subset of vertices v ∈ V for which | I v | > .Then the Q -corrected R -closure L mono sf Γ ( Y Γ ) R of the monotone stratum of L sf Γ ( Y Γ ) is ofdimension | I Γ | and deformation retracts onto an ( | I Γ | − | V | ) -dimensional embedded torus, (13) L mono sf Γ ( Y Γ ) R (cid:32) (cid:81) v ∈ V T | I v |− (cid:44) → (cid:81) v ∈ V ( R ∪ {∞} ) | I v | sf v , projecting to embedded tori T | I v |− (cid:44) → ( R ∪ {∞} ) | I v | sf v parallel to B sf v ⊂ ( R ∪ {∞} ) | I v | sf v . Non-monotone strata, when they exist, change the topology of the total L-space region andhave implications for “boundedness from below” in the sense of N´emethi and Gorsky [11], butwe defer the study of non-monotone regions to later work, whether by this author or others. New tools. In fact, the propositions in Sections 6 and 7 provide many tools for analyzingquestions not addressed in this paper. For example, in the absence of an exceptional spliceat v , Proposition 6.2(+). iii precisely characterizes when y v ( e )0+ = p ∗ v q v , defining the right-handboundary of the non-monotone stratum at that component. These tools can also be used tocharacterize the non-product components of the monotone stratum more explicitly. New features. Even so, Theorems 1.6 and 1.7 already reveal more interesting behavior thanappears for nondegenerate torus-link satellites. In particular, the boundary of the S -slopeL-space region need not occur at infinity . For example, if(a) Γ = v specifies a single torus-link satellite of the unknot, and p v = q v = 1,(b) Γ specifies an iterated torus-link satellite, and m + v < 0, or(c) Γ specifies an iterated or algebraic-link satellite, and we restrict to an appropriatepiece of L mono sf Γ ( Y Γ ) | sf v outside the product region,then we encounter regions of the form ψ (cid:32)(cid:40) y v ∈ Q | I v | (cid:12)(cid:12)(cid:12)(cid:12) (cid:80) i ∈ I v (cid:98) y vi (cid:99) ≥ m + v (cid:41)(cid:33) = Λ v · (cid:104) p v q v , + ∞ ] | I v | ∪ Λ v · S | I v | (cid:16) (cid:104)−∞ , p v q v − | m + v | × (cid:104) p v q v , + ∞ ] | I v |−| m + v | (cid:17) for −| I v | ≤ m + v < 0, with additional components added onto the unit-radius neighborhoodof R v if m v < −| I v | . (An analogous phenomenon occurs in the negative direction when m − v > n − , n + ∈ Z ≥ , it is possible to construct an iterated satellite by torus links for which the S v component of some stratum of the L-space region fills up the quadrant (cid:104)−∞ , p v q v − (cid:105) n − × (cid:104) p v q v + 1 , + ∞(cid:105) n + , n − + n + = | I v | . There likewise exist iterated torus-link satellite exteriors Y Γ with u, v ∈ Vert(Γ) for whichthe projections of L S ( Y Γ ) to the positive quadrant (cid:104) p u q u + 1 , + ∞(cid:105) | I u | and to the negativequadrant (cid:104)−∞ , p v q v − (cid:105) | I v | are both empty.We therefore feel that the notion of “L-space link” should be broadened to encompass anylink whose L-space surgery region contains an open neighborhood in the space of slopes, ratherthan defining this notion in terms of large positive slopes in the L-space surgery region.1.5. Organization. Section 2 establishes basic Seifert fibered space conventions and elabo-rates on the distinguished slope subsets Λ, R , and Z . Section 3 introduces notation for L-spaceintervals and proves a new gluing theorem for knot exteriors with graph manifolds. Section 4introduces machinery developed by the author in [27] to compute L-space intervals for fiberexteriors in graph manifolds, and applies this to prove Theorem 4.5, an sf -slope version ofthe torus-link satellite results in Theorem 1.2. Section 5 addresses the topology of L-spaceregions and proves Theorem 1.3. Section 6 describes the graph Γ associated to an iteratedtorus-link satellite, computes various estimates useful for bounding L-space surgery regions foriterated-torus-link and algebraic link satellites, and proves Theorem 1.7 for iterated-torus-linksatellites. Section 7 describes adaptations of this graph Γ to accomodate algebraic link exteri-ors, discusses monotonicity, and proves Theorems 1.6 and 1.8. Section 8 proves results relatedto conjectures of Boyer-Gordon-Watson and Juh´asz, including generalizations of Theorem 1.4.Readers interested in constructing their own L-space regions for algebraic link satellitesor iterated torus-link satellites should refer to the L-space interval technology introduced forgraph manifolds in Section 4, and to the analytical tools developed in Section 6. Acknowledgements. It is a pleasure to thank Eugene Gorsky, Gordana Mati´c, Andr´asN´emethi, and Jacob Rasmussen and for helpful conversations. I am especially indebted toMaciej Borodzik, for his hospitality at the University of Warsaw, for extensive feedback, andfor his encouraging me to write something down about L-space surgeries on algebraic links. ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 11 Basis conventions and the slope subsets Λ , R , and Z Suppose Y := M \ ◦ ν ( L ), with boundary ∂Y = (cid:96) i =1 ∂ i Y , ∂ i Y = ∂ν ( L i ), is the link exteriorof an n -component link L = (cid:96) ni =1 L i ⊂ M in a closed oriented 3-manifold M . Then, up tochoices of sign, the Dehn filling M of Y specifies a ( multi -) meridional class ( µ , . . . , µ n ) ∈ H ( ∂Y ; Z ) = (cid:76) ni =1 H ( ∂ i Y ; Z ), where each meridian µ i ∈ H ( ∂ i Y ; Z ) is the class of a curvebounding a compressing disk of the solid torus ν ( L i ). Any choice of classes λ , . . . , λ n ∈ H ( ∂Y ; Z ) satisfying µ i · λ i = 1 for each i then produces a surgery basis ( µ , λ , . . . , µ n , λ n )for H ( ∂ i Y ; Z ). We call these λ i surgery longitudes , or just longitudes if the context is clear.When M = S , H ( ∂Y ; Z ) has a conventional basis given by taking each λ i to be the rationallongitude ; that is, each λ i generates the kernel of the homomorphism ι i ∗ : H ( ∂ i Y ; Q ) → H ( M \ ◦ ν ( L i ); Q ) induced by the inclusion ι : ∂ i Y (cid:44) → M \ ◦ ν ( L i ). For M = S , the rationallongitude coincides with Seifert-framed longitude.It is important to keep in mind that for knots and links in S , the conventional homologybasis is not always the most natural surgery basis. In particular, any cable or satellite ofa knot in S determines a surgery basis for which the surgery longitude corresponds to theSeifert longitude of the associated torus knot or companion knot. This cable surgery basis or satellite surgery basis does not coincide with the conventional basis for S .For Y a compact oriented 3-manifold with boundary ∂Y = (cid:96) ni =1 ∂ i Y a disjoint union oftori, any basis B = (cid:81) ni =1 ( m i , l i ) for H ( ∂Y ; Z ) determines a map π b : n (cid:89) i =1 P ( H ( ∂ i Y ; Z )) −→ ( Q ∪ {∞} ) n b , (14) (cid:81) ni =1 [ a i m i + b i l i ] (cid:55)−→ ( a b , . . . , a n b n ) , which associates b - slopes a i b i ∈ Q ∪ {∞} to nonzero elements a i m i + b i l i ∈ H ( ∂ i Y ; Z ). Each b -slope ( a b , . . . , a n b n ) ∈ ( Q ∪ {∞} ) n b specifies a Dehn filling Y b ( a b , . . . , a n b n ), which is the closed3-manifold given by attaching a compressing disk, for each i , to a simple closed curve in theprimitive homology class corresponding to [ a i m i + b i l i ] ∈ P ( H ( ∂ i Y ; Z )), and then gluing in a3-ball to complete this solid torus filling of ∂ i Y . Notationally, we write A b ( Y ) := π b ( A ( Y )) forany subset A ( Y ) ⊂ (cid:81) ni =1 P ( H ( ∂ i Y ; Z )) of slopes for Y . Thus, L S ( Y ( np,nq ) ) ⊂ ( Q ∪ {∞} ) nS realizes L ( Y ( np,nq ) ) with respect to the conventional homology basis for link exteriors in S Seifert fibered basis. For Y Seifert fibered over an n -times punctured S , there isa conventional Seifert fibered basis sf = ( ˜ f , − ˜ h , . . . , ˜ f n , − ˜ h n ) for H ( ∂Y ; Z ) which makesslopes correspond to Seifert data for Dehn fillings of Y . That is, each − ˜ h i is the meridian ofthe i th excised regular fiber, and each ˜ f i is the lift of the regular fiber class f ∈ H ( Y ; Z ) to aclass ˜ f i ∈ H ( ∂ i Y ; Z ) satisfying − ˜ h i · ˜ f i = 1. Note that this makes ( ˜ f i , − ˜ h i ) a reverse-oriented basis for each H ( ∂ i Y ; Z ), but this choice is made so that if Y is trivially Seifert fibered,then with respect to our Seifert fibered basis, the Dehn filling Y sf ( β α , . . . , β n α n ) coincides withthe genus zero Seifert fibered space M := M S ( β α , . . . , β n α n ) with (non-normalized) Seifertinvariants ( β α , . . . , β n α n ) and first homology(15) H ( M ; Z ) = (cid:104) f, h . . . , h n | (cid:80) ni =1 h i = 0; ι ∗ ( µ ) = · · · = ι n ∗ ( µ n ) = 0 (cid:105) , where(16) µ i := β i ˜ f i − α i ˜ h i ∈ H ( ∂ i Y ; Z ) , ι i : ∂ i Y (cid:44) → Y, h i := ι i ∗ (˜ h i ) , and ι i ∗ ( ˜ f i ) = f. Action of Λ. The relation (cid:80) ni =1 h i = 0 in (15) comes from regarding the meridianimages − h i ∈ H ( Y ; Z ) as living in some global section S \ (cid:96) ni =1 D i (cid:44) → Y of the S fibration,so that each − h i = − ∂D i can be regarded as − h i = ∂ i ( S \ (cid:96) nj D j ), making the total class − (cid:80) ni =1 h i bound a disk in S \ (cid:96) ni D i . This choice of global section is not canonical, however.Any new choice of global section would correspond to a new choice of meridians,(17) ˜ h i (cid:55)−→ ˜ h (cid:48) i := ˜ h i − l i ˜ f i , i ∈ { , . . . , n } for some l ∈ Z n , n (cid:80) i =1 l i = 0 . Writing µ i = β (cid:48) i ˜ f i − α (cid:48) i ˜ h (cid:48) i to express µ i in terms of this new basis yields(18) β (cid:48) i = β i + l i α i , α (cid:48) i = α i , β (cid:48) i α (cid:48) i = β i α i + l i , i ∈ { , . . . , n } . In other words, the lattice of global section reparameterizations(19) Λ( Y ) := { l ∈ Z n | (cid:80) ni =1 l i = 0 } acts on Seifert data, hence on sf -slopes, by addition, without changing the underlying manifoldor S fibration. Moreover, for any choice of boundary-homology basis b , the change of basisfrom sf -slopes to b slopes induces an action of Λ on b -slopes.2.3. Action of Λ on torus-link-exterior slopes. As occurs in the case when Y = Y n ( p,q ) isthe exterior of the torus link T ( np, nq ) ⊂ S (see (42)), the Λ-action on S -slopes(20) α (cid:55)→ l · α = ψ ( l + ψ − ( α )) , l ∈ Λ sf ( Y n ( p,q ) ) , α ∈ ( Q ∪ {∞} ) nS , induced by the transformation(21) ψ : ( Q ∪ {∞} ) n sf −→ ( Q ∪ {∞} ) nS , y (cid:55)→ (cid:16) pq + y , . . . , pq + y n (cid:17) , is of particular interest to us. To aid in the introduction’s discussion of the role of Λ in Theo-rem 1.2, we temporarily introduce the sets L , L , L ⊂ ( Q ∪ {∞} ) nS of S -slopes, as follows: L := { ∞ } ⊂ ( Q ∪ {∞} ) nS , ∞ := ( ∞ , . . . , ∞ ) , (22) L := S n · ([ N, + ∞ ] × {∞} n − ) ⊂ ( Q ∪ {∞} ) nS , (23) L := (cid:0) [ −∞ , pq (cid:105) n \ [ −∞ , N (cid:105) n (cid:1) ∪ (cid:98) R S ∪ (cid:104) pq, + ∞ ] n ⊂ ( Q ∪ {∞} ) nS , (24)for some N ∈ Z , where we have temporarily introduced the notation (cid:98) R S to denote the union(25) (cid:98) R S := S n · (cid:0) { pq } × ([ −∞ , pq (cid:105)∪(cid:104) pq, + ∞ ]) n − (cid:1) ⊂ ( Q ∪ {∞} ) nS of sets of S -slopes. Lastly, for ε > 0, we take U npq ( ε ) to be the radius- ε punctured neighborhood(26) U npq ( ε ) := (cid:83) ni =1 { α | α i ∈ [ pq − ε, pq (cid:105) ∪ (cid:104) pq, pq + ε ] } ⊂ ( Q ∪ {∞} ) nS of the union of hyperplanes R S = (cid:83) ni =1 { α | α i = pq } ⊂ ( Q ∪ {∞} ) nS discussed in Section 2.4. Proposition 2.1. The action of Λ on ( Q ∪ {∞} ) nS in (20) satisfies the following properties: ( a ) Λ S ∩ Q nS ⊂ (cid:0) [ pq − , pq (cid:105) ∪ (cid:104) pq, pq + 1] (cid:1) n ; Λ S = Λ · { ∞ } S = Λ · L . ( b ) (Λ · L i ) \ L i ⊂ U npq (1) for i = 0 always, for i = 1 when N > pq , and for i = 2 when N ≤ pq . ( c ) For ε > , each of the following sets of S -slopes can be realized as a union of finitelymany rectangles of dimensions 0, 1, and { n − , n } , respectively: (Λ · L ) \ U npq ( ε ) , (Λ · L ) \ U npq ( ε ) for N > pq, and (Λ · L ) \ U npq ( ε ) for N ≤ pq .Proof. Part ( a ). The first statement follows from the fact that m ∈ [ − , (cid:105) ∪ (cid:104) , ∪ {∞} forall m ∈ Z . The second statement is due to the fact that ∈ Λ sf implies ∞ = ψ ( ) ∈ Λ S .Part ( b ). First note that the action of Z on ( Q ∪ {∞} ) by addition fixes both ∞ = ψ − j ( pq )as a point and its complement Q = ψ − j (cid:0) [ −∞ , pq (cid:105) ∪ (cid:104) pq, + ∞ ] (cid:1) as a set, for any j ∈ { , . . . , n } .The action of Λ on ( Q ∪ {∞} ) nS therefore fixes setwise the union (cid:98) R S of products of such sets. ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 13 Since (Λ · X ) \ X ⊂ (Λ sf \ { } ) · X for any subset X ⊂ ( Q ∪ {∞} ) nS of S -slopes, and since L i ⊂ ([ −∞ , pq (cid:105) n ∪ (cid:104) pq, + ∞ ] n ) S for i = 0 always, for i = 1 when N > pq , and for i = 2 when N ≤ pq , it is sufficient to show that (Λ sf \ { } ) · ([ −∞ , pq (cid:105) n ∪ (cid:104) pq, + ∞ ] n ) S ⊂ U npq (1). To seethis, we first note that l ∈ (Λ sf \ { } ) must have at least one positive and at least one negativecomponent, say l i + ∈ [1 , + ∞(cid:105)∩ Z and l i − ∈ (cid:104)−∞ , − ∩ Z , implying ( l +[0 , + ∞(cid:105) n ) | i + ⊂ [1 , + ∞(cid:105) and ( l + (cid:104)−∞ , n ) | i − ⊂ (cid:104)−∞ , − l · (cid:104) pq, + ∞ ] n ) | i + = ψ ( l + ψ − ( (cid:104) pq, + ∞ ] n )) | i + = ψ ( l + [0 , + ∞(cid:105) n ) | i + ⊂ (cid:104) pq, pq + 1] , (27) ( l · (cid:104)−∞ , pq ] n ) | i − = ψ ( l + ψ − ( (cid:104)−∞ , pq ] n )) | i − = ψ ( l + (cid:104)−∞ , n ) | i + ⊂ [ pq − , pq (cid:105) , (28)and so we conclude that l · ([ −∞ , pq (cid:105) n ∪ (cid:104) pq, + ∞ ] n ) S ⊂ U npq (1), completing the proof of ( b ).Part ( c ). In the sf basis, the complement of U npq ( ε ) within the set of S -slopes is given by(29) ψ − (cid:0) ( Q ∪ {∞} ) nS \ U npq ( ε ) (cid:1) = (cid:10) − ε , + ε (cid:11) n ∪ (cid:83) ni =1 { y | y i = ∞} ⊂ ( Q ∪ {∞} ) n sf . Since both (cid:83) ni =1 { y | y i = ∞} sf = ψ − ( R S ) and ψ − ( (cid:98) R S ) are fixed setwise by Λ, and since ψ − ( L i ) ∩ ψ − ( R S ) = ∅ for i = 0 and for i = 1 with N > pq , but ψ − ( L i ) ∩ ψ − ( R S ) = ψ − ( (cid:98) R S ) when i = 2 and N ≤ pq , it follows that(30) (Λ sf + ψ − ( L i )) ∩ (cid:83) ni =1 { y | y i = ∞} sf = (cid:40) ∅ i = 0 or ( i = 1 and N > pq ) ψ − ( (cid:98) R S ) i = 2 and N ≤ pq . Thus, since ψ − ( (cid:98) R S ) is already a finite union of ( n − sf + ψ − ( L i )) ∩ (cid:10) − ε , + ε (cid:11) n sf is a union of finitely many rectangles of dimensions0, 1, or n in the respective cases that i = 0, i = 1 with N > pq , or i = 2 with N ≤ pq . Theproof of this latter statement is straightforward, however, since ψ − ( L i ) is already a finiteunion of rectangles of dimensions 0, 1, or n , respectively for the three above respective cases,and only finitely many distinct rectangles can be formed by intersecting Z n translates of theserectangles with (cid:10) − ε , + ε (cid:11) n sf ⊂ ( Q ∪ {∞} ) n sf . (cid:3) Reducible and exceptional sets R and Z . Like the above action of Λ, the followingfacts about reducible fillings are well known in low dimensional topology, but for the benefitof a diverse readership we provide some details. Proposition 2.2. Let ˆ Y denote the trivial S fibration over S \ (cid:96) ni =0 D i , and let Y denotethe Dehn filling of ˆ Y along the S -fiber lift ˜ f ∈ H ( ∂ ˆ Y ; Z ) , i.e. , along the ∞ sf -slope of ∂ ˆ Y . Then Y is a connected sum Y = ni =1 ( S i \ S f × D i ) (where S f is the fiber), and eachexterior S i \ S f × D i has meridian − ˜ h i and rational longitude ˜ f i .Proof. Choose a global section S \ (cid:96) ni =0 D i (cid:44) → ˆ Y which respects the sf basis. We shall stretchthe disk S \ D into a (daisy) flower shape, with one D i contained in each petal. Embed2 n points p − , p +1 , . . . , p − n , p + n (cid:44) → ∂D , in that order with respect to the orientation of − ∂D .For each i ∈ { , . . . , n } , let δ i and ε i denote the respective arcs from p − i to p + i and from p + i to p − i +1(mod n ) along − ∂D , and properly embed an arc γ i (cid:44) → S \ (cid:96) ni =0 D i from p + i to p − i which winds once positively around D i and winds zero times around the other D j , withoutintersecting any of the other γ j arcs. Holding the p ± i points fixed while stretching the δ i arcsoutward and pulling the γ i arcs tight realizes our global section as the punctured flower shape(31) S \ n (cid:96) i =0 D i = ˜ D ∪ n (cid:96) i =1 ( ˜ D i \ D i ) , where ˜ D denotes the central disk of the flower, bounded by ∂ ˜ D = ( (cid:96) ni =1 − γ i ) ∪ ( (cid:96) ni =1 ε i ),and each ˜ D i denotes the petal-shaped disk bounded by ∂ ˜ D i = δ i ∪ p ± i γ i .The Dehn filling Y is formed by multiplying the above global section with the fiber S f andthen gluing a solid torus D f × ∂D (with ∂D f = S f ) along S f × ∂D . Since(32) − ∂D = ( n (cid:96) i =1 δ i ) ∪ ( n (cid:96) i =1 ε i ) , ∂ ˜ D = ( n (cid:96) i =1 − γ i ) ∪ ( n (cid:96) i =1 ε i ) , and ∂ ˜ D i = δ i ∪ p ± i γ i , we can decompose the solid torus D f × ∂D along the disks D f × p ± i , and distribute thesesolid-torus components among the boundaries of ˜ D and the ˜ D i , so that(33) − ∂D f × ∂D = ( D f × ∂ ˜ D ) \ (cid:96) ni =1 ( D f × − ◦ γ i ) ∪ (cid:96) ni =1 (cid:16) ( D f × ∂ ˜ D i ) \ ( D f × ◦ γ i ) (cid:17) , where the union is along the boundary 2-spheres(34) S i := ( D f × p − i ) ∪ ( S f × γ i ) ∪ ( D f × p + i ) = ∂ ( D f × ◦ γ i )of the balls D f × ± ◦ γ i . Thus, if we set S i := ( D f × ∂ ˜ D i ) ∪ ( S f × ˜ D i ) for i ∈ { , . . . , n } , then(35) Y = S n (cid:96) i =1 ( S i \ S f × D i ) , with the connected sum taken along the spheres S i . (cid:3) Corollary 2.3. If ˆ Y is as above, and if ( y , . . . , y n ) ∈ {∞} k × Q n − k for some k ∈ { , . . . , n } ,then ˆ Y sf ( ∞ , y , . . . , y n ) = ( ki =1 S × S ) ni = k +1 M S ( y i )) . This motivates the following terminology. Definition 2.4. Suppose that ∂Y = (cid:96) ni =1 T i , and that Y has a Seifert fibered JSJ componentcontaining ∂Y . Then the reducible slopes R ( Y ) and exceptional slopes Z ( Y ) ⊂ R ( Y ) aregiven by R sf ( Y ) := S n · ( {∞} × ( Q ∪ ∞ ) n − ) and Z sf ( Y ) := S n · ( {∞} × ( Q ∪ ∞ ) n − ) , where S n acts by permutation of slopes. Note that occasionally, slopes in R ( Y ) yield Dehn fillings which are connected sums of alens space with 3-spheres, hence are not reducible. Definition 2.5. Suppose Y is as above. If the Seifert fibered component containing ∂Y has no exceptional fibers, then the false reducible slopes R ( Y ) ⊂ R ( Y ) of Y are given by R sf ( Y ) = S n · ( {∞} × Q × { } n − ) . Any reducible slopes which are not false reducible arecalled truly reducible . Equivalently, the truly reducible slopes are those slopes which yieldreducible Dehn fillings. Rational longitudes B . Our last distinguished slope set of interest, the set B of rationallongitude slopes, makes sense for Y of any geometric type. Definition 2.6. Suppose Y is a compact oriented 3-manifold with ∂Y a union of n > toroidal boundary components and with at least one rational-homology-sphere Dehn filling.The set of rational longitudes B ⊂ (cid:96) ni =1 P ( H ( ∂ i Y ; Z )) of Y is the set of slopes (36) B := { β ∈ (cid:96) ni =1 P ( H ( ∂ i Y ; Z )) | b ( Y ( β )) > } . Note that this implies Z = R ∩ B . Moreover, when Y is Seifert fibered, B sf ( Y ) is theclosure of a linear subspace of sf -slopes. That is, by Section 5 of [26] (for example), we have(37) b (cid:16) M S (cid:0) β α , . . . , β n α n (cid:1)(cid:17) > ⇐⇒ (cid:80) ni =1 β i α i = 0 . ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 15 In particular, since the T ( np, nq )-exterior Y n ( p,q ) := M S ( − q ∗ p , p ∗ q , ∗ , . . . , n ∗ ), constructed in (42),has β − α − + β α = pq , the next proposition follows immediately from line (37). Proposition 2.7. If Y = S \ ◦ ν ( T ( np, nq )) is the exterior of the ( np, nq ) torus link, then B sf ( Y ) = (cid:110) y ∈ ( Q ∪ {∞} ) n sf (cid:12)(cid:12)(cid:12) pq + (cid:80) ni =1 y i = 0 (cid:111) , is the closure in ( Q ∪{∞} ) n sf of the hyperplane (cid:110) y ∈ Q n (cid:12)(cid:12)(cid:12) pq + (cid:80) ni =1 y i = 0 (cid:111) in Q n ⊂ ( Q ∪{∞} ) n sf .Moreover, if B sf denotes the real closure of B sf in (cid:96) ni =1 P ( H ( ∂ i Y ; R )) ∼ → ( R ∪ {∞} ) n sf , then B sf ∼ = T n − ∼ = ( R ∪ {∞} ) n − (cid:44) → ( R ∪ {∞} ) n sf ∼ = T n . L-space intervals and gluing L-space interval notation. For the following discussion, Y denotes a compact oriented3-manifold with torus boundary ∂Y , and B is a basis for H ( ∂Y ; Z ), inducing an identification π B : P ( H ( ∂Y ; Z )) → ( Q ∪ {∞} ) b =: P ( H ( ∂Y ; Z )) b . Definition 3.1. We introduce the notation [[ · , · ]] , so that for y − , y + ∈ P ( H ( ∂Y ; Z )) b , thesubset [[ y − , y + ]] ⊂ P ( H ( ∂Y ; Z )) b is defined as follows. [[ y − , y + ]] := (cid:40) P ( H ( ∂Y ; Z )) b \ { y − } = P ( H ( ∂Y ; Z )) b \ { y − } y − = y + P ( H ( ∂Y ; Z )) b ∩ I ( y − ,y + ) y − (cid:54) = y + , where I ( y − ,y + ) ⊂ P ( H ( ∂Y ; R )) b indicates the closed interval with left-hand endpoint y − andright-hand endpoint y + . By Proposition 1.3 and Theorem 1.6 of J. Rasmussen and the author’s [26], the L-spaceinterval L ( Y ) ⊂ P ( H ( ∂Y ; Z )) of L-space Dehn filling slopes of Y can only take certain forms. Proposition 3.2 (J. Rasmussen, S. Rasmussen [26]) . One of the following is true:(i) L ( Y ) = ∅ ,(ii) L ( Y ) = { η } , for some η ∈ P ( H ( ∂Y ; Z )) ,(iii) L b ( Y ) = [[ l, l ]] , with l ∈ P ( H ( ∂Y ; Z )) b the rational longitude of Y , or(iv) L b ( Y ) = [[ y − , y + ]] with y (cid:54) = y + . It is for this reason that we refer to the space of L-space Dehn filling slopes as an interval .It also makes sense to speak of the the interior of this interval. Definition 3.3. The L-space interval interior L ◦ ( Y ) ⊂ L ( Y ) of Y satisfies L ◦ b ( Y ) := ∅ L ( Y ) = ∅ or L ( Y ) = { η } [[ l, l ]] L b ( Y ) = [[ l, l ]] P ( H ( ∂Y ; Z )) b ∩ ◦ I ( y − ,y + ) L b ( Y ) = [[ y − , y + ]] with y − (cid:54) = y + , where ◦ I ( y − ,y + ) ⊂ P ( H ( ∂Y ; R )) b indicates the open interval with left-hand endpoint y − andright-hand endpoint y + . This gives us a new way to characterize the property of Floer simplicity for Y . Proposition 3.4 (J. Rasmussen, S. Rasmussen [26]) . The following are equivalent: • Y has more than one L-space Dehn filling, • L b ( Y ) = [[ y − , y + ]] for some y − , y + ∈ P ( H ( ∂Y ; Z )) b , • L ◦ ( Y ) (cid:54) = ∅ .In the case that any, and hence all, of these three properties hold, we say that Y is Floer simple . Both Floer simple manifolds and graph manifolds have predictably-behaved unions withrespect to the property of being an L-space. Theorem 3.5 (Hanselman, J. Rasmussen, S. Rasmussen, Watson [26, 12, 27]) . If the manifold Y ∪ ϕ Y , with gluing map ϕ : ∂Y → − ∂Y , is a closed union of 3-manifolds, each with incom-pressible single-torus boundary, and with Y i both Floer simple or both graph manifolds, then Y ∪ ϕ Y is an L-space ⇐⇒ ϕ P ∗ ( L ◦ ( Y )) ∪ L ◦ ( Y ) = P ( H ( ∂Y ; Z )) . Unfortunately, this theorem fails to encompass the case in which an L-space knot exterior isglued to a non-Floer simple graph manifold, and our study of surgeries on iterated or algebraicsatellites will certainly require this case. We therefore prove the following result. Theorem 3.6. If Y ∪ ϕ Y , with gluing map ϕ : ∂Y → − ∂Y , is a closed union of 3-manifolds,such that Y is the exterior of a nontrivial L-space knot K ⊂ S , and Y is a graph manifold,or connected sum thereof, with incompressible single torus boundary, then (38) Y ∪ ϕ Y is an L-space ⇐⇒ ϕ P ∗ ( L ◦ ( Y )) ∪ L ◦ ( Y ) = P ( H ( ∂Y ; Z )) . Moreover, if Y is not Floer simple, then (38) holds for K ⊂ S an arbitrary nontrivial knot.Proof. We first reduce to the case prime Y . Let Y (cid:48) denote the connected summand of Y containing ∂Y , and recall that hat Heegaard Floer homology tensors over connected sums.Thus, if Y has any non-L-space closed connected summands, then Y ∪ ϕ Y is a non-L-spaceand L ( Y ) = ∅ , regardless of the union Y ∪ ϕ Y (cid:48) . On the other hand, if all the closed connectedsummands of Y are L-spaces, then L ◦ ( Y ) = L ◦ ( Y (cid:48) ), and Y ∪ ϕ Y is an L-space if and onlyif Y ∪ ϕ Y (cid:48) is an L-space. We therefore henceforth assume Y is prime.If Y is Floer simple, then when K ⊂ S is an L-space knot, Y is Floer simple, and so thedesired result is already given by Theorem 3.5. We therefore assume Y is not Floer simple,implying N L ( Y ) = P ( H ( ∂Y ; Z )) or N L ( Y ) = P ( H ( ∂Y ; Z )) \ { y } for some single slope y .In either case, L ◦ ( Y ) = ∅ , and so it remains to show that Y ∪ ϕ Y is not an L-space.In [27], the author showed for any (prime) graph manifold Y with single torus bound-ary that if F ( Y ) (called F d ( Y ) in that paper’s notation) denotes the set of slopes α ∈ P ( H ( ∂Y ; Z )) for which Y admits a co-oriented taut foliation restricting to a product folia-tion of slope α on ∂Y , then F ( Y ) = N L ( Y ) \ R ( Y ). Since Y is prime, R ( Y ) = {∞} ∈ P ( H ( ∂Y ; Z )) sf . In particular, F ( Y ) is the complement of a finite set in P ( H ( ∂Y ; Z )).On the other hand, Li and Roberts show in [20] that for the exterior of an arbitrarynontrivial knot in S , such as Y , one has F S ( Y ) ⊃ (cid:104) a, b (cid:105) for some a < < b . In particular, F ( Y ) is infinite, implying ϕ P ∗ ( F ( Y )) ∩ F ( Y ) is nonempty. Thus we can construct a co-oriented taut foliation F on Y ∪ ϕ Y by gluing together co-oriented taut foliations restrictingto a matching product foliation of some slope α ∈ ϕ P ∗ ( F ( Y )) ∩ F ( Y ) on ∂Y .Eliashberg and Thurston showed in [8] that a C co-oriented taut foliation can be perturbedto a pair of oppositely oriented tight contact structures, each with a symplectic semi-fillingwith b +2 > 0. Ozsv´ath and Szab´o [24] showed that one can associate a nonzero class in reducedHeegaard Floer homology to such a contact structure. This result was recently extended to C co-oriented taut foliations by Kazez and Roberts [19] and independently by Bowden [2].Thus, our co-oriented taut foliation F on Y ∪ ϕ Y implies that Y ∪ ϕ Y is not an L-space. (cid:3) ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 17 Torus-link satellites T ( np, nq ) ⊂ S and Seifert structures on S . Since S is a lens space, any Seifertfibered realization of S can have at most 2 exceptional fibers:(39) S = M S ( − q ∗ p , p ∗ q ) , p ∗ p − q ∗ q = 1 , where the right-hand constraint on p, q, p ∗ , q ∗ ∈ Z is necessary (and sufficient) to achieve H ( M S ( − q ∗ p , p ∗ q ); Z ) = 0. The one-exceptional-fiber Seifert structures for S are exhausted bythe cases (cid:8) − q ∗ p , p ∗ q (cid:9) = (cid:8) n , (cid:9) , n ∈ Z . The above Seifert structure exhibits S as a union S = ν ( λ − ) ∪ (cid:0) S \ ( D (cid:113) D − ) (cid:1) × S ∪ ν ( λ )(40) = ( D − × S ) ∪ [ − ε, + ε ] × T ∪ ( D × S ) , (41)where λ − and λ are exceptional fibers of meridian-slopes [ µ − ] sf = − q ∗ p and [ µ ] sf = p ∗ q ,respectively, forming a Hopf link λ − , λ ⊂ S .Regular fibers in this Seifert fibration are confined to some neighborhood [ − ε, + ε ] × T of a torus T , and they foliate this T with fibers all of the same slope. Since λ − and λ are of multiplicities p and q , respectively, any regular fiber f wraps p times around thecore λ − of the solid torus neighborhood ν ( λ − ), and wraps q times around the core λ of ν ( λ ), or equivalently, winds q times along the core of ν ( λ − ). That is, any regular fiber f is a ( p, q ) curve in the boundary T = ∂ν ( λ − ) of the solid torus ∂ν ( λ − ) of core λ − .(See Proposition 4.2 for a more careful treatment of framings and orientations.) Thus, anycollection f , . . . , f n of regular fibers in M S ( − q ∗ p , p ∗ q ) allows us to realize the exterior(42) Y n ( p,q ) := M S ( − q ∗ p , p ∗ q ) \ ◦ ν (cid:0)(cid:96) ni =1 f i (cid:1) =: M S ( − q ∗ p , p ∗ q , ˆ ∗ , . . . , n ˆ ∗ ) = S \ ◦ ν ( T ( np, nq ))of T ( np, nq ) ⊂ S . As a link in the solid torus, T ( np, nq ) ⊂ ν ( λ − ) inhabits the exterior(43) ˆ Y ( p,q ) := M S ( − q ∗ p , p ∗ q ) \ ◦ ν ( λ ) =: M S ( − q ∗ p , ˆ ∗ ) = ν ( λ − )of the fiber λ of meridian-slope p ∗ q . This solid-torus link T ( np, nq ) ⊂ ˆ Y ( p,q ) then has exterior(44) ˆ Y n ( p,q ) := ˆ Y ( p,q ) \ ◦ ν (cid:0)(cid:96) ni =1 f i (cid:1) =: M S ( − q ∗ p , ˆ ∗ , ˆ ∗ , . . . , n ˆ ∗ ) = ˆ Y ( p,q ) \ ◦ ν ( T ( np, nq )) . To make this association ( p, q ) (cid:55)→ ˆ Y n ( p,q ) well defined, we adopt the following convention. Definition 4.1. To any ( p, q ) ∈ Z with gcd( p, q ) = 1 , we associate the pair ( p ∗ , q ∗ ) ∈ Z : (45) ( p, q ) (cid:55)→ ( p ∗ , q ∗ ) ∈ Z , p ∗ p − q ∗ q = 1 , q ∗ ∈ { , . . . , p − } , where we demand p > without loss of generality (since p = 0 -satellites are unlinks). Construction of satellites. For a knot K ⊂ M in a closed oriented 3-manifold M ,we define the T ( np, nq )-torus-link satellite K ( np,nq ) ⊂ M to be the image of the torus link T ( np, nq ) embedded in the boundary of ν ( K ), composed with the inclusion ν ( K ) (cid:44) → M .Thus, if we write Y := M \ ◦ ν ( K ) for the exterior of K ⊂ M and take ˆ Y n ( p,q ) as defined in(44), then for an appropriate choice of gluing map ¯ ϕ : ∂Y → − ∂ ˆ Y n ( p,q ) , we expect the union(46) Y ( np,nq ) := Y ∪ ¯ ϕ ˆ Y n ( p,q ) , ¯ ϕ : ∂Y → − ∂ ˆ Y n ( p,q ) to be the exterior of K ( np,nq ) ⊂ M . Proposition 4.2. Suppose p, q, n ∈ Z with n, p > and gcd( p, q ) = 1 . Choose a surgery basis ( µ, λ ) for the boundary homology H ( ∂Y ; Z ) of the knot exterior Y := M \ ◦ ν ( K ) , and take ˆ Y n ( p,q ) as in (44) and Y ( np,nq ) as in (46). If the gluing map ¯ ϕ : ∂Y → − ∂ ˆ Y n ( p,q ) induces thehomomorphism ¯ ϕ ∗ : H ( ∂Y ; Z ) → H ( ∂ ˆ Y n ( p,q ) ; Z ) , (47) ¯ ϕ ∗ ( µ ) := − q ∗ ˜ f + p ˜ h , ¯ ϕ ∗ ( λ ) := p ∗ ˜ f − q ˜ h , on homology, and hence the orientation-preserving linear fractional map ¯ ϕ P ∗ : P ( H ( ∂Y ; Z )) surg → P ( H ( ∂ ˆ Y n ( p,q ) ; Z )) sf , ¯ ϕ P ∗ (cid:16) ab (cid:17) = aq ∗ − bp ∗ ap − bq = q ∗ p − bp ( ap − bq ) , (48) on slopes, then Y ( np,nq ) sf ( ) = M , and Y ( np,nq ) is the exterior of of the T ( np, nq ) satellite K ( np,nq ) ⊂ M of K ⊂ M .Proof. The Dehn filling Y ( np,nq ) sf ( ) is given by the union Y ( np,nq ) sf ( ) = Y ∪ ¯ ϕ ˆ Y ( p,q ) , for the solidtorus ˆ Y ( p,q ) = M S ( − q ∗ p , ∗ ) defined in (43). The boundary of the compressing disk of ˆ Y ( p,q ) isgiven by the rational longitude l = − (cid:80) − i = − β i α i = q ∗ p of ˆ Y ( p,q ) (see the last line of Theorem 4.3).Thus, since [ ¯ ϕ ( µ )] sf = q ∗ p , Y ( np,nq ) sf ( ) is in fact the Dehn filling Y ( np,nq ) sf ( ) = Y ( µ ) = M .Since Y ( np,nq ) = M \ (cid:96) ni =1 ◦ ν ( f i ) is the exterior of n regular fibers from ˆ Y ( p,q ) = M S ( − q ∗ p , ∗ )we must verify that our lift ˜ f ∈ H ( ∂ ˆ Y ( p,q ) ; Z ) of a regular fiber class to the boundary ∂ ˆ Y ( p,q ) = ∂ ˆ Y n ( p,q ) of the solid torus ˆ Y ( p,q ) is represented by a ( p, q ) torus knot on ∂ ˆ Y ( p,q ) relative to the framing specified by µ and λ . Indeed, from (47), we have˜ f = ( pp ∗ − qq ∗ ) ˜ f = p ¯ ϕ ∗ ( λ ) + q ¯ ϕ ∗ ( µ ) , as required. The induced map ¯ ϕ P ∗ on slopes preserves orientation, because the map ¯ ϕ isorientation reversing, but the surgery basis and Seifert fibered basis are positively orientedand negatively oriented, respectively. (cid:3) Computing L-space intervals. The primary tool we shall use is a result of the authorwhich computes the L-space interval for the exterior of a regular fiber in a closed 3-manifoldwith a Seifert fibered JSJ component. Theorem 4.3 (S. Rasmussen (Theorem 1.6) [27]) . Suppose M is a closed oriented 3-manifoldwith some JSJ component ˆ Y which is Seifert fibered over an n bi -times-punctured S , so thatwe may express M as a union M = ˆ Y ∪ ¯ ϕ (cid:96) n bi j =1 Y j , ¯ ϕ j : ∂Y j → − ∂ j ˆ Y, where each Y j is boundary incompressible ( i.e. is not a solid torus or a connected sum thereof ).Write ( y , . . . , y m ) for the Seifert slopes of ˆ Y , so that ˆ Y is the partial Dehn filling of S × ( S \ (cid:96) m + n bi i =1 D i ) by ( y , . . . , y m ) in our Seifert fibered basis. Further suppose that each Y j isFloer simple, so that we may write ¯ ϕ P ∗ ( L ( Y j )) sf = [[ y bi j − , y bi j + ]] ⊂ P ( H ( ∂ j ˆ Y ; Z )) sf for each j ∈ { , . . . , n bi } . Let Y denote the exterior Y = M \ ◦ ν ( f ) of a regular fiber f ⊂ ˆ Y .If L ( Y ) is nonempty, then ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 19 L sf ( Y ) = (cid:40) { y − } = { y + } Y Floer not simple [[ y − , y + ]] Y Floer simple , where y − := sup k> − k (cid:32) m (cid:80) i =0 (cid:98) y i k (cid:99) + n bi (cid:80) j =1 (cid:16) (cid:100) y bi j + k (cid:101) − (cid:17)(cid:33) ,y + := inf k> − k (cid:32) − m (cid:80) i =0 (cid:100) y i k (cid:101) + n bi (cid:80) j =1 (cid:16) (cid:98) y bi j − k (cid:99) + 1 (cid:17)(cid:33) . The above extrema are realized for finite k if and only if Y is boundary incompressible. When Y is boundary compressible, y − = y + = l = − (cid:80) mi =1 y i is the rational longitude of Y . Remarks. In the above, we define y − := ∞ or y + := ∞ , respectively, if any infinite termsappear as summands of y − or y + , respectively. For x ∈ R , the notations (cid:98) x (cid:99) and (cid:100) x (cid:101) respec-tively indicate the greatest integer less than or equal to x and the least integer greater thanor equal to x , as usual. In addition, we always take k to be an integer. Thus the expression“ k > 0” always indicates k ∈ Z > .In order to use the above theorem, we first have to know whether L ( Y ) is nonempty andwhether Y is Floer simple. The author provides a complete (and lengthy) answer to thisquestion in [27]. Here, we restrict to the cases of most relevance to the current question. Theorem 4.4 (S. Rasmussen [27]) . Assuming the hypotheses of Theorem 4.3, set n ∞ := |{ i ∈ { , . . . , n }| y i = ∞}| ,n < bi := |{ j ∈ { , . . . , n bi } | − ∞ < y bi j − < y bi j + < + ∞}| . If ∞ / ∈ { y bi − , y bi , . . . , y bi n bi − , y bi n bi + } , then the following are true.(i) If n ∞ > , then any Dehn filling of Y is a connected sum with S × S , and L sf ( Y ) = ∅ .(ii) If n ∞ = 1 , then L sf ( Y ) = (cid:40) (cid:104)−∞ , + ∞(cid:105) n < bi = 0 ∅ n < bi > . (iii) If n ∞ = 0 , then L sf ( Y ) = [[ y − , y + ]] with y − > y + n < bi = 0[ y − , y + ] n < bi = 1 and y − < y + { y − } = { y + } n < bi = 1 and y − = y + ∅ n < bi = 1 and y − > y + ∅ n < bi > . Suppose instead that ∞ ∈ { y bi − , y bi , . . . , y bi n bi − , y bi n bi + } .(iv) If either ∞ / ∈ { y bi − , . . . , y bi n bi − } or ∞ / ∈ {{ y bi , . . . , y bi n bi + } , then L sf ( Y ) = [[ y − , y + ]] n < bi = 0 and n ∞ = 0 (cid:104)−∞ , + ∞(cid:105) n < bi = 0 and n ∞ = 1 ∅ n < bi = 0 and n ∞ > ∅ n < bi (cid:54) = 0 . Proof. See Proposition 4.7 in [27]. (cid:3) To state the below theorem efficiently, we need to introduce one last notational convention. Notation. When the brackets [ · ] are applied to a real number, they always indicate the map(49) [ · ] : R → [0 , (cid:105) , [ x ] := x − (cid:98) x (cid:99) . Note that the maps (cid:98)·(cid:99) , (cid:100)·(cid:101) , and [ · ] satisfy the useful identities,(50) − (cid:98)− x (cid:99) = (cid:100) x (cid:101) , x = (cid:98) x (cid:99) + [ x ] = (cid:100) x (cid:101) − [ − x ] for all x ∈ R . We are now ready to classify L-space surgeries on torus-link satellites of L-space knots. Theorem 4.5. Let Y := S \ ◦ ν ( K ) denote the exterior of a positive L-space knot K ⊂ S ofgenus g ( K ) , and let Y ( np,nq ) := S \ ◦ ν ( K ( np,nq ) ) denote the exterior of the ( np, nq ) -torus-linksatellite K ( np,nq ) ⊂ S of K ⊂ S , for n, p, q ∈ Z with n, p > , and gcd( p, q ) = 1 .Construct ˆ Y n ( p,n ) , ˆ Y ( p,q ) , and Y ( np,nq ) := Y ∪ ¯ ϕ ˆ Y n ( p,q ) as in Proposition 4.2, with the Seifertstructure sf on ˆ Y n ( p,q ) as specified by Proposition 4.2. ( ψ ) There is a change of basis map ψ : ( Q ∪ {∞} ) n sf −→ ( Q ∪ {∞} ) nS , y (cid:55)→ (cid:16) pq + y , . . . , pq + y n (cid:17) , which converts the above-specified Seifert basis slopes ( Q ∪ {∞} ) n sf into conventional link ex-terior slopes in S , so that L S ( Y ( np,nq ) ) = ψ ( L sf ( Y ( np,nq ) )) . ( i.a ) If N := 2 g ( K ) − > qp , K ⊂ S is nontrivial, and p > , then L sf ( Y ( np,nq ) ) = Λ sf ( Y ( np,nq ) ) := { l ∈ Z n ⊂ ( Q ∪ {∞} ) n | (cid:80) ni =1 l i = 0 } . ( i.b ) If g ( K ) − > qp , K ⊂ S is nontrivial, and p = 1 , then L sf ( Y ( np,nq ) ) = Λ sf ( Y ( np,nq ) ) + S n (cid:16)(cid:104) , N − q (cid:105) × { } n − (cid:17) . ( ii ) If g ( K ) − ≤ qp with K ⊂ S nontrivial, or if p, q > with K ⊂ S trivial, then N L sf ( Y ( np,nq ) ) = Z sf ( Y ( np,nq ) ) ∪ { y ∈ Q n | y + < < y − } , where Z sf ( Y ( np,nq ) ) = { y ∈ ( Q ∪ {∞} ) n | { i ∈ { , . . . , n }| y i = ∞} > } ,y − = − (cid:80) ni =1 (cid:98) y i (cid:99) ,y + = − (cid:80) ni =1 (cid:100) y i (cid:101) − (cid:40) p + q − g ( K ) p (cid:80) ni =1 (cid:98) [ − y i ]( p + q − g ( K ) p ) (cid:99) = 00 otherwise . ( iii ) If K ⊂ S is the unknot, with p = 1 and q > , then L sf ( Y ( n,nq ) ) = R sf ( Y ( n,nq ) ) \ Z sf ( Y ( n,nq ) ) ∪ { y ∈ Q n | M S ( q , y ) a SF L-space } , or equivalently, N L sf ( Y ( n,nq ) ) = Z sf ( Y ( n,nq ) ) ∪ (cid:110) y ∈ Q n (cid:12)(cid:12)(cid:12) − (cid:108) kq (cid:109) − (cid:80) ni =1 (cid:100) y i k (cid:101) < < − − (cid:106) kq (cid:107) − (cid:80) ni =1 (cid:98) y i k (cid:99) ∀ k ∈ Z > (cid:111) . Remarks. A knot K ⊂ S is called a positive (respectively negative ) L-space knot if K admitsan L-space surgery for some finite S -slope m > m < L S ( Y ( np,nq ) ) = −L S ( ¯ Y ( np, − nq ) ) for ¯ Y ( np, − nq ) , the ( np, − nq )-torus-link satellite of the mirror knot ¯ K ⊂ S ,the above theorem and Theorem 1.2 are easily adapted to satellites of negative L-space knotsor to negative torus links. Any p = 0 satellite is just the n -component unlink, with L S = (cid:81) ni =1 [ −∞ , (cid:105) ∪ (cid:104) , + ∞ ]. Note that while Theorem 1.2 excludes the case of torus links proper(satellites of the unknot) which are “degenerate,” i.e. , which have 1 ∈ { p, q } , this case istreated in ( iii ) above, setting p = 1 without loss of generality. If q = 0 in this case, weagain have the n -component unlink. For any nontrivial degenerate torus link, part ( iii )above implies that the boundary of L sf follows a piece-wise-constant chaotic pattern, similarto the boundary of the region of Seifert fibered L-spaces. This is unsurprising, since the ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 21 irreducible surgeries on T ( n, n ) consist of all oriented Seifert fibered spaces over S of n orfewer exceptional fibers. Lastly, if K ( np,nq ) ⊂ S is any nontrivial-torus-link satellite of a non -L-space knot in S , then the L-space gluing result conjectured in [26] for arbitrary closedoriented 3-manifolds with single-torus boundary—which the authors of [13] have announcedthey expect to prove in the near future—would imply that L ( Y ( np,nq ) ) = Λ( Y ( np,nq ) ). Proof of ( ψ ). Let Y j := Y ( np,nq ) sf (0 , . . . , , j ˆ ∗ , , . . . , 0) denote the partial Dehn filling of Y ( np,nq ) which fills in all n boundary components except ∂Y j = ∂ j Y ( np,nq ) with regular fiberneighborhoods, so that, by the definitions of Y ( np,nq ) and ˆ Y n ( p,q ) in (46) and (44), we have(51) Y j = Y ∪ ¯ ϕ ˆ Y n ( p,q ) = Y ∪ ¯ ϕ M S ( − q ∗ p , ˆ ∗ , , . . . , , j ˆ ∗ , . . . , , ¯ ϕ : ∂Y → − ∂ ˆ Y n ( p,q ) , with ¯ ϕ as defined in Proposition 4.2, where we recall from Definition 4.1 that pp ∗ − qq ∗ = 1with 0 ≤ q ∗ < p . Observing that Y j has the Dehn filling Y j ( − ˜ h j ) = S , we take µ j := − ˜ h j ∈ H ( ∂Y j ; Z ) for the meridian in our S surgery basis for H ( ∂Y j ; Z ).As shown, for example, in [27], any homology class λ j ∈ H ( ∂Y j ; Z ) representing the rationallongitude of Y j has sf -slope π sf ( λ j ) given by the negative sum of the sf -slope images ofthe rational longitudes of the manifolds glued into the boundary components of ˆ Y n ( p,q ) , plusthe negative sum of Seifert-data slopes (which are just the sf -slope images of the rationallongitudes of the corresponding fiber neighborhoods), as follows:(52) π sf ( λ j ) = − (cid:16) − q ∗ p + ¯ ϕ P ∗ ( π S ( λ )) + (cid:80) i (cid:54) = j (cid:17) = − (cid:16) − q ∗ p + p ∗ q (cid:17) = − pq . This uses the definition in (48) of the induced map ¯ ϕ P ∗ : P ( H ( ∂Y ; Z )) S → P ( H ( ∂ ˆ Y n ( p,q ) Z )) sf on slopes, to calculate the slope ¯ ϕ P ∗ ( π S ( λ )) = ¯ ϕ P ∗ ( π S (0)) = p ∗ q ∈ P ( H ( ∂ ˆ Y n ( p,q ) Z )) sf .To obtain µ j · λ j = 1, we are constrained by the choice µ j = − h j to select the representative λ j := ˜ f j + pq ˜ h j for the sf -slope π sf ( λ j ) = − pq . The resulting homology change of basis(53) µ j (cid:55)→ f j − h j , λ j (cid:55)→ f j + pq ˜ h j for H ( ∂ j Y ( np,nq ) ; Z ) then induces a map on slopes with inverse(54) ψ j : P ( H ( ∂ j Y ( np,nq ) ; Z )) sf → P ( H ( ∂ j Y ( np,nq ) ; Z )) S , y j (cid:55)→ pq + y j . Setup for ( i ) and ( ii ). We begin with the case in which K ⊂ S is nontrivial, so that itsexterior Y = S \ ◦ ν ( K ) is boundary incompressible. It is easy to show (see “example” in [26,Section 4]) that such Y has L-space interval(55) L S ( Y ) = [ N, + ∞ ] , N := deg(∆( K )) − g ( K ) − , where ∆( K ) and g ( K ) are the Alexander polynomial and genus of K . Writing(56) ¯ ϕ P ∗ ( L ( Y )) sf = [[ y − , y ]] , we then use (48) to compute that(57) y − := Nq ∗ − p ∗ Np − q = q ∗ p + p ( q − Np ) , y := q ∗ p . For a given sf -slope y := ( y , . . . , y n ) ∈ ( Q ∪ {∞} ) n sf , we verify whether the Dehn filling Y ( np,nq ) sf ( y ) is an L-space by examining the L-space interval, computed via Theorem 4.3, of aregular fiber exterior in Y ( np,nq ) sf ( y ). That is, if we let ˆ Y ( np,nq ) denote the regular fiber exterior(58) ˆ Y ( np,nq ) := Y ( np,nq ) \ ◦ ν ( f ) for a regular fiber f ⊂ ˆ Y n ( p,q ) , then Y ( np,nq ) ( y ) is an L-space if and only if the meridionalslope 0 ∈ P ( H ( ˆ Y ( np,nq ) sf ( y ); Z )) sf satisfies 0 ∈ L sf ( ˆ Y ( np,nq ) sf ( y )). Since Y is Floer simple andboundary incompressible, Theorem 4.3 tells us that if L ( ˆ Y ( np,nq ) sf ( y )) is nonempty, then it isdetermined by y − , y + ∈ P ( H ( ˆ Y ( np,nq ) sf ( y ); Z )) sf , where y − := max k> y − ( k ) , y + := min k> y + ( k ) , (59) y − ( k ) := − k (cid:16) (cid:106) − q ∗ p k (cid:107) + (cid:80) ni =1 (cid:98) y i k (cid:99) + ( (cid:100) y k (cid:101) − (cid:17) (60) = k (cid:16)(cid:108) q ∗ p k (cid:109) − (cid:100) y k (cid:101) − (cid:80) ni =1 (cid:98) y i k (cid:99) (cid:17) ,y + ( k ) := − k (cid:16) − (cid:108) − q ∗ p k (cid:109) + (cid:80) ni =1 (cid:100) y i k (cid:101) + ( (cid:98) y − k (cid:99) + 1) (cid:17) (61) = k (cid:16)(cid:106) q ∗ p k (cid:107) − (cid:98) y − k (cid:99) − (cid:80) ni =1 (cid:100) y i k (cid:101) (cid:17) . Thus, since y = q ∗ p , we have y − ( k ) = − k (cid:80) ni =1 (cid:98) ( (cid:98) y i (cid:99) + [ y i ]) k (cid:99) (62) = − k (cid:80) ni =1 (cid:98) [ y i ] k (cid:99) − (cid:80) ni =1 (cid:98) y i (cid:99)≤ − (cid:80) ni =1 (cid:98) y i (cid:99) for all k > 0, which, since y − (1) = − (cid:80) ni =1 (cid:98) y i (cid:99) , implies(63) y − = − (cid:80) ni =1 (cid:98) y i (cid:99) . For y + , there are multiple cases. Proof of ( i ) : N = 2 g ( K ) − > qp . Since q − N p < 0, we have y − < q ∗ p = y , which, byTheorem 4.4, implies L ( ˆ Y ( np,nq ) ( y )) (cid:54) = ∅ if and only if y ∈ Q n and y − ≤ y + . Case (a): p > 1. Since 0 < y − < q ∗ p < 1, we have(64) y + ≤ y + (1) = − (cid:80) ni =1 (cid:100) y i (cid:101) , so that y − ≤ y + if and only if (cid:80) ni =1 ( (cid:100) y i (cid:101) − (cid:98) y i (cid:99) ) ≤ 0, which, for y ∈ Q n , occurs if and only if y ∈ Z n . If y ∈ Z n , then y + ( k ) ≤ y + (1) for all k > 0, implying(65) y + = y + (1) = − (cid:80) ni =1 y i = y − , so that Theorem 4.4 tells us(66) L sf ( ˆ Y ( np,nq ) ( y )) = { y − } = { y + } = {− (cid:80) ni =1 y i } . Thus, since Y ( np,nq ) ( y ) is an L-space if and only if 0 ∈ L sf ( ˆ Y ( np,nq ) ( y )), we have(67) L sf ( Y ( np,nq ) ) = Λ sf ( Y ( np,nq ) ) := { y ∈ Z n ⊂ ( Q ∪ {∞} ) n | (cid:80) ni =1 y i = 0 } . Case (b): p = 1. In this case, q ∗ p = 0 and y − = − N − q , so that(68) y + ( k ) = k (cid:16)(cid:108) kN − q (cid:109) + (cid:80) ni =1 (cid:98) [ − y i ] k (cid:99) (cid:17) − (cid:80) ni =1 (cid:100) y i (cid:101) . In particular, since y + (1) = 1 − (cid:80) ni =1 (cid:100) y i (cid:101) , the condition y − ≤ y + ≤ y + (1) implies(69) (cid:80) ni =1 ( (cid:100) y i (cid:101) − (cid:98) y i (cid:99) ) ≤ y + + (cid:80) ni =1 (cid:100) y i (cid:101) ≤ , ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 23 which, for y ∈ Q n , occurs only if for some j ∈ { , . . . , n } we have y i ∈ Z for all i (cid:54) = j . Forsuch a y , we then have(70) y + ( k ) = k (cid:16)(cid:108) kN − q (cid:109) + (cid:98) [ − y j ] k (cid:99) (cid:17) − (cid:80) ni =1 (cid:100) y i (cid:101) . For k > 0, set s := (cid:108) kN − q (cid:109) and write k = s ( N − q ) − t with 0 ≤ t < N − q .If [ − y i ] ≥ N − q − N − q , then(71) y + ( k ) + (cid:80) ni =1 (cid:100) y i (cid:101) ≥ s ( N − q ) − (cid:100) [ − y j ] t (cid:101) s ( N − q ) − t ≥ k > 0, which, since y + (1) + (cid:80) ni =1 (cid:100) y i (cid:101) = 1, implies(72) y + = 1 − (cid:80) ni =1 (cid:100) y i (cid:101) = − (cid:80) ni =1 (cid:98) y i (cid:99) = y − , so that Y ( np,nq ) ( y ) is an L-space if and only if (cid:80) ni =1 (cid:98) y i (cid:99) = 0.If [ − y i ] < N − q − N − q , then(73) y + + (cid:80) ni =1 (cid:100) y i (cid:101) ≤ y + ( N − q ) + (cid:80) ni =1 (cid:100) y i (cid:101) ≤ N − q − N − q < . The left half of (69) then tells us that (cid:80) ni =1 ( (cid:100) y i (cid:101) − (cid:98) y i (cid:99) ) < 1, implying y ∈ Z n and (cid:80) ni =1 (cid:100) y i (cid:101) = (cid:80) ni =1 (cid:98) y i (cid:99) . Thus, since Z (cid:51) y − ≤ y + < y − +1, we have y − ≤ ≤ y + if and only if (cid:80) ni =1 (cid:98) y i (cid:99) = 0.In total, we have learned that y ∈ L sf ( Y ( np,nq ) ) if and only if (cid:80) ni =1 (cid:98) y i (cid:99) = 0 and there exists j ∈ { , . . . , n } such that y i ∈ Z for all i (cid:54) = j and [ − y j ] ∈ (cid:104) N − q − N − q , (cid:69) ∪ { } , or equivalently,[ y j ] ∈ (cid:104) , N − q (cid:105) . In other words, L sf ( Y ( np,nq ) ) = S n · (cid:110) (cid:104) l + 0 , l + N − q (cid:105) × ( l , . . . , l n ) (cid:12)(cid:12)(cid:12) l ∈ Z n , (cid:80) ni =1 l i = 0 (cid:111) (74) = Λ sf ( Y ( np,nq ) ) + S n (cid:16)(cid:104) , N − q (cid:105) × { } n − (cid:17) . (cid:3) Proof of ( ii ) : N = 2 g ( K ) − ≤ qp and q > 0. We divide this section into three cases: N = qp , N < qp with K ⊂ S nontrivial, and K ⊂ S the unknot p, q > Case N = qp . Here, N > K ⊂ S is nontrivial, and N p = q implies p = 1, sothat y = q ∗ p = 0 and y − = − N − q = ∞ . Theorem 4.4.iv then implies that Z ∞ sf ( Y ( np,nq ) ) ⊂N L sf ( Y ( np,nq ) ), but that(75) (( Q ∪ {∞} ) n \ Q n ) \ Z sf ( Y ( np,nq ) ) ⊂ L sf ( Y ( np,nq ) ) , since these are the slopes with n ∞ = 1, and since 0 ∈ (cid:104)−∞ , + ∞(cid:105) . For y ∈ Q n , Theorem 4.4.ivtells us L sf ( ˆ Y ( np,nq ) ( y )) = [[ y − , y + ]] = [ − (cid:80) ni =1 (cid:98) y i (cid:99) , + ∞ ], so that(76) N L sf ( Y ( np,nq ) ) = Z sf ( Y ( np,nq ) ) ∪ { y ∈ Q n | − ∞ < < y − } . Since p + q − g ( K ) p = q − N p = 0, the definition of y + in part ( ii ) of Theorem 4.5 makes y + = −∞ , and so Theorem 4.5.ii holds. Case N < qp . Here, q − N p > ≤ y = q ∗ p < y − ≤ 1. Thus, since0 ∈ (cid:104)−∞ , + ∞(cid:105) , the n ∞ = 1 case of Theorem 4.4.ii implies (75) holds, whereas Theo-rem 4.4.i implies Z sf ( Y ( np,nq ) ) ⊂ N L sf ( Y ( np,nq ) ). For y ∈ Q n , Theorem 4.4.iii tells us that L sf ( ˆ Y ( np,nq ) ( y )) = [[ y − , y + ]] with y − ≥ y + , so that Y ( np,nq ) ( y ) is not an L-space if and onlyif y + < < y − . We therefore have(77) N L sf ( Y ( np,nq ) ) = Z sf ( Y ( np,nq ) ) ∪ { y ∈ Q n | y + < < y − } , where, the definitions of y − and y + are appropriately adjusted in the case that K ⊂ S is theunknot. Case N < qp with K ⊂ S nontrivial . We already know that y − = − (cid:80) ni =1 (cid:98) y i (cid:99) when K ⊂ S is nontrivial and y ∈ Q n . Thus, it remains to compute y + for y ∈ Q n .Since x = (cid:100) x (cid:101) − [ − x ] for all x ∈ R , we have y + ( k ) = k (cid:16)(cid:106) q ∗ p k (cid:107) − (cid:98) y − k (cid:99) + (cid:80) ni =1 (cid:98) [ − y i ] k (cid:99) (cid:17) − (cid:80) ni =1 (cid:100) y i (cid:101) (78)for all k > 0. Write k = s ( q − N p ) + t for s, t ∈ Z ≥ with s := (cid:106) kq − Np (cid:107) and t < q − N p .Using the facts that q ∗ ( q − N p ) = q ∗ q − N pq ∗ = p ∗ p − − N pq ∗ = p ( p ∗ − N q ∗ ) − (cid:98) w (cid:99) − (cid:98) x (cid:99) ≥ (cid:98) w − x (cid:99) and (cid:98)− x (cid:99) = −(cid:100) x (cid:101) for all w, x ∈ R (in (80)), we obtain¯ y + ( k ) := k (cid:16)(cid:106) q ∗ p k (cid:107) − (cid:98) y − k (cid:99) (cid:17) = k (cid:16)(cid:106) s ( p ∗ − N q ∗ ) + − s + tq ∗ p (cid:107) − (cid:106) s ( p ∗ − N q ∗ ) + tq ∗ + t/ ( q − Np ) p (cid:107)(cid:17) = k (cid:16)(cid:106) − sp + tq ∗ p (cid:107) − (cid:106) tq ∗ p (cid:107)(cid:17) (79) ≥ − s ( q − Np ) + t (cid:108) sp (cid:109) (80) = − q − Np + ( s − (cid:100) s/p (cid:101) )( q − Np ) + tk ( q − Np ) ≥ − q − Np . (81)If (cid:80) ni =1 (cid:98) [ − y i ]( q − N p ) (cid:99) = 0, then, writing k = 1( q − N p ) + 0, we can use line (79) tocompute ¯ y + ( q − N p ), so that we obtain(82) y + ( q − N p ) = ¯ y + ( q − N p ) + 0 − (cid:80) ni =1 (cid:100) y i (cid:101) = − q − Np − (cid:80) ni =1 (cid:100) y i (cid:101) . Thus, since (81) implies y + ( k ) ≥ − q − Np − (cid:80) ni =1 (cid:100) y i (cid:101) for all k > 0, we conclude that(83) y + = − q − Np − (cid:80) ni =1 (cid:100) y i (cid:101) . On the other hand, if (cid:80) ni =1 (cid:98) [ − y i ]( q − N p ) (cid:99) > 0, then we know there exists i ∗ ∈ { , . . . , n } for which y i ∗ ≥ ( q − N p ) − . Thus, writing k = s ( q − N p ) + t and using line (80), we obtainthe lower bound y + ( k ) ≥ ¯ y + ( k ) + k (cid:106) q − Np k (cid:107) − (cid:80) ni =1 (cid:100) y i (cid:101)≥ − k (cid:108) sp (cid:109) + k s − (cid:80) ni =1 (cid:100) y i (cid:101)≥ − (cid:80) ni =1 (cid:100) y i (cid:101) . (84)Since this bound is realized by y + (1) = − (cid:80) ni =1 (cid:100) y i (cid:101) , we deduce that y + = − (cid:80) ni =1 (cid:100) y i (cid:101) . Thus,since q − N p = p + q − g ( K ) p , the last line of Theorem 4.5.ii holds. Case N < qp with p, q > and K ⊂ S the unknot . Since Y := S \ ◦ ν ( K ) satisfies(85) L S ( Y ) = [[0 , Q ∪ {∞} \ { } , we use (48) to compute, for ¯ ϕ P ∗ ( L ( Y )) sf = [[ y − , y ]], that(86) y − = y = q ∗ − p ∗ p − q = p ∗ q = q ∗ p + pq . ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 25 Thus, applying Theorem 4.3 and mildly simplifying, we obtain that y − = sup k> y − ( k ) and y + = inf k> y + ( k ) , for(87) y − ( k ) := k (cid:16) − (cid:108) q ∗ p k (cid:109) − (cid:106) p ∗ q k (cid:107) − (cid:80) ni =1 (cid:98) [ y i ] k (cid:99) (cid:17) − (cid:80) ni =1 (cid:98) y i (cid:99) , (88) y + ( k ) := k (cid:16) (cid:106) q ∗ p k (cid:107) − (cid:108) p ∗ q k (cid:109) + (cid:80) ni =1 (cid:98) [ − y i ] k (cid:99) (cid:17) − (cid:80) ni =1 (cid:100) y i (cid:101) . (89)For y − ( k ), we (again) obtain the bound y − ( k ) = k (cid:16)(cid:108) q ∗ p k (cid:109) − (cid:16)(cid:106)(cid:16) q ∗ p + pq (cid:17) k (cid:107) + 1 (cid:17) − (cid:80) ni =1 (cid:98) [ y i ] k (cid:99) (cid:17) − (cid:80) ni =1 (cid:98) y i (cid:99)≤ − (cid:80) ni =1 (cid:98) y i (cid:99) for all k ∈ Z > , (90)which, for p, q > y − (1) = − (cid:80) ni =1 (cid:98) y i (cid:99) , so that y − = − (cid:80) ni =1 (cid:98) y i (cid:99) .To compute y + , we note that since p, q > 1, we can invoke Lemma 4.7 (below), so that(91) 1 + (cid:106) q ∗ p k (cid:107) + (cid:106) kp + q (cid:107) ≥ (cid:108) p ∗ q k (cid:109) for all k ∈ Z > . (Here, we multiplied the original inequality by k and then observed that the integer on theleft hand side must be bounded by an integer.) In particular,(92) k (cid:16) (cid:106) q ∗ p k (cid:107) − (cid:108) p ∗ q k (cid:109) + (cid:106) kp + q (cid:107)(cid:17) − (cid:80) ni =1 (cid:100) y i (cid:101) ≥ − (cid:80) ni =1 (cid:100) y i (cid:101) for all k ∈ Z > . Thus, when (cid:80) ni =1 (cid:98) [ − y i ]( p + q ) (cid:99) > 0, so that at least one y i satisfies [ − y i ] ≥ p + q , line (92)tells us that y + ( k ) ≥ − (cid:80) ni =1 (cid:100) y i (cid:101) , a bound which is realized by y + (1) when p, q > 1. On theother hand, if (cid:80) ni =1 (cid:98) [ − y i ]( p + q ) (cid:99) = 0, then (92) implies that(93) y + ( k ) ≥ − k (cid:106) kp + q (cid:107) − (cid:80) ni =1 (cid:100) y i (cid:101) ≥ − p + q − (cid:80) ni =1 (cid:100) y i (cid:101) for all k ∈ Z k> , a bound which is realized by y + ( p + q ). We therefore have(94) y + = − (cid:80) ni =1 (cid:100) y i (cid:101) − (cid:40) p + q (cid:80) ni =1 (cid:98) [ − y i ]( p + q ) (cid:99) = 00 (cid:80) ni =1 (cid:98) [ − y i ]( p + q ) (cid:99) > , completing the proof of part ( ii ). (cid:3) Proof of ( iii ) : K ⊂ S , p = 1, q > 0. Here, we have the same case as above, but with p = 1 and q > 0, implying q ∗ p = 0 and p ∗ q = q . Thus, the Dehn filling Y ( n,nq ) sf ( y ) is the Seifertfibered space M S ( q , y ), and we have N L sf ( Y ( n,nq ) ) = Z sf ( Y ( n,nq ) ) ∪ { y ∈ Q n | y + < < y − } (95) = Z sf ( Y ( n,nq ) ) ∪ (cid:110) y ∈ Q n (cid:12)(cid:12)(cid:12) − (cid:108) kq (cid:109) − (cid:80) ni =1 (cid:100) y i k (cid:101) < < − − (cid:106) kq (cid:107) − (cid:80) ni =1 (cid:98) y i k (cid:99) ∀ k ∈ Z > (cid:111) . (cid:3) The above theorem leads to the following Corollary 4.6. Theorem 1.2 holds.Proof. For the p > i ), we simply replace Λ sf with Λ S , which contains the S -slope ( ∞ , . . . , ∞ ) in its orbit. For part ( ii ) and for the p = 1 case of part ( i ), it isstraightforward to show that both the expression in the bottom line of part ( ii ) of Theorem 1.2and the expression S n · (cid:0) [ N, + ∞ ] × {∞} n − (cid:1) in part ( i ) contain fundamental domains (underthe action of Λ) of the respective L-space regions specified above. (cid:3) We now return to the lemma cited in the proof of Theorem 4.5. ii . Lemma 4.7. If p, q > , then k (cid:16) (cid:106) q ∗ p k (cid:107) + (cid:106) kp + q (cid:107)(cid:17) ≥ p ∗ q for all k ∈ Z > . Proof. Using the notation(96) [ x ] m := x − | m | (cid:106) x | m | (cid:107) for any x ∈ R , m ∈ Z (cid:54) =0 , we define z ( k ) ∈ Q for all k ∈ Z > , as follows: z ( k ) := k (cid:16) k (cid:16) (cid:106) q ∗ p k (cid:107) + (cid:106) kp + q (cid:107)(cid:17) − p ∗ q (cid:17) = 1 + kqq ∗ − kpp ∗ pq − [ kq ∗ ] p p + (cid:106) kp + q (cid:107) = [ kq − ] p p − kpq + (cid:106) kp + q (cid:107) + (cid:40) k ] p = 00 [ k ] p (cid:54) = 0(97) = q [ kq − ] p − [ k ] p + q pq + (cid:16) pq − ( p + q ) pq (cid:17)(cid:106) kp + q (cid:107) + (cid:40) k ] p = 00 [ k ] p (cid:54) = 0 , (98)where we note that(99) pq − ( p + q ) = ( p − q − − ≥ p, q > . We next claim that z ( k ) ≥ z ( k − ( p + q )) ≥ 0. First, for [ kq − ] p / ∈ { , } , we have z ( k ) − z ( k − ( p + q )) = q · pq + pq − ( p + q ) pq > , (100)so that z ( k ) > z ( k − ( p + q )) ≥ 0. If [ kq − ] p = 0, implying [ k ] p = 0, then line (97) gives(101) z ( k ) = − kpq + (cid:106) kp + q (cid:107) + 1 ≥ kp + q − kpq ≥ . This leaves us with the case in which [ kq − ] p = 1, so that line (97) yields(102) z ( k ) = (cid:106) kp + q (cid:107) − k − qpq . Since [ kq − ] p = 1 implies k ≡ q (mod p ), we can write k = ( sq + t ) p + q , with s = (cid:106) k − qpq (cid:107) and t ∈ { . . . , q − } . When t = 0, we obtain(103) z ( k ) = z ( spq + q ) = (cid:106) spq + qp + q (cid:107) − s = (cid:106) spq + qp + q − s (cid:107) ≥ p, q > . On the other hand, when t ≥ 1, we have(104) z ( k ) ≥ (cid:98) z ( k ) (cid:99) = (cid:106) spq + tp + qp + q (cid:107) − ( s + 1) = (cid:106) s ( pq − ( p + q )) + ( t − pp + q (cid:107) ≥ , completing the proof of our claim. Since the case k = p + q is subsumed in the case [ kq − ] p = 1,we also have z ( p + q ) ≥ 0, and so by the induction, it suffices to prove the lemma for k < p + q .Suppose that 0 < k < p + q and k / ∈ p Z (since z ( k ) ≥ k ] p = 0), so that we now have(105) z ( k ) = pq (cid:0) q [ kq − ] p − k (cid:1) . Since z ( aq ) = pq ( q · a − aq ) = 0 for a ∈ Z , we may also assume k / ∈ q Z . Now, the ChineseRemainder Theorem tells us that(106) k = (cid:2) q [ kq − ] p + p [ kp − ] q (cid:3) pq , ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 27 but since 0 < k < p + q and k / ∈ p Z ∪ q Z , we also have(107) k < p + q ≤ q [ kq − ] p + p [ kp − ] q < pq, (108) requiring that q [ kq − ] p + p [ kp − ] q = k + pq, (109) so that z ( k ) = pq (cid:0) pq − p [ kp − ] q (cid:1) > . (cid:3) L-space region topology Topologizing L-space regions. To clarify the sense in which we interpret topologicalproperties of L-space and non-L-space regions, we introduce the following notion. Definition 5.1. For any subset A ⊂ ( Q ∪ {∞} ) n (cid:44) → ( R ∪ {∞} ) n with complement A c :=( Q ∪ {∞} ) n \ A and real closure A ⊂ ( R ∪ {∞} ) n , we define the Q -corrected R -closure A R := A \ A c ⊂ ( R ∪ {∞} ) n of A . Note that this implies A R ∩ ( Q ∪ {∞} ) n = A The Q -corrected R -closure is a particularly natural construction for L-space regions, dueto the following fact. Proposition 5.2. If L ⊂ ( Q ∪ {∞} ) n and NL ⊂ ( Q ∪ {∞} ) n are the respective L-space andnon-L-space regions for some compact oriented 3-manifold Y with ∂Y = (cid:96) ni =1 T i , then (110) L R (cid:113) NL R = ( R ∪ {∞} ) n . Proof. This is mostly due to the structure of L-space intervals (L-space regions for n = 1)described in Section 3. In particular, when n = 1, Proposition 3.2 implies that the pair( L R , NL R ) takes precisely one of the following forms:( i ) L R = ∅ , NL R = ( R ∪ {∞} ) ;( ii ) L R = { y } , NL R = ( R ∪ {∞} ) \ { y } , for some y ∈ ( Q ∪ {∞} ) ;( iii ) L R = ( R ∪ {∞} ) \ { l } , NL R = { l } , for l ∈ ( Q ∪ {∞} ) the rational longitude;( iv ) L R = I ( y − ,y + ) , NL R = I ◦ ( y + ,y − ) , for some y − , y + ∈ ( Q ∪ {∞} ) with y − (cid:54) = y + ,where I ( y − ,y + ) ⊂ ( R ∪ {∞} ) denotes the real closed interval with left-hand endpoint y − andright-hand endpoint y + , and I ◦ ( y + ,y − ) ⊂ ( R ∪ {∞} ) is the interior in ( R ∪ {∞} ) of I ( y + ,y − ) .In particular, we always have L R (cid:113) NL R = ( R ∪ {∞} ) , and each of L R and NL R is either asingle rational point, a single interval with rational endpoints, empty, or the whole set.Since intervals form a basis for the topology on ( R ∪ {∞} ) , and since the product topologyon ( R ∪ {∞} ) k − × ( R ∪ {∞} ) coincides with the usual topology on ( R ∪ {∞} ) k for any k ∈ Z ≥ , the proposition follows from induction on n . (cid:3) L-space region topology for torus links. There are five qualitatively different topolo-gies possible for the L-space region of a torus-link-satellite of a knot in S . Theorem 5.3. For n, p, q > Z > , let K ( np,nq ) ⊂ S , with exterior Y ( np,nq ) := S \ ◦ ν ( K ( np,nq ) ) ,be the T ( np, nq ) -satellite of a positive L-space knot K ⊂ S . Associate L , NL , Λ , and B to Y ( np,nq ) as usual, with B the set of rational longitudes of Y ( np,nq ) as discussed in Section 2.5. ( i.a ) If g ( K ) − > qp and p > , or if g ( K ) − > qp + 1 and p = 1 ,then L R deformation retracts onto Λ . ( i.b ) If g ( K ) − qp + 1 and n > ,then L R is connected, π ( L R sf ) (cid:39) ker( δ ) as in (117) , and rank H ( L R ) = (cid:0) n (cid:1) − . ( i.c ) If g ( K ) − qp + 1 and n ∈ { , } ,then dim( L R ) = 1 and L R is contractible.( ii.a ) If g ( K ) − qp ,then dim( L R ) = n and L R is contractible. ( ii.b ) If g ( K ) − < qp , including the case of K ( np,nq ) = T ( np, nq ) ,then NL R deformation retract onto the ( n − -torus B R = T n − ⊂ ( R ∪ {∞} ) n = T n ,and L R deformation retracts onto a T n − parallel to B R .Proof of ( i.a ) . Since we already have L = Λ for p > 1, assume that p = 1. Theorem 4.5then tells us that L R sf = Λ sf + P ( N, q ), where(111) P ( N, q ) := (cid:96) ni =1 (cid:16) { } i − × (cid:104) , N − q (cid:105) × { } n − i (cid:17) R ⊂ ( R ∪ {∞} ) n . Clearly P ( N, q ) deformation retracts onto ∈ ( Q ∪ {∞} ) n . Since 2 g ( K ) − > qp + 1, we have P ( N, q ) ⊂ (cid:96) ni =1 (cid:2) , (cid:3) R i . Thus all of the translates { l + P ( N, q ) } l ∈ Λ sf are pairwise disjoint,and L R sf deformatioin retracts onto Λ sf . (cid:3) Proof of ( i.b ) . When 2 g ( K ) − qp + 1, and n > 1, line (111) still holds, but this time with (cid:104) , N − q (cid:105) = [0 , π ( L R sf ) = 0, first note that Λ sf is generated by the elements(112) ε ij := ε i − ε j , i, j ∈ { , . . . , n } , with ε i := (0 , . . . , , i , . . . , ∈ Z n the standard basis element for Z n . Then for any such ε ij and any l ∈ Λ sf , the origin l of thetranslate P l := l + P ( N, q ) is path-connected to the origin ε ij + l of the translate P ε ij + l , viathe path γ l ij ( t ) : [0 , → L R sf ,(113) γ l ij (cid:12)(cid:12)(cid:12) [ , ]( t ) = l + 2 t ε i , γ l ij (cid:12)(cid:12)(cid:12) [ , ]( t ) = ( ε ij + l ) + 2(1 − t ) ε j , (114) l (cid:32) { ( l ) + ε i } = P l ∩ P ε ij + l = { ( ε ij + l ) + ε j } (cid:32) ε ij + l . Thus L R sf is path connected (hence connected), and in fact, these basic paths γ l ij from l ∈ P l to ε ij + l ∈ P ε ij + l generate the groupoid G of homotopy classes of paths in L R sf betweenelements of Λ sf . Let G ⊂ G denote the subset of homotopy clases of paths starting at , sothat elements of G are uniquely represented by reduced words(115) g = (cid:16) γ l ( i ,j ,e ) i j (cid:17) e (cid:16) γ l i j (cid:17) e . . . (cid:16) γ l m i m j m (cid:17) e m ∈ G, i k < j k , e k ∈ {± } , l k +1 = l k + ε i k j k ∀ k, with right-multiplication corresponding to concatenation of paths. Note that we have replaced(116) γ l ji (cid:55)→ (cid:16) γ l − ε ij (cid:17) − ij ∀ i < j, and defined l ( i, j, e ) := (cid:40) e = +1 − ε ij e = − , recalling that ε ji = − ε ij .If we introduce the free group F ( n ) and epimorphism δ : F ( n ) → Λ sf ,(117) F ( n ) := (cid:104) x ij (cid:105) ≤ i 1. Moreover,2 g ( K ) − < qp implies q > K is the unknot, in which case we can take the mirror of K if q < 0. Thus we also assume q > N L sf = Z sf ∪ N , N := { y ∈ Q n sf | y + ( y ) < < y − ( y ) } , with y − ( y ) := max k> y − ( y , k ) , y − ( y , k ) := − k (cid:80) ni =1 (cid:98) y i k (cid:99) − c − ( k )(124) y + ( y ) := min k> y + ( y , k ) , y + ( y , k ) := − k (cid:80) ni =1 (cid:100) y i k (cid:101) − c + ( k )(125)for all y ∈ Q n sf and for certain c − ( k ) , c + ( k ) ∈ k Z bounded above and below by linear functionsin k , and determined by p , q , and 2 g ( K ) − 1, and on whether K ⊂ S is trivial. In particular,each of c − ( k ) and c + ( k ) are independent of y ∈ Q n sf .Since Z sf = B sf \ ( B sf ∩ Q n sf ), implying Z R sf = B R sf \ ( B R sf ∩ R n sf ), it remains to construct adeformation retraction from N R to B R sf ∩ R n sf ⊂ N R . Toward that end, we define(126) := (1 , . . . , ∈ Z n sf , l ( y ) := − pq − (cid:80) ni =1 y i ∈ R n sf , so that for y ∈ Q n sf , l ( y ) is the rational longitude of the exterior ˆ Y ( np,nq ) ( y ) := Y ( np,nq ) ( y ) \ ◦ ν ( f )of a regular fiber f in Y ( np,nq ) ( y ). We then claim that the homotopy(127) z : [0 , × N R → R n sf , z t ( y ) := y + t · n l ( y ) provides a deformation retraction from N R to B R sf ∩ R n sf ⊂ N R .First, note that (123) also implies that N L sf ( ˆ Y ( np,nq ) ( z )) = (cid:104) y + ( z ) , y + ( z ) (cid:105) for all z ∈ Q n sf .Thus, for all z ∈ Q n sf , we have l ( z ) ∈ N L sf ( ˆ Y ( np,nq ) ( z )), so that l ( z ) ∈ (cid:104) y + ( z ) , y − ( z ) (cid:105) . Thus,0 < (1 − t ) l ( y ) = l ( z t ( y )) < y − ( z t ( y )) for all t ∈ [0 , (cid:105) ∩ Q if 0 < l ( y );(128) 0 > (1 − t ) l ( y ) = l ( z t ( y )) > y + ( z t ( y )) for all t ∈ [0 , (cid:105) ∩ Q if 0 > l ( y )(129)for all y ∈ Q n sf , where the equivalence (1 − t ) l ( y ) = l ( z t ( y )) follows quickly from the definitionsof l and z . Now, either by Proposition 5.2 and the structure of the structure of L-spaceintervals, or by Calegari and Walker’s studies of “ziggurats” [5], we know that as functions on R n sf , y − and y + are piecewise constant, with rational endpoints, in each coordinate direction.Thus, since l and z are linear and the above inequalities are strict, we have0 < y − ( z t ( y )) if 0 < l ( y ) , > y + ( z t ( y )) if 0 > l ( y )(130)for all y ∈ R n sf and t ∈ [0 , (cid:105) R .On the other hand, our definitions of y ± and z imply that for any y ∈ N R , we have y + ( z t ( y )) ≤ y + ( y ) < t ∈ [0 , R if 0 < l ( y );(131) y − ( z t ( y )) ≥ y − ( y ) > t ∈ [0 , R if 0 > l ( y ) . (132)Combining these three lines of inequalities tells us that for any y ∈ N R , we have z t ( y ) ∈ N R for all t ∈ [0 , R . Thus z provides a deformation retraction from N R to B R sf ∩ R n sf ⊂ N R . (cid:3) Topology of monotone strata. Sections 6 and 7 analyze the L-space surgery regionsfor satellites by iterated torus-links and by algebraic links, respectively. While these sectionsprimarily focus on approximation tools, Section 7.6 returns to the question of exact L-spaceregions for such satellites, and describes how to decompose these L-space regions into strataaccording to monotonicity criteria, which govern where the endpoints of local L-space intervalslie, relative to asymptotes of maps on slopes induced by gluing maps.Local monotonicity criteria also help determine the topology of these strata, a phenomenonwe illustrate with Theorem 7.5 in Section 7, where we show that the Q -corrected R -closure ofthe monotone stratum of the L-space surgery region of an appropriate satellite link admits adeformation retraction onto an embedded torus analogous to that in Theorem 5.3 .ii.b above.6. Iterated torus-link satellites Just as one can construct a torus-link-satellite exterior from a knot exterior by gluing anappropriate Seifert fibered space to the knot exterior (as described in Proposition 4.2), oneconstructs an iterated torus-link-satellite exterior by gluing an appropriate rooted (tree-)graphmanifold to the knot exterior, where this graph manifold is formed by iteratively performingthe Seifert-fibered-gluing operations associated to individual torus-link-satellite operations.6.1. Construction of iterated torus-link-satellite exteriors. An iterated torus-link-satellite of a knot exterior Y = M \ ◦ ν ( K ) is specified by a weighted, rooted tree Γ, corre-sponding to the minimal JSJ decomposition of the graph manifold glued to the knot exteriorto form the satellite. We weight each vertex v ∈ Vert(Γ) by the 3-tuple ( p v , q v , n v ) ∈ Z corresponding to the pattern link T v := T ( n v p v , n v q v ). As usual, we demand that p v , n v > q v (cid:54) = 0. (If any vertex had p v = 0 or q v = 0, then our satellite-link exterior would bea nontrivial connected sum, in which case we might as well have considered the irreduciblecomponents of the exterior separately. Moreover, the links of complex surface singularitiesare irreducible, so the algebraic links we consider later on will necessarily be irreducible.)The weight ( p v , q v , n v ) also specifies the JSJ component Y v as the Seifert fibered exterior(133) Y v := ˆ Y n v ( p v ,q v ) = M S ( − q ∗ v p v , ˆ ∗ , ˆ ∗ , . . . , n ˆ ∗ ) = ˆ Y ( p v ,q v ) \ ◦ ν ( T v )of T v ⊂ ˆ Y ( p v ,q v ) as a link in the solid torus(134) ˆ Y ( p v ,q v ) := M S ( − q ∗ v p v , p ∗ v q v ) \ ◦ ν ( λ ) = S \ ◦ ν ( λ ) = ν ( λ − ) , for λ − the multiplicity- p v fiber of merdional sf -slope y v − = − q ∗ v p v , and λ the multiplicity- q v fiber of meridional sf -slope y v = p ∗ v q v , as in Section 4.1. As usual, ( p ∗ v , q ∗ v ) ∈ Z denotes theunique pair of integers satisfying vp v p ∗ v − q v q ∗ v = 1 with q ∗ v ∈ { , . . . , p v − } . ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 31 Specifying a root r for the tree Γ determines an orientation on edges, up to over-all sign. Wechoose to direct edges towards the root r , and write E in ( v ) for the set of edges terminating ona vertex v . On the other hand, each non-root vertex v has a unique edge emanating from it,and we call this outgoing edge e v . We additionally declare one edge e r to emanate from the theroot vertex r towards a null vertex null / ∈ Vert(Γ), which we morally associate (with no hat)to our original knot exterior, Y null := Y = M \ ◦ ν ( K ). For any (directed) edge e ∈ Edge(Γ),we write v ( e ) for the destination vertex of e , so that v = v ( − e v ) for all v ∈ Vert(Γ).For notational convenience, we also associate an “index” j ( e ) ∈ { , . . . , n v ( e ) } to each edge e ,specifying the boundary component ∂ j ( e ) Y v ( e ) of Y v ( e ) to which ∂ Y v ( − e ) is glued when weembed the pattern link T v ( − e ) in a neighborhood of ∂ j ( e ) Y v ( e ) . As such, each edge e ∈ Edge(Γ)corresponds to a gluing map(135) ϕ e : ∂ Y v ( − e ) → − ∂ j ( e ) Y v ( e ) , along the incompressible torus joining Y v ( − e ) to Y v ( e ) . This ϕ e is the inverse of the map ¯ ϕ used in the satellite construction of Proposition 4.2. We express its induced map on slopes(136) ϕ P e ∗ : P ( H ( ∂ Y v ( − e ) ; Z )) sf −→ P ( H ( ∂ j ( e ) Y v ( e ) ; Z )) sf , y (cid:55)→ yp v ( − e ) − q ∗ v ( − e ) yq v ( − e ) − p ∗ v ( − e ) , in terms of sf -slopes on both sides. Thus ϕ P e ∗ is orientation reversing. Note that for any v ∈ Vert(Γ), the map ϕ P e v ∗ is determined by ( p v , q v , n v ) and j ( e v ). We additionally define(137) J v := { j ( e ) | e ∈ E in ( v ) } , and I v := { , . . . , n v } \ J v for each v ∈ Vert(Γ), so that the space of Dehn filling slopes of Y Γ is given by(138) (cid:89) v ∈ Vert(Γ) (cid:89) i ∈ I v P ( H ( ∂ i Y v ; Z )) . Writing Γ v for the subtree of Γ of which v is the root, let Y Γ v denote the graph manifoldwith JSJ decomposition given by the Seifert fibered spaces Y u and gluing maps ϕ e ( u ) for u ∈ Vert(Γ v ), so that Y Γ v is constructed recursively as(139) Y Γ v = Y v ∪ { ϕ e } (cid:97) e ∈ E in( v ) Y Γ v ( − e ) . The exterior Y Γ := M \ ◦ ν ( K Γ ) of the iterated torus-link-satellite K Γ ⊂ M of K ⊂ M specifiedby Γ is then given by Y Γ = Y ∪ ϕ er Y Γ .6.2. Dehn fillings of Y Γ . For v ∈ Vert(Γ), any sf Γ v -slope(140) y Γ v := (cid:89) v ∈ Vert(Γ v ) y v , y v ∈ ( Q ∪ {∞} ) | I v | sf v := (cid:89) i ∈ I v P ( H ( ∂ i Y v ; Z )) sf v , determines an L-space interval L ( Y Γ v ( y Γ v )) for the sf Γ v -Dehn-filling Y Γ v ( y Γ v ). If Y Γ v ( y Γ v ) isFloer simple, then we write(141) [[ y v − , y v ]] := L sf v ( Y Γ v ( y Γ v )) , [[ y v ( e v ) j ( e v ) − , y v ( e v ) j ( e v )+ ]] = ϕ P e v ∗ ( L sf v ( Y Γ v ( y Γ v )))to express L sf v ( Y Γ v ( y Γ v )) in terms of sf v -slopes and sf v ( e v ) -slopes, respectively. Note thatsince ϕ P e v ∗ is orientation-reversing, we have(142) y v ( e v ) j ( e v ) − = ϕ P e v ∗ ( y v ) , y v ( e v ) j ( e v )+ = ϕ P e v ∗ ( y v − ) . If we focus instead on the incoming edges of v , then any Dehn filling of the boundarycomponents ∂Y Γ v \ ∂Y v of Y Γ v allows us to partition the graph manifolds incident to v , labeledby J v = { j ( e ) | E in ( v ) } , according to whether they are boundary compressible ( bc )—a solidtorus or connected sum thereof—or boundary incompressible ( bi ): J bc v := { j ( e ) | Y Γ v ( − e ) is bc , e ∈ E in ( v ) } , (143) J bi v := J v \ J bc v . (144)Setting y vj := y vj ± when y vj + = y vj + , we additionally define the sets J bi + v Z , J bi − v Z and J bc v Z :(145) J bi ± v Z := { j ∈ J bi v | y vj ± ∈ Z } , J bc v Z := { j ∈ J bc v | y vj ∈ Z } . For v ∈ Vert(Γ), k ∈ Z > , define ¯ y v ∓ Σ ( k ) := 0 if ∞ ∈ { y vj ± } j ∈ J v ∪ { y vi } i ∈ I v , and otherwise set¯ y v − Σ ( k ) := (cid:80) j ∈ J bi v (cid:0)(cid:6) [ y vj + ] k (cid:7) − (cid:1) + (cid:80) i ∈ I v ∪ J bc v (cid:98) [ y vi ] k (cid:99) , (146) ¯ y v ( k ) := (cid:80) j ∈ J bi v (cid:0)(cid:6) [ − y vj − ] k (cid:7) − (cid:1) + (cid:80) i ∈ I v ∪ J bc v (cid:98) [ − y vi ] k (cid:99) . (147)In addition, define ¯ y v − := sup k> ¯ y v − ( k ), ¯ y v := inf k> ¯ y v ( k ), where¯ y v − ( k ) := k (cid:16) − (cid:108) q ∗ v p v k (cid:109) − ¯ y v − Σ ( k ) (cid:17) − | J bi + v Z | , (148) ¯ y v ( k ) := k (cid:16) (cid:106) q ∗ v p v k (cid:107) + ¯ y v ( k ) (cid:17) + | J bi − v Z | . (149)The “sup” and “inf” account for cases in which ¯ y v ± ( k ) = 0. The above notation provides aconvenient way to repackage our computation of L-space interval endpoints. Proposition 6.1. If y v − , y v ∈ P ( H ( ∂ Y v ; Z )) sf v are the (potential) L-space interval end-points for Y Γ v ( y Γ v ) as defined in Theorem 4.3, then y v − = ¯ y v − − (cid:80) j ∈ J bi v (cid:0) (cid:100) y vj + (cid:101)− (cid:1) − (cid:80) j ∈ J bc v (cid:98) y vj (cid:99) − (cid:80) i ∈ I v (cid:98) y vi (cid:99) ,y v = ¯ y v − (cid:80) j ∈ J bi v (cid:0) (cid:98) y vj − (cid:99) +1 (cid:1) − (cid:80) j ∈ J bc v (cid:100) y vj (cid:101) − (cid:80) i ∈ I v (cid:100) y vi (cid:101) . Moreover, ¯ y v ∓ = q ∗ v p v when ∞ ∈ { y vj ± } j ∈ J v ∪ { y vi } i ∈ I v , but ¯ y v − = (cid:108) q ∗ v p v (cid:109) − if J bi + v Z (cid:54) = ∅ and ∞ / ∈ { y vj + } j ∈ J v ∪ { y vi } i ∈ I v ;(150) ¯ y v = 1 if J bi − v Z (cid:54) = ∅ and ∞ / ∈ { y vj − } j ∈ J v ∪ { y vi } i ∈ I v . (151) Proof. The displayed equations in Proposition 6.1 come directly from the definitions of y v ± specified by Theorem 4.3, but subjected to some mild manipulation of terms using the factsthat x = (cid:98) x (cid:99) + [ x ] and x = (cid:100) x (cid:101) − [ − x ] for all x ∈ R , and that(152) | J bi + v Z | = (cid:80) j ∈ J bi v (cid:0) (cid:98) y vj + (cid:99) − ( (cid:100) y vj + (cid:101) − (cid:1) , −| J bi − v Z | = (cid:80) j ∈ J bi v (cid:0) (cid:100) y vj − (cid:101) − ( (cid:98) y vj − (cid:99) + 1) (cid:1) . For the second half of the proposition, first note that the ¯ y v ∓ = q ∗ v p v result follows directlyfrom taking the k → ∞ limit. In the case of ∞ / ∈ { y vj + , y vj − , y vi | j ∈ J v , i ∈ I v } we temporarilyset ˆ y v ∓ ( k ) := ¯ y v ∓ ( k ) ± | J bi ± v Z | , so thatˆ y v − ( k ) ≤ k (cid:16)(cid:108) q ∗ v p v k (cid:109) + ( | J bi + v Z | − (cid:17) ≤ (cid:108) q ∗ v p v (cid:109) + k ( | J bi + v Z | − , (153) ˆ y v ( k ) ≥ k (cid:16)(cid:106) q ∗ v p v k (cid:107) − ( | J bi − v Z | − (cid:17) ≥ − k ( | J bi − v Z | − ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 33 for all k ∈ Z > . If J bi + v Z (cid:54) = ∅ (respectively J bi − v Z (cid:54) = ∅ ), then the above bound for ˆ y v − ( k )(respectively ˆ y v ( k )) is nonincreasing (respectively nondecreasing) in k , so thatˆ y v − ( k ) ≤ (cid:108) q ∗ v p v (cid:109) − | J bi + v Z | , ˆ y v ( k ) ≥ − | J bi − v Z | (155)for all k ∈ Z > . Since these bounds are each realized when k = 1, this completes the proof ofthe bottom line of the proposition. (cid:3) The above method of computation for L-space interval endpoints helps us to prove someuseful bounds for these endpoints. Proposition 6.2. Suppose ¯ y v ( k ) , ¯ y v − , and ¯ y v are as defined in (147), (148), and (149),and that ∞ / ∈ { y vj + , y vj − , y vi | j ∈ J v , i ∈ I v } . Then ¯ y v − and ¯ y v satisfy the following properties. ¯ y v − = q ∗ v p v ⇐⇒ ¯ y v = q ∗ v p v ⇐⇒ J bi v = ∅ and { y vj + , y vj − , y vi | j ∈ J v , i ∈ I v } ⊂ Z (=) = ⇒ Y Γ v ( y Γ v ) is bc . ( − ) If Y Γ v ( y Γ v ) is bi , then ¯ y v − ∈ (cid:104)(cid:108) q ∗ v p v (cid:109) − , q ∗ v p v (cid:69) = (cid:40)(cid:2) , q ∗ v p v (cid:11) p v (cid:54) = 1 (cid:2) − , (cid:11) p v = 1 . (+) If Y Γ v ( y Γ v ) is bi , then ¯ y v ∈ (cid:68) q ∗ v p v , (cid:106) q ∗ v p v (cid:107) +1 (cid:105) = (cid:68) q ∗ v p v , (cid:105) . If in addition, J bi − v Z = ∅ ,then for q v > and m ∈ Z (possibly negative or zero) with m < q v p v , ¯ y v satisfies ( i ) ¯ y v ∈ (cid:68) q ∗ v p v , p ∗ v − mq ∗ v q v − mp v (cid:105) ¯ y v ( q v − mp v ) = 0 (cid:68) p ∗ v − mq ∗ v q v − mp v , (cid:105) ¯ y v ( q v − mp v ) > , where we note that for a, b ∈ Z with a, b < q v p v , one has ( ii ) p ∗ v − aq ∗ v q v − ap v < p ∗ v − bq ∗ v q v − bp v ⇐⇒ a < b. Lastly, if q v > p v > , then ( iii ) ¯ y v = p ∗ v q v ⇐⇒ ¯ y v ( q v ) = 0 , ¯ y v ( q v + p v ) > , and J bi − v Z = ∅ . To aid in the proof of (=), we first prove the following Claim. If the hypotheses of Proposition 6.2 hold, then J bi v = ∅ and { y vj + , y vj − , y vi | j ∈ J v , i ∈ I v } ⊂ Z ⇐⇒ Y Γ v ( y Γ v ) is bc and ¯ y v − = ¯ y v − = q ∗ v p v . Proof of Claim. For the ⇒ direction, the lefthand side implies the irreducible componentof Y Γ v ( y Γ v ) containing ∂Y Γ v ( y Γ v ) is Seifert fibered over the disk with one or fewer exceptionalfibers, hence is bc , and direct computation shows that ¯ y v − = ¯ y v − = q ∗ v p v .For the ⇐ direction, suppose the righthand side holds. Then J bi v = ∅ , and the irreduciblecomponent of Y Γ v ( y Γ v ) containing ∂Y Γ v ( y Γ v ) is Seifert fibered over the disk with one or fewerexceptional fibers, implying (cid:12)(cid:12)(cid:8) q ∗ v p v , y vj + , y vj − , y vi (cid:12)(cid:12) j ∈ J v , i ∈ I v (cid:9) ∩ Z (cid:12)(cid:12) ≤ 1. If q ∗ v p v / ∈ Z , then weare done, but if q ∗ v p v ∈ Z and (cid:12)(cid:12)(cid:8) q ∗ v p v , y vj + , y vj − , y vi (cid:12)(cid:12) j ∈ J v , i ∈ I v (cid:9) ∩ Z (cid:12)(cid:12) = 1, then the longitude l satisfies l = − (cid:80) i ∈ I v ∪ J v y vi / ∈ Z , contradicting the fact that l = ¯ y v − = ¯ y v − = q ∗ v p v = 0. (cid:3) Proof of ( − ) and part of (=) . When J bi + v Z (cid:54) = ∅ , ( − ) follows from Proposition 6.1, which tellsus ¯ y v − = ¯ y v − (1) = (cid:108) q ∗ v p v (cid:109) − 1. When J bi + v Z = ∅ , we have the bounds(156) ¯ y v − ≥ ¯ y v − (1) = (cid:40) p v (cid:54) = 1 − p v = 1 , ¯ y v − ( k ) ≤ k (cid:16)(cid:106) q ∗ v p v k (cid:107) +1 (cid:17) < q ∗ v p v ∀ k ∈ Z > . Thus, either ¯ y v − ∈ (cid:104)(cid:108) q ∗ v p v (cid:109) − , q ∗ v p v (cid:69) or ¯ y v − = q ∗ v p v . The latter case implies sup k → + ∞ ¯ y v − ( k ) is notattained for finite k , and so Theorem 4.3 tells us that Y Γ v ( y Γ v ) is bc and ¯ y v − = ¯ y v . (cid:3) Proof of (+) and remainder of (=) . Proposition 6.1 tells us that ¯ y v = 1 when J bi − v Z (cid:54) = ∅ ,so we henceforth assume J bi − v Z = ∅ . In this case, we have(157) ¯ y v ≤ ¯ y v (1) = 1 , ¯ y v ( k ) ≥ k (cid:16)(cid:108) q ∗ v p v k (cid:109) − (cid:17) > q ∗ v p v ∀ k ∈ Z > . Thus, either ¯ y v ∈ (cid:10) q ∗ v p v , (cid:3) or ¯ y v = q ∗ v p v . The latter case implies inf k → + ∞ ¯ y v ( k ) is not attainedfor finite k , which Theorem 4.3 tells us implies that Y Γ v ( y Γ v ) is bc and ¯ y v = ¯ y v − .For (+ .i ), fix some m ∈ Z with m < q v p v . If ¯ y v ( q v − mp v ) = 0, then(158) ¯ y v ≤ ¯ y v ( q v − mp v ) = q v − mp v (cid:16) (cid:106) q ∗ v p v ( q v − mp v ) (cid:107) + 0 (cid:17) = p ∗ v − mq ∗ v q v − mp v . Next, suppose that ¯ y v ( q v − mp v ) > 0, so that either [ − y vj − ] > ( q v − mp v ) − for some j ∈ J bi v , or [ − y vi ] ≥ ( q v − mp v ) − for some i ∈ I v ∪ J bi v (or both occur). Since for anyrational x > ( q v − mp v ) − , we have (cid:100) xk (cid:101) − ≥ (cid:106) kq v − mp v (cid:107) for all k ∈ Z > , the condition¯ y v ( q v − mp v ) > y v ( k ) ≥ (cid:106) kq v − mp v (cid:107) for all k ∈ Z > . We then have k (cid:16) ¯ y v ( k ) − p ∗ v − mq ∗ v q v − mp v (cid:17) ≥ k (cid:16) k (cid:16) (cid:106) q ∗ v p v k (cid:107) + (cid:106) kq v − mp v (cid:107)(cid:17) − p ∗ v − mq ∗ v q v − mp v (cid:17) (159) = 1 + (cid:106) kq v − mp v (cid:107) − [ q ∗ v k ] p v p v + q ∗ v p v k − p ∗ v − mq ∗ v q v − mp v k = 1 − [ q ∗ v k ] p v p v + (cid:106) kq v − mp v (cid:107) − kp v ( q v − mp v ) = 1 − [ q ∗ v k ] p v + [ k ] q v − mp v / ( q v − mp v ) p v + (cid:16) p v − p v (cid:17)(cid:106) kq v − mp v (cid:107) > k ∈ Z > , and so ¯ y v > p ∗ v − mq ∗ v q v − mp v .Statement (+ .ii ) is a simple consequence of the fact that(160) p ∗ v − bq ∗ v q v − bp v − p ∗ v − aq ∗ v q v − ap v = b − a ( q v − bp v )( q v − ap v ) . This leaves us with (+ .iii ). Since q v > p v > p ∗ v q v < 1, but Proposition 6.1 tells us¯ y v = 1 if J bi − v Z (cid:54) = ∅ , we henceforth assume J bi − v Z = ∅ . Setting m = 0 in (+ .i ) then gives us(161) ¯ y v ≤ p ∗ v q v ⇐⇒ ¯ y v ( q v ) = 0 . Similarly, setting m = − .i ) yields the relation(162) ¯ y v ≤ p ∗ v + q ∗ v q v + p v < p ∗ v q v ⇐⇒ ¯ y v ( q v + p v ) = 0 , ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 35 where we used (+ .ii ) for the right-hand inequality. Lastly, suppose that ¯ y v ( q v + p v ) > .i ), this implies that ¯ y v ( k ) ≥ (cid:106) kp v + q v (cid:107) for all k ∈ Z > . We then have(163) ¯ y v ( k ) ≥ k (cid:16) (cid:106) q ∗ v p v k (cid:107) + (cid:106) kp v + q v (cid:107)(cid:17) ≥ p ∗ v q v ∀ k ∈ Z > , with the right-hand inequality coming from Lemma 4.7, and so ¯ y v ≥ p ∗ v q v . (cid:3) There is one more collection of estimates that will be particularly useful in the case ofgeneral iterated torus-link satellites. Proposition 6.3. The following bounds hold. ( i ) If q v > and ¯ y v − Σ ([ p v ] q v ) > , then ¯ y v − < p ∗ v q v − p v ] q v ( q v ) , ϕ P e v ∗ (¯ y v − ) > (cid:108) p v q v (cid:109) − ii ) If q v < and ¯ y v − Σ ([ − p v ] q v ) > , then ¯ y v − < p ∗ v q v + − p v ] q v ( q v ) , ϕ P e v ∗ (¯ y v − ) > (cid:108) p v q v (cid:109) − iii ) If q v > and ¯ y v ([ − p v ] q v ) > , then ¯ y v > p ∗ v q v + − p v ] q v ( q v ) , ϕ P e v ∗ (¯ y v ) < (cid:106) p v q v (cid:107) + 1;( iv ) If q v < and ¯ y v ([ p v ] q v ) > , then ¯ y v > p ∗ v q v − p v ] q v ( q v ) , ϕ P e v ∗ (¯ y v ) < (cid:106) p v q v (cid:107) + 1 . Proof of ( i ). If [ p v ] q v ∈ { , } , then ¯ y v − Σ ([ p v ] q v ) ≤ p v ] q v ≥ 2, implying p v ≥ q v ≥ 3, so that(164) ϕ P e v ∗ (cid:16) p ∗ v q v − p v ] q v ( q v ) (cid:17) = (cid:108) p v q v (cid:109) − . By reasoning similar to that used in the proof of (+ .i ) above, the hypothesis ¯ y v − Σ ([ p v ] q v ) > y v − Σ ( k ) ≥ (cid:106) k [ p v ] q v (cid:107) for all k ∈ Z > . Thus, if we set(165) m := (cid:106) p v q v (cid:107) , so that [ p v ] q v = p v − mq v and p ∗ v q v − p v ] q v ( q v ) = q ∗ v − mp ∗ v p v − mq v , then it suffices to prove negativity, for all k ∈ Z > , of the difference k (cid:16) ¯ y v − ( k ) − q ∗ v − mp ∗ v p v − mq v (cid:17) ≤ k (cid:16) k (cid:16) − (cid:108) q ∗ v p k (cid:109) − (cid:106) kp v − mq v (cid:107)(cid:17) − q ∗ v − mp ∗ v p v − mq v (cid:17) = − [ − q ∗ v k ] p v p v − (cid:106) kp v − mq v (cid:107) + q ∗ v p k − q ∗ v − mp ∗ v p v − mq v k = − p v q v + q v [ q − v k ] p v p v q v − (cid:106) k [ p v ] q v (cid:107) + (cid:16) mp v (cid:17) k [ p v ] q v . (166)Now, mp v already satisfies the bound(167) mp v = (cid:98) p v /q v (cid:99) p v = p v − [ p v ] q v p v q v ≤ p v − p v q v < q v ≤ . Thus, if (cid:106) k [ p v ] qv (cid:107) ≥ 1, then(168) (cid:106) k [ p v ] q v (cid:107) ≥ (cid:16)(cid:106) k [ p v ] q v (cid:107) + 1 (cid:17) > (cid:16) k [ p v ] q v (cid:17) > (cid:16) mp v (cid:17) k [ p v ] q v , making the right-hand side of (166) negative.We therefore henceforth assume that (cid:106) k [ p v ] qv (cid:107) = 0, implying k < [ p v ] q v . Thus, since(169) q v [ q − v k ] p v + p v [ p − v k ] q v = p v q v + k, and since mq v = p v − [ p v ] q v , we obtain k (cid:16) ¯ y v − ( k ) − q ∗ v − mp ∗ v p v − mq v (cid:17) ≤ k − p v [ p − v k ] q v + ( p v − [ p v ] q v ) k [ p v ] qv p v q v = − [ p − v k ] q v + k [ p v ] qv q v < . (170) (cid:3) Proof of ( ii ). The claim holds vacuously for [ − p v ] q v ∈ { , } , so we assume [ − p v ] q v ≥ q v ≤ − 3, in which case(171) ϕ P e v ∗ (cid:16) p ∗ v q v + − p v ] q v ( q v ) (cid:17) = (cid:108) p v q v (cid:109) − . Since ¯ y v − Σ ([ − p v ] q v ) > y v − Σ ( k ) ≥ (cid:106) k [ − p v ] q v (cid:107) for all k ∈ Z > , we set(172) m := − (cid:106) p v q v (cid:107) = (cid:108) p v | q v | (cid:109) , with [ − p v ] q v = − p v − mq v and p ∗ v q v + − p v ] q v ( q v ) = − q ∗ v − mp ∗ v − p v − mq v . Using arguments similar to those in part ( i ), it is straightforward to derive the bound k (cid:16) ¯ y v − ( k ) − − q ∗ v − mp ∗ v − p v − mq v (cid:17) ≤ − p v | q v | + | q v | [ | q | − v ( − k )] p v p v | q v | − (cid:106) k [ − p v ] q v (cid:107) + (cid:16) mp v (cid:17) k [ − p v ] q v for all k ∈ Z > , and to show that the right-hand side is negative if (cid:106) k [ − p v ] qv (cid:107) ≥ 1, allowing usto assume that (cid:106) k [ − p v ] qv (cid:107) = 0 and k < [ − p v ] q v . Thus, since(173) | q v | [ | q v | − ( − k )] p v + p v [ p − v ( − k )] q v = p v q v − k, and since mq v = p v + [ − p v ] q v , we obtain k (cid:16) ¯ y v − ( k ) − − q ∗ v − mp ∗ v − p v − mq v (cid:17) ≤ − k − p v [ p − v ( − k )] q v + ( p v + [ − p v ] q v ) k [ − p v ] qv p v | q v | (174) = − [ p − v ( − k )] q v + k [ − p v ] qv | q v | < . (175) (cid:3) Proofs of ( iii ) and ( iv ). Respectively similar to proofs of ( ii ) and ( i ). (cid:3) L-space surgery regions for iterated satellites: Proof of Theorem 1.7. We havefinally done enough preparation to prove Theorem 1.7 from the introduction. Proof of Theorem 1.7. The bulk of part ( i ) is proven in “Claim 1” in the proof of Theorem 8.1.Since the right-hand condition of (262) is equivalent to the condition that Y Γ ( y Γ ) be an L-space, Claim 1 proves that Y Γ ( y Γ ) is an L-space if and only if Y Γ ( y Γ ) = S . Thus, if wedefine Λ Γ as in (11), then the statement L ( Y Γ ) = Λ Γ holds tautologically.The proof of part ( ii ) begins similarly to the proof of Theorem 4.5.( i.b ), except that insteadof deducing that (cid:80) i ∈ I r ( (cid:100) y ri (cid:101) − (cid:98) y ri (cid:99) ) ≤ 1, we deduce that(176) (cid:80) i ∈ I r ( (cid:100) y ri (cid:101) − (cid:98) y ri (cid:99) ) + (cid:80) j ∈ J bi r (cid:0)(cid:0) (cid:98) y rj − (cid:99) +1 (cid:1) − (cid:0) (cid:100) y rj + (cid:101)− (cid:1)(cid:1) − (cid:80) j ∈ J bc r ( (cid:100) y rj (cid:101) − (cid:98) y rj (cid:99) ) ≤ , with y rj − ≥ y rj + for all j ∈ J bc r . In the case that (cid:80) i ∈ I r ( (cid:100) y ri (cid:101) − (cid:98) y ri (cid:99) ) = 1 and the other sumsvanish, we are reduced to the original case of Theorem 4.5.( i.b ), obtaining the component(177) Λ r · S | I r | ([ N, + ∞ ] ×{∞} | I r |− ) × (cid:89) e ∈ E in ( r ) Λ Γ v ( − e ) ⊂ L S ( Y Γ ) . In the case that (cid:80) i ∈ I r ( (cid:100) y ri (cid:101) − (cid:98) y ri (cid:99) ) = 0, we have that y r ∈ Z | I r | , and all but one incomingedge of r , say e , descend from trees with trivial fillings. Performing these trivial fillings reduces Y Γ to the exterior of a Γ v ( − e ) -satellite of the (1 , q r )-cable of K ⊂ S , but the (1 , q r )-cable is ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 37 just the identity operation, so we are left with the exterior Y Γ v ( − e ) of the Γ v ( − e ) -satellite of K ⊂ S . Considering this for all edges e ∈ E in ( r ) then gives the remaining component(178) (cid:97) e ∈ E in ( r ) (cid:16) L S ( Y Γ v ( − e ) ) × Λ Γ \ Γ v ( − e ) (cid:17) ⊂ L S ( Y Γ ) . Part ( iii ) . Before proceeding with the main inductive argument in this section, we attend tosome bookkeeping issues. In particular, our inductive proof requires each I v to be nonempty.For any w ∈ Vert(Γ) with I w = ∅ , we repair this situation artificially, as follows. First, redefine I w := { } . Next, if 0 ≥ m + w , then set L min + sf w = { } , and declare R sf w \Z sf w = L min − sf w = ∅ . Finally,if 0 < m + w , then 0 < m + w < m − w , so set L min − sf w = { } , and declare R sf w \ Z sf w = L min + sf w = ∅ .For a vertex v ∈ Vert(Γ), inductively assume, for each incoming edge e ∈ E in ( v ), that forany y Γ v ( − e ) ∈ (cid:81) u ∈ Vert(Γ v ( − e ) ) (cid:0) L min + sf u ∪ R sf u \ Z sf u ∪ L min − sf u (cid:1) , we have (cid:24) p v ( − e ) q v ( − e ) (cid:25) − ≤ y vj ( e )+ ≤ y vj ( e ) − ≤ (cid:22) p v ( − e ) q v ( − e ) (cid:23) + 1 , (179) (cid:24) p v ( − e ) q v ( − e ) (cid:25) − < y vj ( e )+ , y vj ( e ) − < (cid:22) p v ( − e ) q v ( − e ) (cid:23) + 1 if Y Γ v ( − e ) ( y Γ v ( − e ) ) is bi , (180)where, again, y vj ( e ) ± := ϕ P e ∗ ( y v ( − e )0 ∓ ), with L sf ( Y Γ v ( − e ) ( y Γ v ( − e ) )) = [[ y v ( − e )0 − , y v ( − e )0+ ]]. Note thatthis inductive assumption already holds vacuously if v is a leaf.If y v ∈ R sf v \ Z sf v , then y v − = y v = ∞ , implying that(181) y v ( e v ) j ( e v ) ± = ϕ P e v ∗ ( y v ∓ ) = p v q v ∈ (cid:68)(cid:108) p v q v (cid:109) − , (cid:106) p v q v (cid:107) + 1 (cid:69) . If y v ∈ L min − sf v ∪ L min + sf v ⊂ Q | I v | , then applying (179) to Proposition 6.1 yields y v − ≤ ¯ y v − − (cid:88) e ∈ E in ( v ) (cid:18)(cid:24) p v ( − e ) q v ( − e ) (cid:25) − (cid:19) − (cid:80) i ∈ I v (cid:98) y vi (cid:99) , (182) y v ≥ ¯ y v − (cid:88) e ∈ E in ( v ) (cid:18)(cid:22) p v ( − e ) q v ( − e ) (cid:23) + 1 (cid:19) − (cid:80) i ∈ I v (cid:100) y vi (cid:101) , (183)and assuming (179) for Theorem 4.4 implies that(184) y v ≤ y v − . Suppose y v ∈ L min + sf v , so that the bound (cid:80) i ∈ I v (cid:98) y vi (cid:99) ≥ m + v , together with (182), implies that y v − ≤ ¯ y v − − q v = − J v (cid:54) = ∅ ; q v < − p v q v > 10 otherwise . (185)Since Proposition 6.2 tells us ¯ y v − ≤ q ∗ v p v , we then have(186) y v ≤ y − ≤ q ∗ v p v < p ∗ v q v = ( φ P e v ∗ ) − ( ∞ ) . Since ϕ P e v ∗ is locally monotonically decreasing in the complement of its vertical asymptote at( φ P e v ∗ ) − ( ∞ ) = p ∗ v q v , line (186) implies(187) y v ( e v ) j ( e v )+ := ϕ P e v ∗ ( y v − ) ≤ y v ( e v ) j ( e v ) − := ϕ P e v ∗ ( y v ) < φ P e v ∗ ( ∞ ) = p v q v < (cid:106) p v q v (cid:107) + 1 , where φ P e v ∗ ( ∞ ) = p v q v is the location of the horizontal asymptote of ϕ P e v ∗ . Thus, since (cid:108) p v q v (cid:109) − < p v q v = φ P e v ∗ ( ∞ ), we deduce that to finish establishing our inductive hypotheses for e v in the y v ∈ L min + sf v case, it suffices to show that(188) y v − ≤ ( ϕ P e v ∗ ) − (cid:16)(cid:108) p v q v (cid:109) − (cid:17) , with equality only if Y Γ v ( y Γ v ) is bc . If | q v | = 1, it is straightforward to compute that(189) ( ϕ P e v ∗ ) − (cid:16)(cid:108) p v q v (cid:109) − (cid:17) = p ∗ v q v − − p v , q v ) = (1 , − − p v , q v ) (cid:54) = (1 , − , q v = − q v = 1 . Proposition 6.2 tells us ¯ y − ≤ q ∗ v p v , with equality only if Y Γ v ( y Γ v ) is boundary compressible.Thus, since q ∗ v p v < 1, with q ∗ v p v = 0 when p v = 1, it follows from (185) and (189) that (188) holds.Next suppose that | q v | > 1, so that(190) ( ϕ P e v ∗ ) − (cid:16)(cid:108) p v q v (cid:109) − (cid:17) = p ∗ v q v − ± ± p v ] q v ( q v ) for ± q v > . If 0 < p v q v < 1, this makes ( ϕ P e v ∗ ) − (cid:16)(cid:108) p v q v (cid:109) − (cid:17) = q ∗ v p v , so that (188) follows from (185) andthe fact that ¯ y − ≤ q ∗ v p v , with equality only if Y Γ v ( y Γ v ) is boundary compressible. Thisleaves the cases in which p v q v > q v < − 1. If J v = ∅ , then L min + sf v respectively excludes { (cid:80) (cid:98) y vi (cid:99) = (cid:80) (cid:98) [ y vi ][ − p v ] q v (cid:99) = 0 } or { (cid:80) (cid:98) y vi (cid:99) = (cid:80) (cid:98) [ y vi ][ p v ] q v (cid:99) = 0 } , and so (188) follows frompart ( i ) or ( ii ), respectively, of Proposition 6.3. If J v (cid:54) = ∅ , then y v − ≤ q ∗ v p v − 1, and it is easyto show that(191) q ∗ v p v − < p ∗ v q v − ± ± p v ] q v ( q v ) for ± q v > , completing our inductive step for the case of y v ∈ L min + sf v .The proof of our inductive step for the case of y v ∈ L min − sf v follows from symmetry underorientation reversal.Recall that we regard the root vertex r at the bottom of the tree Γ = Γ r as having anoutgoing edge e r pointing to the empty vertex v ( e r ) := null , where this null vertex v ( e r )corresponds to the exterior Y := S \ ◦ ν ( K ) of the companion knot K ⊂ S , from which thesatellite exterior Y Γ := Y Γ ∪ Y is formed. Recursing down to this null vertex v ( e r ), our aboveinduction shows that for any y Γ ∈ (cid:81) v ∈ Vert(Γ r ) (cid:0) L min + sf v ∪ R sf v \ Z sf v ∪ L min − sf v (cid:1) , we have(192) (cid:108) p r q r (cid:109) − ≤ y v ( e r ) j ( e r )+ ≤ y v ( e r ) j ( e r ) − ≤ (cid:106) p r q r (cid:107) + 1 . This final L-space interval ϕ P e r ∗ ( L sf ( Y Γ r ( y Γ ))) = [[ y v ( e r ) j ( e r ) − , y v ( e r ) j ( e r )+ ]] S expresses slopes in termsof the reversed S -slope basis, “ S .” That is, the meridian and longitude of K ⊂ S haverespective S -slopes 0 = ∞ and ∞ = . If K ⊂ S is the unknot, then its S -longitude ∞ satisfies ∞ ∈ [[ y v ( e r ) j ( e r ) − , y v ( e r ) j ( e r )+ ]], so we deduce that y Γ ∈ L sf Γ ( Y Γ ) in this case.If K ⊂ S is nontrivial, then its exterior Y has L-space interval L S ( Y ) = [0 , N ] S . Assumethat Γ satisfies hypothesis ( iii ) of the theorem. Then since q r p r ≥ N := 2 g ( K ) − ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 39 < p r q r ≤ 1, (179) and (180) tell us that 0 ≤ y v ( e r ) j ( e r )+ ≤ y v ( e r ) j ( e r ) − , with y v ( e r ) j ( e r )+ = 0 only if Y Γ r ( y Γ )is bc . Thus, it remains to show that either y v ( e r ) j ( e r ) − < N , or y v ( e r ) j ( e r ) − ≤ N and Y Γ ( y Γ ) is bc .If y Γ | r ∈ L min + r , then (187) tells us y v ( e r ) j ( e r ) − < p r q r < N . If y Γ | r ∈ R r \ Z r , then Y Γ ( y Γ ) is bc , and since y r − = y r = ∞ , we have y v ( e r ) j ( e r ) − = y v ( e r ) j ( e r )+ = p r q r ∈ (cid:10) , N (cid:3) . This leaves us withthe case of y Γ | r ∈ L min − r , for which we have(193) y r ≥ ¯ y r + (cid:40) p (cid:54) = 12 p = 1 ≥ q ∗ p + (cid:40) p (cid:54) = 12 p = 1 > q ∗ p + p ( q − pN ) = ( ϕ P e r ∗ ) − (cid:16) N (cid:17) , implying y v ( e r ) j ( e r ) − < N , and thereby completing the proof of the theorem. (cid:3) Remark. It is only in (193) that we use the hypothesis of part ( iii ) that q r > g ( K ) − N .In the case that we do have p r = 1 and q r = N , this implies ( ϕ P e r ∗ ) − (cid:16) N (cid:17) = ∞ , requiring L min − sf r to be empty, but we still have the modified result that(194) L sf Γ ( Y Γ ) ⊃ (cid:89) v ∈ Vert(Γ) (cid:0) L min − sf v ∪ R v \ Z v ∪ L min + sf v (cid:1) \ L min − sf r . Satellites by algebraic links Smooth and Exceptional splices. As mentioned in the introduction, the class ofalgbraic link satellites is slightly more general than the class of iterated torus link satellites,in that the JSJ decomposition graph of the exterior must allow one extra type of edge.To describe how this new type of edge is different, we first address the notion of splice maps.Suppose K ⊂ M and K ⊂ M are knots in compact oriented 3-manifolds M and M , witheach ∂M i a possibly-empty disjoint union of tori. Let Y i := M i \ ◦ ν ( K ) denote the exterior ofeach knot K i ⊂ M i , with ∂ Y i := − ∂ ◦ ν ( K i ), and choose a surgery basis ( µ i , λ i ) ∈ H ( ∂ Y i ; Z )for each exterior, with λ i the Seifert longitude if M i is either an integer homology sphere ora link exterior in a specified integer homology sphere.A gluing map φ : ∂ Y → − ∂ Y is then called a splice if the induced map on homologysends µ (cid:55)→ λ and µ (cid:55)→ λ . Gluings via splice maps are minimally disruptive to homology.For instance, if M is an integer homology sphere, then H ( Y ∪ φ Y ; Z ) ∼ = H ( M ; Z ). If M = S and K is an unknot, then we in fact have Y ∪ φ Y = M . In particular, if M is the exterior M = S \ ◦ ν ( L ) of some link L ⊂ S , and if K ⊂ M is an unknot in thecomposition K (cid:44) → M (cid:44) → S , then Y ∪ φ Y is the exterior of the satellite link of the companionknot K ⊂ M by the pattern link L ⊂ ( S \ ◦ ν ( K )). In particular, for satellites by T ( np, nq ),this unknot K ⊂ M is the multiplicity- q fiber λ ⊂ Y n ( p,q ) in the T ( np, nq )-exterior(195) Y n ( p,q ) := S \ ◦ ν ( T ( np, nq ) = M S ( − qp , pq ) \ ◦ ν ( (cid:96) ni =1 f n ) , c.f. (42) . In an iterated torus-link satellite, we only perform satellites on components of the compan-ion link we are building. That is, for an edge e ∈ Edge(Γ) from v ( − e ) to v ( e ), we always forma T v ( e ) -torus-link satellite that splices the multiplicity- q v ( − e ) -fiber λ v ( − e )0 , with exterior(196) Y v ( − e ) := Y n v ( − e ) ( p v ( − e ) ,q v ( − e ) ) \ ◦ ν ( λ v ( − e )0 ) , to the j ( e ) th component of the T v ( e ) torus link. Since this j ( e ) th link component is representedby the smooth fiber f j ( e ) ⊂ M S ( − q ∗ v p v , p ∗ v q v ), we call this operation a smooth splice . In an algebraic link exterior, however, an edge e can also specify an exceptional splice map,in which we splice the q v ( − e ) -fiber λ v ( − e )0 to the exceptional p v ( e ) -fiber λ v ( e ) − ⊂ Y v ( e ) . Thismultiplicity- p v ( e ) fiber λ v ( e ) − is not a component of our original companion link or its iteratedsatellites, but is rather the core of the solid torus ν ( λ v ( e ) − ) hosting T v ( e ) ⊂ ν ( λ v ( e ) − ) , of which Y v ( e ) = ν ( λ v ( e ) − ) \ ◦ ν ( T v ( e ) ) is the exterior . Thus an exceptional-splice satellite embeds the solidtorus hosting T v ( − e ) inside the solid torus hosting T v ( e ) . Since an exceptional splice at e takesthe satellite of the p v ( e ) -fiber λ v ( e ) − ⊂ Y v ( e ) , we set j ( e ) = − Slope maps induced by splices. For the induced maps on slopes, we haveSmooth splice ϕ e : ∂ Y v ( − e ) → − ∂ j ( e ) Y v ( e ) , (197) [ ϕ P e ∗ ] (cid:18) p ∗ v ( − e ) − q ∗ v ( − e ) q v ( − e ) − p v ( − e ) (cid:19) = (cid:18) (cid:19) = ⇒ ϕ P e ∗ ( y ) = p v ( − e ) y − q ∗ v ( − e ) q v ( − e ) y − p ∗ v ( − e ) for an edge e corresponding to a smooth splice, andExceptional splice σ e : ∂ Y v ( − e ) → − ∂ − Y v ( e ) , (198)[ σ P e ∗ ] (cid:18) p ∗ v ( − e ) − q ∗ v ( − e ) q v ( − e ) − p v ( − e ) (cid:19) = (cid:18) p ∗ v ( e ) − q ∗ v ( e ) − q v ( e ) p v ( e ) (cid:19) = ⇒ [ σ P e ∗ ] = (cid:18) p ∗ v ( e ) − q ∗ v ( e ) − q v ( e ) p v ( e ) (cid:19)(cid:32) p v ( − e ) − q ∗ v ( − e ) q v ( − e ) − p ∗ v ( − e ) (cid:33) for an edge e corresponding to an exceptional splice. To accommodate our notation to thesetwo different types of maps, we define(199) φ e := (cid:40) σ e j ( e ) = − ϕ e j ( e ) (cid:54) = − . In addition, since the exceptional fiber at ∂ − Y v is only exceptional if p v > 1, we adopt theconvention that exceptional splice edges only terminate on vertices v with p v > Algebraic links. Eisenbud and Neumann show in [7] that a graph Γ with such edges andvertices specifies an algebraic link exterior if and only if Γ satisfies the algebraicity conditions( i ) p v , q v , n v > v ∈ Vert(Γ) , ( ii ) ∆ e > e ∈ Edge(Γ) , (200) ∆ e := (cid:40) p v ( e ) q v ( − e ) − p v ( − e ) q v ( e ) j ( e ) = − q v ( − e ) − p v ( e ) p v ( − e ) q v ( e ) j ( e ) (cid:54) = − . ensuring negative definiteness. Eisenbud and Neumann also prove that any algebraic linkexterior can be realized by such a graph. Note that the above algebraicity conditions imply(201) 0 < p v ( − e ) q v ( − e ) < j ( e ) (cid:54) = − . For notational convenience, we adopt the convention that J v remains the same, only index-ing incoming edges corresponding to smooth splices. That is, we define(202) J v := { j ( e ) | e ∈ E in ( v ) } ∩ { , . . . , n v } , and its complement I v still indexes the remaining boundary components,(203) I v := { , . . . , n v } ∩ J v . ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 41 Lastly, since φ e is always orientation-reversing, the induced map φ P e ∗ is still decreasing withrespect to the circular order on sf -slopes, and the impact of φ P e ∗ on the linear order of finite sf -slopes still depends on the positions of the horizontal and vertical asymptotes(204) ξ v ( e ) := φ P e ∗ ( ∞ ) ∈ ( Q ∪ {∞} ) sf v ( e ) and η v ( − e ) := ( φ P e ∗ ) − ( ∞ ) ∈ ( Q ∪ {∞} ) sf v ( − e ) , respectively, of the graph of φ P e ∗ . More explicitly, we have(205) ξ v ( e ) = p v ( − e ) q v ( − e ) j ( e ) (cid:54) = − − q ∗ v ( e ) p v ( e ) + p v ( − e ) p v ( e ) ∆ e j ( e ) = − , η v ( − e ) = p ∗ v ( − e ) q v ( − e ) j ( e ) (cid:54) = − q ∗ v ( − e ) p v ( − e ) + p v ( e ) p v ( − e ) ∆ e j ( e ) = − . Adapting Propositions 6.1 and 6.2 for algebraic link exteriors. In the case ofalgebraic link exteriors, we must incorporate the possiblity of exceptional splices into theexpressions ¯ y v − ( k ) and ¯ y v ( k ) originally defined in (148) and (149), from which(206) ¯ y v − := sup k> ¯ y v − ( k ) and ¯ y v := inf k> ¯ y v ( k )are defined. The only changes that arise are localized to the summands (cid:108) q ∗ v p v k (cid:109) and (cid:106) q ∗ v p v k (cid:107) in¯ y v − ( k ) and ¯ y v ( k ), respectively. We perform such modifications as follows.First, for v ∈ Vert(Γ), with Γ specifying an algebraic link exterior, set(207) y v − ± := (cid:40) − q ∗ v p v − / ∈ j ( E in ( v )) y vj ( e (cid:48) ) ± = φ P e (cid:48) ∗ (cid:16) y v ( − e (cid:48) )0 ∓ (cid:17) − j ( e (cid:48) ) , e (cid:48) ∈ E in ( v ) , and for k ∈ Z > , define y v (cid:48)− ( k ) and y v (cid:48)− − ( k ) by setting y v (cid:48)− ( k ) := (cid:40) − (cid:0)(cid:6) y v − k (cid:7) − (cid:1) Y Γ v ( − e (cid:48) ) ( y Γ v ( − e (cid:48) ) ) bi − (cid:4) y v − k (cid:5) Y Γ v ( − e (cid:48) ) ( y Γ v ( − e (cid:48) ) ) bc , (208) y v (cid:48)− − ( k ) := (cid:40) − (cid:0)(cid:4) y v − − k (cid:5) + 1 (cid:1) Y Γ v ( − e (cid:48) ) ( y Γ v ( − e (cid:48) ) ) bi − (cid:6) y v − − k (cid:7) Y Γ v ( − e (cid:48) ) ( y Γ v ( − e (cid:48) ) ) bc , (209)where again, e (cid:48) ∈ E in ( v ) is the unique incoming edge with j ( e (cid:48) ) = − 1, if such e (cid:48) exists. If − / ∈ j ( E in ( v )), then we take Y Γ v ( − e (cid:48) ) ( y Γ v ( − e (cid:48) ) ) to be boundary-compressible.Next, we define ¯ y v (cid:48) − ( k ) and ¯ y v (cid:48) ( k ) to be respective results of replacing the summand (cid:108) q ∗ v p v k (cid:109) with y v (cid:48)− ( k ) in the definition of ¯ y v − ( k ) in (148), and replacing the summand (cid:106) q ∗ v p v k (cid:107) with y v (cid:48)− − ( k ) in the definition of ¯ y v ( k ) in (149). That is, we set¯ y v (cid:48) − ( k ) := ¯ y v − ( k ) + k (cid:16) y v (cid:48)− ( k ) − (cid:108) q ∗ v p v k (cid:109)(cid:17) , (210) ¯ y v (cid:48) ( k ) := ¯ y v − ( k ) + k (cid:16) y v (cid:48)− − ( k ) − (cid:106) q ∗ v p v k (cid:107)(cid:17) , (211)and by analogy with the definition of ¯ y v ± in (206), we define(212) ¯ y v (cid:48) − := sup k> ¯ y v (cid:48) − ( k ) and ¯ y v (cid:48) := inf k> ¯ y v (cid:48) ( k ) . We are now ready to state and prove an analog of Proposition 6.1 and a supplement toProposition 6.2. Proposition 7.1. Suppose v ∈ Vert(Γ) for a graph Γ specifying the exterior of an algebraiclink. If y v − , y v ∈ P ( H ( ∂ Y v ; Z )) sf v are the (potential) L-space interval endpoints for Y Γ v ( y Γ v ) as defined in Theorem 4.3, then y v − = ¯ y v (cid:48) − − (cid:80) j ∈ J bi v (cid:0) (cid:100) y vj + (cid:101)− (cid:1) − (cid:80) j ∈ J bc v (cid:98) y vj (cid:99) − (cid:80) i ∈ I v (cid:98) y vi (cid:99) ,y v = ¯ y v (cid:48) − (cid:80) j ∈ J bi v (cid:0) (cid:98) y vj − (cid:99) +1 (cid:1) − (cid:80) j ∈ J bc v (cid:100) y vj (cid:101) − (cid:80) i ∈ I v (cid:100) y vi (cid:101) , for J bc v and J bi v as defined in (143) and (144).Proof. This follows directly from Theorem 4.3. (cid:3) Proposition 7.2. Suppose that Γ specifies the exterior of an algebraic link, and that v ∈ Vert(Γ) has an incoming edge e (cid:48) with j ( e (cid:48) ) = − . If (213) − q ∗ v p v ≤ y vj ( e (cid:48) )+ ≤ y vj ( e (cid:48) ) − ≤ − q ∗ v p v + m e (cid:48) − , for some m e (cid:48) − ∈ Z > , then ¯ y v (cid:48) − and ¯ y v (cid:48) satisfy (214) ¯ y v (cid:48) − ≤ ¯ y v − , ¯ y v (cid:48) ≥ ¯ y v + m e (cid:48) − . Proof. It is straightforward to show that the bounds in (213) imply that(215) y v (cid:48)− ( k ) ≤ (cid:108) q ∗ v p v k (cid:109) , y v (cid:48)− − ( k ) ≥ (cid:106) q ∗ v p v k (cid:107) + m e (cid:48) − for all k ∈ Z > . Thus, since m e (cid:48) − ∈ Z implies that k (cid:98) m e (cid:48) − k (cid:99) = k (cid:100) m e (cid:48) − k (cid:101) = m e (cid:48) − , the desiredresult follows directly from (210), (211), and the definitions of ¯ y v (cid:48) ∓ in (212). (cid:3) L-space surgery regions for algbraic link satellites: Proof of Theorem 1.6. If Γspecifies a one-component algebraic link, i.e. , a knot, then the L-space region is just an interval,determined by iteratively computing the genus of successive cables. For multi-componentlinks, we can bound the L-space region as described in Theorem 1.6 in the introduction. Proof of Theorem 1.6 .The proofs of parts ( i ) and ( ii ) are the same as those in the iterated torus link satellitecase, if one keeps in mind that the p r = 1 condition for ( ii.b ) and an explicit hypothesis for( ii.a ) each rule out the possibility of an incoming exceptional splice at the root vertex.The proof of part ( iii ) also adapts the proof used for iterated torus satellites, but we providemore details in this case. Again, for bookkeeping convenience, we redefine I w := { } and set L min + sf w := { } and R sf w \ Z sf w := L min − sf w := ∅ , for any w ∈ Vert(Γ) with I w = ∅ .For a vertex v ∈ Vert(Γ), we inductively assume, for each incoming edge e ∈ E in ( v ), thatfor any y Γ v ( − e ) ∈ (cid:81) u ∈ Vert(Γ v ( − e ) ) (cid:0) L min + sf u ∪ R sf u \ Z sf u ∪ L min − sf u (cid:1) , we have µ e + m e + ≤ y vj ( e )+ ≤ y vj ( e ) − ≤ µ e + m e − , (216) µ e + m e + < y vj ( e )+ , y vj ( e ) − < µ e + m e − if Y Γ v ( − e ) ( y Γ v ( − e ) ) is bi , (217)(218) where µ e := φ P e ∗ (cid:18) − q ∗ v ( − e ) − p v ( − e ) (cid:19) = (cid:40) j ( e ) (cid:54) = − − q ∗ v p v j ( e ) = − ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 43 is the meridian slope of the fiber of Y v to which Y v ( − e ) is spliced along e (so the image of thelongitude of slope − q ∗ v ( − e ) − p v ( − e ) paired with the meridian of slope p ∗ v ( − e ) q v ( − e ) ), and where m e + := (cid:40)(cid:108) p v ( − e ) q v ( − e ) (cid:109) − j ( e ) (cid:54) = − j ( e ) = − , (219) m e − := (cid:106) p v ( − e ) q v ( − e ) (cid:107) + 1 j ( e ) (cid:54) = − (cid:108) p v ( − e ) p v ∆ e (cid:109) + 1 j ( e ) = − . (220)We then set m + v := − (cid:88) e ∈ E in ( v ) m e + + 0 = 0 , (221) m − v := − (cid:88) e ∈ E in ( v ) m e − − J v (cid:54) = ∅ ; j ( e v ) (cid:54) = − (cid:108) p v ( ev ) p v ∆ ev (cid:109) + 1 j ( e v ) = − 10 otherwise . (222)The statement of Theorem 1.6 makes the substitution (cid:106) p v ( − e ) q v ( − e ) (cid:107) + 1 (cid:55)→ j ( e ) (cid:54) = − m e − , in its role as a summand of of m − v . However, this substitution is an equivalence forall v ∈ Vert(Γ) and e ∈ E in ( v ), since the algebraicity condition (200) implies 0 < p v ( − e ) q v ( − e ) < j ( e ) (cid:54) = − 1. Note that this does not imply 0 < p r q r < r = v ( − e r ), becauseof our declared convention that v ( e r ) = null / ∈ Vert(Γ). The algebraicity condition (200)also implies that the conditions q = +1, q < 0, and p v q v > +1 are never met for j ( e v ) (cid:54) = − v (cid:54) = r . Thus, the j ( e ) (cid:54) = − L min − sf v and L min + sf v in Theorem 1.6. iii coincide with the respective definitions of L min − sf v and L min + sf v in Theorem 1.7. iii .If y v ∈ R sf u \ Z sf u , then y v − = y v = ∞ . Thus, y v ( e v ) j ( e v ) ± = φ P e v ∗ ( y v ∓ ) = φ P e v ∗ ( ∞ ) =: ξ v ( e v ) ,and referring to (205) for the computation of ξ v ( e v ) , we have(223) y v ( e v ) j ( e v ) ± = ξ v ( e v ) = µ e v + p v q v j ( e v ) (cid:54) = − p v p v ( e v ) ∆ e v j ( e v ) = − ∈ (cid:10) µ e v + m e v + , µ e v + m e v − (cid:11) . We assume y v ∈ L min − sf v ∪ L min + sf v for the remainder. This assumption, together with ourinductive assumptions, makes Theorem 4.4 yield(224) y v ≤ y v − , and Proposition 6.2 tells us(225) ¯ y v − ≤ q ∗ v p v ≤ ¯ y v , with equality only if Y Γ v ( y Γ v ) is bc . We furthermore already know that ¯ y v ± = ¯ y v (cid:48) ± when − / ∈ j ( E in ( v )). Combining this fact withProposition 7.2, given our inductive assumptions, yields(226) ¯ y v (cid:48) − ≤ ¯ y v − , ¯ y v (cid:48) ≥ ¯ y v + (cid:40) m e (cid:48) − ∃ e (cid:48) ∈ E in ( v ) , j ( e (cid:48) ) = − − / ∈ j ( E in ( v )) . Suppose y v ∈ L min + sf v . Then from Proposition 7.1, we have y v − ≤ ¯ y v (cid:48) − − (cid:80) j ∈ J v y vj + − (cid:80) i ∈ I v (cid:98) y vi (cid:99) (227) ≤ ¯ y v (cid:48) − − (cid:80) j ∈ J v m e + − m + v (228) = ¯ y v (cid:48) − ≤ ¯ y v − ≤ q ∗ v p v . (229)Thus, altogether we have(230) y v ≤ y v − ≤ q ∗ v p v < η v , with equality only if Y Γ v ( y Γ v ) is bc . Here, η v := ( φ P e v ∗ ) − ( ∞ ) is the location of the vertical asymptote of φ P e v ∗ . The inequality q ∗ v p v < η v follows directly from the computation of η in (205), plus the fact that q ∗ v p v < p ∗ v q v . Since φ P e v ∗ is locally monotonically decreasing on the complement of η v , this implies that all theexpressions on the left-hand side of (230) have φ P e v ∗ -images below the horizontal asymptoteat ξ v ( e v ) , but in reverse order. That is, we have(231) φ P e v ∗ (cid:16) q ∗ p v (cid:17) := µ e v ≤ y v ( e v ) j ( e v )+ := φ P e v ∗ ( y v − ) ≤ y v ( e v ) j ( e v ) − := φ P e v ∗ ( y v ) < ξ v ( e v ) . Thus, since m e v + = 0 and since (223) shows that ξ v ( e v ) < µ e v + m e v − , we obtain(232) µ e v + m e v + ≤ y v ( e v ) j ( e v )+ ≤ y v ( e v ) j ( e v ) − < µ e v + m e v − , with equality only if Y Γ v ( y Γ v ) is bc .Lastly, suppose y v ∈ L min − sf v . Then combining Proposition 7.1 (for line (233)) with therighthand inequality of (226), the inductive upper bounds on y vj for j ∈ J v := j ( E in ( v )) | > ,and the upper bound (cid:80) i ∈ I v (cid:100) y vi (cid:101) ≤ m − v for y v ∈ L min − sf v (for line (234)), we obtain y v ≥ ¯ y v (cid:48) − (cid:80) j ∈ J v y vj − − (cid:80) i ∈ I v (cid:100) y vi (cid:101) , (233) ≥ ¯ y v − (cid:80) e ∈ E in ( v ) m e − − m − v . (234)Combining (225) from Proposition 6.2 with the definition (222) of m − v then gives y v ≥ q ∗ v p v + J v (cid:54) = ∅ ; j ( e v ) (cid:54) = − (cid:108) p v ( ev ) p v ∆ ev (cid:109) + 1 j ( e v ) = − 10 otherwise , (235)with equality only if Y Γ v ( y Γ v ) is bc . When j ( e v ) (cid:54) = − 1, the desired inductive result is estab-lished in the y v ∈ L min − sf v case of the proof of Theorem 1.7. We henceforth assume j ( e v ) = − j ( e v ) = − q ∗ v p v + (cid:108) p v ( e v ) p v ∆ e v (cid:109) + 1 > η v , we have y v − ≥ y v > η v , i.e. , to the rightof the vertical asymptote of φ P e v ∗ at η v . The respective φ P e v ∗ -images y v ( e v ) j ( e v )+ and y v ( e v ) j ( e v ) − of y v − and y v therefore lie above the horizontal asymptote at ξ v ( e v ) , but with reversed order:(236) µ e v + m e v + = − q ∗ v ( e v ) p v ( e v ) < ξ v ( e v ) < y v ( e v ) j ( e v )+ ≤ y v ( e v ) j ( e v ) − . It remains to show that µ e v + m e v − ≥ y v ( e v ) j ( e v ) − := φ P e v ∗ ( y v ) (with equality only if Y Γ v ( y Γ v ) is bc ), for which it suffices to show that ( µ e v + m e v − ) − φ P e v ∗ (cid:16) q ∗ v p v + (cid:108) p v ( e v ) p v ∆ e v (cid:109) + 1 (cid:17) ≥ ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 45 Recall that any edge e with j ( e ) = − φ P e ∗ = σ P e ∗ . If we write (cid:2) σ P e ∗ (cid:3) =: (cid:18) α e ∆ (cid:48) e ∆ e β e (cid:19) forthe entries of the matrix (cid:2) σ P e ∗ (cid:3) as computed in (198), then the relations p ∗ u p u − q ∗ u q u = 1 for each u ∈ Vert(Γ), particularly for u ∈ { v, v ( e v ) } , produce simplifications, incuding the identities(237) p v ( e v ) α e v + q ∗ v ( e v ) ∆ e v = p v , p v β e v + q ∗ v ∆ e v = − p v ( e v ) , q ∗ v α e v + p v ∆ (cid:48) e v = q ∗ v ( e v ) , used in the intermediate steps suppressed in the following calculation. We compute that( µ e v + m e v − ) − φ P e v ∗ (cid:16) q ∗ v p v + (cid:108) p v ( e v ) p v ∆ e v (cid:109) + 1 (cid:17) = (cid:32) − q ∗ v ( e v ) p v ( e v ) + (cid:24) p v p v ( e v ) ∆ e v (cid:25) + 1 (cid:33) − α e v (cid:16) p v ∆ e v + (cid:2) − p v ( e v ) (cid:3) p v ∆ ev (cid:17) + p v ∆ e v (cid:16) p v ∆ e v + (cid:2) − p v ( e v ) (cid:3) p v ∆ ev (cid:17) (238) = 1 + (cid:20) p v p v ( e v ) ∆ e v (cid:21) − ∆ − e v (cid:18) (cid:20) − p v ( e v ) p v ∆ e v (cid:21)(cid:19) − (239) ≥ , (240)since [ x ] := x − (cid:98) x (cid:99) implies 0 ≤ [ x ] < x ∈ Q , and this completes our inductive argument.Since j ( e r ) (cid:54) = − 1, the proof that these inductive bounds cause the Dehn-filled satelliteexterior Y Γ ( y Γ ) := Y Γ ( y Γ ) ∪ ( S \ ◦ ν ( K )) to form an L-space whenever(241) y Γ ∈ (cid:89) v ∈ Vert(Γ r ) (cid:0) L min + sf v ∪ R sf v \ Z sf v ∪ L min − sf v (cid:1) is the same as the corresponding argument in the proof of Theorem 1.7. (cid:3) Monotone strata. Whether for iterated torus satellites and for algebraic link satellites,our inner approximations L min sf Γ ( Y Γ ) := (cid:81) v ∈ Vert(Γ) (cid:0) L min − sf v ∪ R v \ Z v ∪ L min + sf v (cid:1) each involveinductive bounds, namely, (179) and (216), respectively, which make y v ( e ) j ( e )+ ≤ y v ( e ) j ( e ) − , as afunction of y Γ | Γ v ( − e ) , for each edge e ∈ Edge(Γ) and slope y Γ ∈ L min sf Γ ( Y Γ ). This implies that L min sf Γ ( Y Γ ) is confined to a particular substratum of L sf Γ ( Y Γ ), called the monotone stratum . Definition 7.3. For any y Γ ∈ ( Q ∪ {∞} ) | I Γ | sf Γ and v ∈ Vert(Γ) , we call y Γ monotone at v if (242) ∞ ∈ φ P e ∗ L ◦ v ( − e ) ( y ) ∀ e ∈ E in ( v ) and ∞ ∈ φ P e v ∗ L ◦ v ( y ) . The monotone stratum L mono sf Γ ( Y Γ ) of L sf Γ ( Y Γ ) is then the set of slopes y Γ ∈ L sf Γ ( Y Γ ) suchthat y Γ is monotone at all v ∈ Vert(Γ) . In the above, for brevity, we have adopted the following Notation 7.4. For any slope y Γ ∈ ( Q ∪ {∞} ) | I Γ | sf Γ and vertex v ∈ Vert(Γ) , we shall write (243) L v ( y ) := L sf v ( Y Γ v ( y Γ | Γ v )) , L ◦ v ( y ) := L ◦ sf v ( Y Γ v ( y Γ | Γ v )) . Remark. Note that if L ◦ v ( − e ) ( y ) (cid:54) = ∅ for all e ∈ E in ( v ) and L ◦ v ( y ) (cid:54) = ∅ , then φ P e ∗ L v ( − e ) ( y ) = [[ y vj ( e ) − , y vj ( e )+ ]] ∀ e ∈ E in ( v ) , φ P e v ∗ L v ( y ) = [[ y v ( e v ) j ( e v ) − , y v ( e v ) j ( e v )+ ]] , and the monotonicity condition (242) at v is equivalent to the condition that y vj ( e )+ ≤ y vj ( e ) − ∀ e ∈ E in ( v ) , y v ( e v ) j ( e v )+ ≤ y v ( e v ) j ( e v ) − , corresponding to the endpoint-ordering consistent with that for generic Seifert fibered L-spaceintervals. We call this condition “monotonicity” because of its preservation of this ordering.The tools developed in Sections 6 and 7 can be used in much more general settings thanthat of the inner approximation theorems we proved, so long as one first decomposes L sf Γ ( Y Γ )into strata according to monotonicity conditions, similar to how torus link satellites must firstbe classified according to whether 2 g ( K ) − ≤ qp . Monotonicity conditions also impact thetopology of strata. Theorem 7.5. Suppose that K Γ ⊂ S is an algebraic link satellite, specified by Γ , of a positiveL-space knot K ⊂ S , where either K is trivial, or K is nontrivial with q r p r > g ( K ) − . Let V ⊂ Vert(Γ) denote the subset of vertices v ∈ V for which | I v | > .Then the Q -corrected R -closure L mono sf Γ ( Y Γ ) R of the monotone stratum of L sf Γ ( Y Γ ) is ofdimension | I Γ | and deformation retracts onto an ( | I Γ | − | V | ) -dimensional embedded torus, (244) L mono sf Γ ( Y Γ ) R → (cid:81) v ∈ V T | I v |− (cid:44) → (cid:81) v ∈ V ( R ∪ {∞} ) | I v | sf v , projecting to an embedded torus T | I v |− (cid:44) → ( R ∪ {∞} ) | I v | sf v parallel to B sf v ⊂ ( R ∪ {∞} ) | I v | sf v ateach v ∈ V .Proof. We argue by induction, recursing downward from the leaves of Γ towards its root.Observe that for any v ∈ Vert(Γ), we have the fibration(245) L mono sf Γ v ( Y Γ v ) −→ (cid:81) e ∈ E in ( v ) L mono sf Γ v ( − e ) ( Y Γ v ( − e ) )with fiber(246) T v y ∗ := (cid:40) y Γ v ∈ L mono sf Γ v ( Y Γ v ) (cid:12)(cid:12) y Γ v (cid:12)(cid:12)(cid:12) (cid:81) e ∈ E in( v ) Γ v ( − e ) = y ∗ (cid:41) over y ∗ ∈ (cid:81) e ∈ E in ( v ) L mono sf Γ v ( − e ) ( Y Γ v ( − e ) ) for I v (cid:54) = ∅ , with T v y ∗ regarded as a point when I v = ∅ .For v ∈ Vert(Γ), inductively assume the theorem holds for Γ v ( − e ) for all e ∈ E in ( v ). (Notethat this holds vacuously when v is a leaf, in which case we declare T v ∅ := L mono sf Γ v ( Y Γ v ).)If I v = ∅ , then the fibration in (245) is the identity map, making the theorem additionallyhold for Γ v . Next assuming I v (cid:54) = ∅ , we claim the Q -corrected R -closure ( T v y ∗ ) R of T v y ∗ is ofdimension | I v | and deformation retracts onto an embedded torus T | I v |− (cid:44) → ( R ∪ {∞} ) | I v | sf v parallel to B sf v ⊂ ( R ∪ {∞} ) | I v | sf v . In fact, the proof of this statement is nearly identical to theproof of Theorem 5.3 .ii.b in Section 5, but with the replacement(247) N := { y ∈ Q n sf | y + ( y ) < < y − ( y ) } −→ N v := { y v ∈ Q | I v | sf v | y v ( y v ) < η v < y v − ( y v ) } in line (123) (where η v , computed in (205), is the position of the vertical asymptote of φ P e v ∗ ),along with a few minor analogous adjustments corresponding to this change.It remains to show that the fibration in (245) is trivial, but this follows from the fact that(248) L min sf Γ v ( Y Γ v ) := (cid:89) u ∈ V ∩ Vert(Γ v ) (cid:0) L min − sf u ∪ R u \ Z u ∪ L min + sf u (cid:1) is a product over u ∈ V ∩ Vert(Γ v ) which embeds into L mono sf Γ ( Y Γ v ), and for reasons again similarto the proof of Theorem 5.3 .ii.b , each factor L min − sf u ∪ R u \ Z u ∪ L min + sf u also deformationretracts onto an embedded torus T | I u |− (cid:44) → ( R ∪ {∞} ) | I u | sf u parallel to B sf u ⊂ ( R ∪ {∞} ) | I u | sf u ,completing the proof. (cid:3) ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 47 Extensions of L-space Conjecture Results As mentioned in the introduction, Boyer-Gordon-Watson [4] conjectured several years agothat among prime, closed, oriented 3-manifolds, L-spaces are those 3-manifolds whose funda-mental groups do not admit a left orders (LO). Similarly, Juh´asz [18] conjectured that prime,closed, oriented 3-manifold are L-spaces if and only if they fail to admit a co-oriented tautfoliation (CTF). Procedures which generate new collections of L-spaces or non-L-spaces, suchas surgeries on satellites, provide new testing grounds for these conjectures.For Y a compact oriented 3-manifold with boundary a disjoint union of n > F ( Y ) , LO ( Y ) ⊂ (cid:81) ni =1 P ( H ( ∂ i Y ; Z )) so that F ( Y ) := (cid:40) α ∈ n (cid:89) i =1 P ( H ( ∂ i Y ; Z )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y admits a CTF F such that F | ∂Y is the product foliation of slope α . (cid:41) , (249) LO ( Y ) := (cid:40) α ∈ n (cid:89) i =1 P ( H ( ∂ i Y ; Z )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π ( Y ( α )) is LO. (cid:41) . (250)Note that α ∈ F ( Y ) implies that Y ( α ) admits a CTF, but the converse, while true for Y agraph manifold, is not known in general.8.1. Proof of Theorem 1.4 and Generalizations. The proof of Theorem 1.4 relies on therelated gluing behavior of co-oriented taut foliations, left orders on fundamental groups, andthe property of being an non-L-space, for a pair Y , Y of compact oriented 3-manifolds withtorus boundary glued together via a gluing map ϕ : ∂Y → ∂Y .That is, the contrapositives of Theorems 3.5 and 3.6 tell us that if Y i have incompressibleboundaries and are both Floer simple manifolds, both graph manifolds, or an L-space knotexterior and a graph manifold, then(251) ϕ P ∗ ( N L ( Y )) ∩ N L ( Y ) (cid:54) = ∅ ⇐⇒ Y ∪ ϕ Y not an L-space.The analogous statements for CTFs and LOs, while true for graph manifolds (once an excep-tion is made for reducible slopes in the case of CTFs), are not established in general. However,we still have weak gluing statements in the general case. Since product foliations of matchingslope can always be glued together, we have(252) ϕ P ∗ ( F ( Y )) ∩ F ( Y ) (cid:54) = ∅ = ⇒ Y ∪ ϕ Y , if prime, admits a CTF.Moreover, Clay, Lidman, and Watson [6] built on a result of Bludov and Glass [1] to show that(253) ϕ P ∗ ( LO ( Y )) ∩ LO ( Y ) (cid:54) = ∅ = ⇒ π ( Y ∪ ϕ Y ) is LO. Theorem 8.1. Suppose Y Γ is the exterior of an algebraic link satellite or (possibly-iterated)torus-link satellite of a nontrivial positive L-space knot K ⊂ S of genus g ( K ) and exterior Y , with p r > and − / ∈ j ( E in ( r )) . ( LO ) Suppose LO ( Y ) ⊃ N L ( Y ) . ( lo .i ) If g ( K ) − > q r +1 p r , then LO ( Y Γ ) = N L ( Y Γ ) . ( lo .ii ) If g ( K ) − < q r p r and Γ = r specifies a torus link satellite, then LO ( Y Γ ) ⊃ ( N L ( Y Γ ) \ R ( Y Γ )) \ Λ( Y Γ ) · ([ −∞ , N Γ (cid:105) n \ [ −∞ , N Γ − p r (cid:105) n ) , where N Γ := p r q r − q r − p r + 2 g ( K ) p r . ( CTF ) Suppose F ( Y ) = N L ( Y ) . ( ctf .i ) If g ( K ) − > q r +1 p r , then F ( Y Γ ) = N L ( Y Γ ) \ R ( Y Γ ) . ( ctf .ii ) If g ( K ) − < q r p r and Γ = r specifies a torus link satellite, then F ( Y Γ ) ⊃ ( N L ( Y Γ ) \ R ( Y Γ )) \ Λ( Y Γ ) · ([ −∞ , N Γ (cid:105) n \ [ −∞ , N Γ − p r (cid:105) n ) , Note that the requirement that K be nontrivial is just to simplify the theorem statement. If K is trivial, then any surgery on Y Γ is a graph manifold or a connected sum thereof, in whichcase the L-space conjectures written down by Boyer-Gordon-Watson and Juh´asz are alreadyproven to hold, through the work of Boyer and Clay [3] and of Hanselman, J. Rasmussen,Watson, and the author [12]. In addition, the author explicitly shows in [27] that any graphmanifold Y Γ always satisfies(254) LO (Γ) = N L ( Y Γ ) , F ( Y Γ ) = N L ( Y Γ ) \ R ( Y Γ ) . Proof of Theorem. Suppose that LO ( Y ) ⊃ N L ( Y ) (respectively F ( Y ) = N L ( Y )). Since(255) Y Γ = Y Γ ∪ φ er Y and ( φ P e ∗ ) − ( L ( Y )) = (cid:104) q ∗ r p r − p r ( p r N − q r ) , q ∗ r p r (cid:105) sf , where N := 2 g ( K ) − 1, it follows from (253) (respectively (252)) that in order to prove that y Γ ∈ LO ( Y Γ ) (respectively y Γ ∈ F ( Y Γ ) ∪ Z ( Y Γ )), it suffices to show that(256) NL sf ( Y Γ ( y Γ )) ∩ (cid:16)(cid:104) −∞ , q ∗ r p r − p r ( p r N − q r ) (cid:69) ∪ (cid:68) q ∗ r p r , + ∞ (cid:105)(cid:17) (cid:54) = ∅ . On the other hand, (251) implies that y Γ ∈ N L ( Y Γ ) if and only if(257) NL sf ( Y Γ ( y Γ )) ∩ (cid:16)(cid:104) −∞ , q ∗ r p r − p r ( p r N − q r ) (cid:105) ∪ (cid:104) q ∗ r p r , + ∞ (cid:105)(cid:17) (cid:54) = ∅ when Y Γ ( y Γ ) is bi , and if and only if (256) holds when Y Γ ( y Γ ) is bc .Fix some slope y Γ ∈ ( Q ∪ {∞} ) (cid:80) v ∈ Vert(Γ) | I v | and write(258) L sf ( Y Γ ( y Γ )) = [[ y − , y ]] , as usual. It is straightforward to show that (257) fails to hold if and only if(259) q ∗ r p r − p r ( p r N − q r ) < y ≤ y − < q ∗ r p r , and that (256) fails to hold if and only if(260) q ∗ r p r − p r ( p r N − q r ) ≤ y ≤ y − ≤ q ∗ r p r . Note that p r > ≤ q ∗ r p r − p r ( p r N − q r ) < q ∗ r p r < , with q ∗ r p r − p r ( p r N − q r ) = 0 ⇐⇒ N = q r + 1 p r . We begin by proving the following claim. Claim. If N := 2 g ( K ) − > q r p r , p r > , and − / ∈ j ( E in ( r )) , then (262) Y Γ ( y Γ ) = S ⇐⇒ (cid:26) (259) holds if Y Γ ( y Γ ) is bi , (260) holds if Y Γ ( y Γ ) is bc , If, in addition, N := 2 g ( K ) − (cid:54) = q r +1 p r , then (263) (260) holds ⇐⇒ (cid:26) (259) holds if Y Γ ( y Γ ) is bi , (260) holds if Y Γ ( y Γ ) is bc , ATIONAL L-SPACE SURGERIES ON SATELLITES BY ALGEBRAIC LINKS 49 Proof of Claim. Suppose N > q r p r , p r > 1, and − / ∈ j ( E in ( r )). Then Proposition 6.1together with Proposition 6.2.(=) imply that(264) y ∈ q ∗ r p r + Z ⇐⇒ y − ∈ q ∗ r p r + Z ⇐⇒ Y Γ ( y Γ ) is bc = ⇒ y − = y , so that(265) (260) holds and Y Γ ( y Γ ) is bc ⇐⇒ y = q ∗ r p r ⇐⇒ y − = q ∗ r p r = ⇒ Y Γ ( y Γ ) = S . Thus, since the fact that S is an L-space makes the = ⇒ implication of (262) automaticallyhold, this exhausts the case when Y Γ ( y Γ ) is bc . Next suppose that Y Γ ( y Γ ) is bi , so thatProposition 6.2.(+) implies y ∈ (cid:68) q ∗ r p r , (cid:105) + Z . Then (261) implies that (259) always fails tohold, and that (260) fails to hold if N (cid:54) = q r +1 p r , completing the proof of the claim.Continuing with the proof of the theorem, since the right-hand condition of (262) and(263) is equivalent to Y Γ ( y Γ ) being an L-space, and since (260) is the negation of (256), wehave shown that (256) holds if and only if y Γ ∈ N L ( Y Γ ), proving that LO ( Y Γ ) ⊃ N L ( Y Γ )(respectively F ( Y Γ ) ⊃ N L ( Y Γ ) \ R ( Y Γ )) if LO ( Y ) = N L ( Y ) (respectively F ( Y ) = N L ( Y )),with N > q r p r , p r > 1, and − / ∈ j ( E in ( r )). Since S has no co-oriented taut foliationsor left-orders on its fundamental group, we then have LO ( Y Γ ) = N L ( Y Γ ) (respectively F ( Y Γ ) = N L ( Y Γ ) \ R ( Y Γ )).This leaves the case in which we have a single T ( np, nq ) torus-link satellite, with N :=2 g ( K ) − < qp . Arguments similar to those above then show that if LO ( Y ) = N L ( Y )(respectively F ( Y ) = N L ( Y )), then y Γ ∈ N L ( Y Γ ) implies that y Γ ∈ LO ( Y Γ ) (respectively y Γ ∈ F ( Y Γ ) ∪ Z ( Y Γ )), provided that(266) y (cid:54) = q ∗ p + p ( q − pN ) (cid:16) = p ∗ − q ∗ Nq − pN (cid:17) . Propositions 6.1 and 6.2.(+) then tell us that y = p ∗ − q ∗ Nq − pN implies(267) n (cid:88) i =1 (cid:100) y i (cid:101) = 0 , n (cid:88) i =1 (cid:98) [ − y i ]( q − N p ) (cid:99) = 0 , n (cid:88) i =1 (cid:98) [ − y i ]( q − ( N − p ) (cid:99) > , which, under change of basis to S -slopes, becomes(268) Λ( Y Γ ) · ([ −∞ , pq − q + pN (cid:105) n \ [ −∞ , pq − q + pN − p (cid:105) n ) . Since pq − q + pN = pq − q − p + 2( K ) p , the theorem follows. (cid:3) Exceptional Symmetries. As mentioned in the introduction, there are instances, forexteriors of iterated torus-link satellites or algebraic link satellites, in which the Λ-type symme-tries for Seifert fibered components have their influence extend across edges. This phenomenonis more relevant in the context of exceptional splices. Proposition 8.2. Suppose Y Γ is the exterior of an algebraic link satellite K Γ ⊂ S of anontrivial positive L-space knot K ⊂ S of genus g ( K ) , with N := 2( g ) K − > q r p r and − ∈ j ( E in ( r )) . Then L ( Y Γ ) = (cid:83) e ∈ E in ( r ) L e , where L e := y Γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y Γ v ( − e (cid:48) ) ( y Γ v ( − e (cid:48) ) ) is bc , with y rj ( e (cid:48) ) ± ∈ Z , for all e (cid:48) ∈ E in ( r ) , q ∗ r p r − p r ( p r N − q r ) < y ≤ y − < q ∗ r p r if Y Γ v ( − e ) ( y Γ v ( − e ) ) is bi , q ∗ r p r − p r ( p r N − q r ) ≤ y ≤ y − ≤ q ∗ r p r if Y Γ v ( − e ) ( y Γ v ( − e ) ) is bc . Proof. This is established by a straightforward but tedious adaptation of the arguments usedto prove Claim 1 in the proof of Theorem 8.1. (cid:3) Note that the latter two conditions place strong constraints on y r as well. In particular,we must have y r ∈ Z | I v | unless Y Γ v ( − e ) ( y Γ v ( − e ) ) is bc with y j ( e ) ± ∈ Z , in which case y ri ∈ Z all but at most one i ∈ I v .This phenomenon also affects Λ Γ as defined in (11). While Λ Γ ⊃ (cid:81) v ∈ Vert(Γ) Λ v , this contain-ment is proper if there is an edge e ∈ Edge(Γ) for which one can have Y Γ v ( − e ) ( y Γ v ( − e ) ) bc with0 (cid:54) = y v ( e ) j ( e ) ± ∈ Z if j ( e ) (cid:54) = − 1, or any integer value y v ( e ) j ( e ) ± ∈ Z if j ( e ) = − 1. For a satellite withoutexceptional splices, the situation is still relatively simple. 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