Seifert fibered and reducible surgeries on hyperbolic fibered knots
aa r X i v : . [ m a t h . G T ] J u l Seifert fibered and reducible surgeries onhyperbolic fibered knots
Yi NI
Department of Mathematics, Caltech, MC 253-371200 E California Blvd, Pasadena, CA 91125
Email : [email protected]
Abstract
Let K ⊂ S be a hyperbolic fibered knot such that S p/q ( K ), the pq –surgery on K , is either reducible or an atoroidal Seifert fibered space. Weprove that if the monodromy of K is right-veering, then 0 ≤ pq ≤ g ( K ).The upper bound 4 g ( K ) cannot be attained if K is an L-space knot. Ifthe monodromy of K is neither right-veering nor left-veering, then | q | = 1and the surgery is irreducible.As a corollary, for any given positive torus knot T , if p/q ≥ g ( T ) + 4,then p/q is a characterizing slope. This improves earlier bounds of Ni–Zhang and McCoy. We also prove that some finite/cyclic slopes are char-acterizing. More precisely, 14 is characterizing for T , , 17 is characterizingfor T , , and 4 n + 1 is characterizing for T n +1 , except when n = 5. Bya recent theorem of Tange, this shows that T n +1 , is the only knot in S admitting a lens space surgery while the Alexander polynomial has theform t n − t n − + t n − + · · · .In the appendix, we prove that if the rank of the second term of theknot Floer homology of a fibered knot is 1, then the monodromy is eitherright-veering or left-veering. Given a knot K ⊂ S , let S p/q ( K ) be the manifold obtained by pq –surgery on K . The main theorem in this paper gives a contraint on Seifert fibered surgerieson hyperbolic fibered knots. Theorem 1.1.
Let K ⊂ S be a hyperbolic fibered knot such that S p/q ( K ) iseither reducible or an atoroidal Seifert fibered space.(1) If the monodromy of K is right-veering, then ≤ pq ≤ g ( K ) . If K is alsoan L-space knot, then pq = 4 g ( K ) .(2) If the monodromy of K is neither right-veering nor left-veering, then | q | = 1 and S p/q ( K ) is irreducible. emark 1.2. If a hyperbolic knot K ⊂ S admits a reducible surgery, thenthe surgery slope is an integer p with 2 ≤ | p | ≤ g ( K ) − Y is atoroidal if it does not contain an incom-pressible torus. Some authors use the term “atoroidal” to mean that π ( Y ) doesnot contain a Z subgroup, which is not the assumption we want.Right-veering diffeomorphisms were defined in [25]. (This concept was origi-nally conceived by Gabai.) Let F be a compact oriented surface with boundary, a, b ⊂ F be two properly embedded arcs with a (0) = b (0) = x ∈ ∂F . We isotope a, b with endpoints fixed, so that | a ∩ b | is minimal. We say b is to the right of a at x , if either b is isotopic to a with endpoints fixed, or ( b ∩ U ) \ { x } lies inthe “right” component of U \ a , where U ⊂ F is a small neighborhood of x . SeeFigure 1. A diffeomorphism φ : F → F with φ | ∂F = id is right-veering if for ev-ery x ∈ ∂F and every properly embedded arc a ⊂ F with x ∈ a , the image φ ( a )is to the right of a at x . Similarly, we can define left-veering diffeomorphisms. a bx Figure 1: The arc b is to the right of a at x .A rational homology sphere Y is an L-space if rank d HF ( Y ) = | H ( Y ) | . Aknot K ⊂ S is an L-space knot , if there exists a slope pq > S p/q ( K )is an L-space. Remark 1.3.
Theorem 1.1 was more or less known to experts. The point wewant to make is that this theorem becomes very useful when combined withHeegaard Floer homology. First of all, knot Floer homology detects whethera knot is fibered [19, 33]. Secondly, the right-veering condition can often bechecked using Heegaard Floer homology. Honda–Kazez–Mati´c [25] proved thatif an open book supports a tight contact structure, then the correspondingmonodromy is right-veering. It is often possible to get tightness from HeegaardFloer homology [40]. For example, when K is an L-space knot, then K is fibered,and the open book with binding K supports the unique tight contact structureon S , thus the monodromy of K is right-veering. In the appendix, we willprove that if rank \ HF K ( S , K, g ( K ) −
1) = 1 for a fibered knot K , then itsmonodromy is either right-veering or left-veering.Theorem 1.1 can be used to get bounds on characterizing slopes for torusknots. A slope pq ∈ Q is a characterizing slope for a knot L ⊂ S , if S p/q ( K ) ∼ = S p/q ( L ) implies K = L , where “ ∼ =” stands for orientation-preserving homeo-morphism. A famous theorem of Kronheimer–Mrowka–Ozsv´ath–Szab´o [27] saysthat all rational numbers are characterizing slopes for the unknot, confirming a2onjecture of Gordon. Special cases of this theorem were proved in [12, 14, 21],and a Heegaard Floer proof of this theorem was given in [41]. Ozsv´ath andSzab´o [42] further proved that all rational numbers are characterizing slopes forthe trefoils and the figure-8 knot. These knots are the only knots for which thesets of characterizing slopes are known.Dehn surgeries on torus knots have been classified in [32]. For the torusknot T r,s , r, s >
0, the rs –surgery is a connected sum of two lens spaces. If pq = rs ± q , the pq –surgery will be the lens space L ( p, qs ), where the conventionfor L ( p, q ) is that it is oriented as the pq –surgery on the unknot. For all otherslopes, the pq –surgery is a Seifert fibered space over S ( r, s, | ps − qr | ).Ni and Zhang [34] proved that pq is characterizing for the torus knot T r,s whenever pq > r − s − , where r, s >
0. McCoy [30] improved this lowerbound to ( rs − r − s ) which is linear in terms of the genus. As a corollary ofTheorem 1.1, we improve the lower bound. Corollary 1.4.
Let r, s > be a pair of relatively prime integers. Then a slope pq is characterizing for the torus knot T r,s whenever pq ≥ g ( T r,s ) + 4 = 2( r − s −
1) + 4 . Note that Corollary 1.4 contains one previous unknown case of finite surgery:the 13–surgery on T , , which is an I –type spherical manifold. Remark 1.5.
The bound 4 g ( T r,s ) + 4 comes from a theorem of McCoy [30,Theorem 1.1], which says g ( K ) = g ( T r,s ) if S p/q ( K ) ∼ = S p/q ( T r,s ) and pq ≥ g ( T r,s ) + 4. As mentioned in [30, Remark 3.9], it is possible to lower thebound 4 g ( T r,s ) + 4. Thus it is conceivable to lower the bound in Corollary 1.4to 4 g ( T r,s ). However, this cannot hold for all torus knots, since S ( T , ) ∼ = S ( T , ) [34, Example 1.1], and 21 = 4 g ( T , ) + 1. So we will not seek asharper bound for simplicity. Instead, we will focus on the surgeries with finite π not covered by Corollary 1.4. Corollary 1.6.
The slope is a characterizing slope for T , , and the slope is a characterizing slope for T , . A particularly interesting case is s = 2, r = 2 n + 1, where n = g ( T r, ). Theslopes 4 n + t , t = 0 , , , ,
4, are special. The (4 n + 2)–surgery is reducible. The(4 n + 1) and (4 n + 3)–surgeries are lens spaces L (4 n + 1 ,
4) and L (4 n + 3 , n and (4 n + 4)–surgeries are prism manifolds. It follows from a theoremof Greene [23] that 4 n + 2 is characterizing. Baker [1] proved that if S p ( K ) isa lens space with p ≥ g ( K ) −
1, then K is a doubly primitive knot. Buildingon Baker’s work, Rasmussen [44] showed that 4 n + 3 is characterizing. The factthat 4 n and 4 n + 4 are characterizing slopes was proved by Ni–Zhang [35].Greene [22] proved that the list of doubly primitive knots given by Berge [3]is complete. Combined with Baker’s work, one can answer the question whether4 n + 1 is a characterizing slope for T n +1 , , but the author did not find such3tatement in the literature. In this paper, we will provide an answer to thisquestion without using Baker’s work. Theorem 1.7.
If the (4 n + 1) –surgery on a knot K ⊂ S is L (4 n + 1 , , theneither K = T n +1 , or n = 5 and K has the same knot Floer homology as T , . Using a recent result of Tange [45], we also get the following characterizationof T n +1 , among all knots with lens space surgeries. Corollary 1.8.
If a knot K ⊂ S admits a positive lens space surgery and ∆ K ( t ) = t n − t n − + t n − + · · · , then K = T n +1 , .Proof. Suppose that the p –surgery on K is a lens space L ( p, q ). By [45], L ( p, q )is also the p –surgery on T n +1 , . Our conclusion follows from Theorem 1.7 and[44].This paper is organized as follows. In Section 2, we briefly recall someresults about essential laminations and changemaker lattices. In Section 3,we prove Theorem 1.1 using essential laminations. Corollaries 1.4 and 1.6 arealso proved. In Section 4, we prove Theorem 1.7 using Greene’s theory ofchangemaker lattices. In the appendix, we prove Theorem A.1, which says thatif the second term of the knot Floer homology of a fibered knot has rank 1, thenthe monodromy is either right-veering or left-veering. Acknowledgements.
The author was partially supported by NSF grant num-bers DMS-1252992 and DMS-1811900. The author wishes to thank David Gabaifor commenting on an earlier draft of this paper, and referring the author tolots of work on laminations.
In this section, we will collect some results about essential laminations andchangemaker lattices we will use.
Essential laminations were introduced by Gabai and Oertel [18] as a gener-alization of incompressible surfaces. There are two special classes of essentiallaminations: very full laminations (defined by Gabai and Mosher, see [10]) andgenuine laminations [17]. We will not give the definitions here. Instead, we willlist the necessary results.The first result is in Gabai and Oertel’s original paper [18, Theorem 6.1].
Theorem 2.1 (Gabai–Oertel) . If a closed oriented –manifold Y contains anessential lamination, then the universal cover of Y is R . Theorem 2.2 (Brittenham, Claus) . If a Seifert fibered space with base orbifold S ( a, b, c ) contains an essential lamination, then it also contains a taut foliation.Moreover, this manifold does not contain any genuine lamination. Let K ⊂ S be a hyperbolic knot. If λ is a very full lamination in the knotcomplement, one can define the degeneracy locus d ( λ ) which is in the form mn .Here m, n are integers which are not necessarily relatively prime, ( m, n ) = (0 , K is fibered, the stable lamination transverse to the fibration is veryfull. In this case, d ( λ ) ∈ Q is also known as the fractional Dehn twist coefficient of the monodromy [25]. The following proposition can be found in [25, Propo-sition 3.1]. Proposition 2.3 (Honda–Kazez–Mati´c) . If K is a hyperbolic fibered knot, thenthe monodromy of K is right-veering if and only if d ( λ ) > . As a result, the monodromy of K is left-veering if and only if d ( λ ) <
0. Ifthe monodromy is neither right-veering nor left-veering, then d ( λ ) = m .The following theorem was due to Gabai [16, Theorem 8.8]. Theorem 2.4 (Gabai) . Let K ⊂ S be a hyperbolic knot, λ be a very fulllamination in its complement, then the degeneracy locus d ( λ ) is either m or m for an integer m with | m | ≤ g ( K ) − . If K is fibered and λ is the stablelamination transverse to the fibration, then | m | ≥ . If pq is a slope, and d ( λ ) = mn , define ∆( d ( λ ) , pq ) = | pn − qm | . The first partof the following theorem can be found in [18, Theorem 5.3], and the same proofyields the second part. Gabai was aware of the result about genuine laminationswhich was explicitly stated by Brittenham [9]. Gabai and Mosher also provedthis theorem for any hyperbolic cusped manifolds, see [10, Theorem 6.48]. Theorem 2.5 (Gabai) . Let K ⊂ S be a hyperbolic fibered knot, λ be the stablelamination transverse to the fibration. If ∆( d ( λ ) , pq ) ≥ , λ will be an essentiallamination in S p/q ( K ) . If ∆( d ( λ ) , pq ) ≥ , λ will be a genuine lamination in S p/q ( K ) . In this subsection, we briefly recall the theory of changemaker lattices ofGreene [22, 23].When Y is a rational homology sphere and t ∈ Spin c ( Y ), Ozsv´ath and Szab´o[36] defined the correction term d ( Y, t ) ∈ Q which is an invariant of the pair5 Y, t ). If Y is the boundary of a smooth, compact, negative definite 4–manifold X , then c ( s ) + b ( X ) ≤ d ( Y, t ) , (1)for any s ∈ Spin c ( X ) that extends t ∈ Spin c ( Y ).Suppose that Y is obtained by p –surgery on a knot K ⊂ S , p >
0, then Y isthe boundary of a 4–manifold W = W p ( K ) which consists of a zero-handle anda two-handle with attaching curve K . If Y is also the boundary of a smooth,compact, negative definite, simply connected 4–manifold X with b ( X ) = n ,then the four-manifold Z := X ∪ Y ( − W ) is a smooth, closed, negative definite,simply connected 4–manifold with b ( Z ) = n + 1. (The simple connectednessis not necessary, but it suffices for our purpose.) Donaldson’s DiagonalizationTheorem [13] implies that the intersection pairing on H ( Z ) is isomorphic to − Z n +1 , negative of the standard ( n + 1)-dimensional Euclidean integer lattice.Consequently, negative of the intersection pairing on X , denoted Λ, embeds asa codimension one sub-lattice of Z n +1 . Definition 2.6.
A smooth, compact, negative definite 4–manifold X is sharp if for every t ∈ Spin c ( Y ), there exists some s ∈ Spin c ( X ) extending t such thatthe equality is realized in Equation (1). Definition 2.7.
A vector σ = ( σ , σ , . . . , σ n ) ∈ Z n +1 that satisfies 0 ≤ σ ≤ σ ≤ · · · ≤ σ n is a changemaker vector if for every k , with 0 ≤ k ≤ σ + σ + · · · + σ n , there exists a subset S ⊂ { , , . . . , n } such that k = P i ∈ S σ i . Asublattice of Z n +1 is a changemaker lattice if it is the orthogonal complementof a changemaker vector.The following important theorem was proved by Greene [23]. Theorem 2.8 (Greene) . Suppose that K ⊂ S is an L-space knot, and Y = S p ( K ) is the boundary of a simply connected sharp –manifold X with b ( X ) = n . Let Z = X ∪ Y ( − W p ( K )) , and let Q Z be the intersection form on Z . Thenthere is a lattice isomorphism ( H ( Z ) , − Q Z ) → Z n +1 , such that the generator of H ( − W p ( K )) is mapped to a changemaker vector σ with h σ, σ i = p , and H ( X ) is mapped to the orthogonal complement ( σ ) ⊥ of σ . One can recover the normalized Alexander polynomial ∆ K ( t ) = P i a i t i fromthe changemaker vector [23, Lemma 2.5]. Lemma 2.9.
The torsion coefficients of K can be determined by t i ( K ) = min c c − n − , for each i ∈ { , , . . . , ⌊ p ⌋} , , for i > p .where c is subject to c ∈ (1 , , . . . ,
1) + 2 Z n +1 , h c , σ i + p ≡ i (mod 2 p ) . he coefficients a i of ∆ K can be determined by the following rule. For i > , a i = t i − − t i + t i +1 , and a = 1 − X i> a i . Although the genus of K can be deduced from ∆ K , it is useful to know thefollowing direct formula [23, Proposition 3.1]:2 g ( K ) = p − | σ | = p − n X j =0 | σ j | . (2) Now we are ready to prove our main theorem.
Proof of Theorem 1.1. (1) If the monodromy of K is right-veering [25], let λ bethe stable lamination transverse to the fibration, then the degeneracy locus ispositive by Proposition 2.3. So d ( λ ) has the form m for a positive integer m with 2 ≤ m ≤ g ( K ) − pq > g ( K ) or pq <
0, ∆( pq , m ) ≥
3, so Theorem 2.5 implies that S p/q ( K ) contains a genuine lamination. ByTheorem 2.1, S p/q ( K ) is irreducible and its π is infinite.If S p/q ( K ) is an irreducible atoroidal Seifert fibered space with infinite π ,then its base orbifold is S with 3 singular points, we get a contradiction byTheorem 2.2.If K is an L-space knot and pq ≥ g ( K ), then S p/q ( K ) is an L-space by [41].Theorem 2.5 implies that S p/q ( K ) contains an essential lamination. Theorem 2.2implies that S p/q ( K ) has a taut foliation, thus it cannot be an L-space [26, 38],a contradiction.(2) If the monodromy of K is neither right-veering nor left-veering, then d ( λ )has the form m for a positive integer m with 2 ≤ m ≤ g ( K ) − | q | ≥
2, ∆( pq , m ) = | qm | ≥
4, so Theorem 2.5 implies that S p/q ( K ) containsa genuine lamination, which is not possible by Theorems 2.1 and 2.2. Proof of Corollary 1.4.
When pq ≥ g ( T r,s ) + 4 and S p/q ( K ) ∼ = S p/q ( T r,s ), weknow that S p/q ( T r,s ) is an L-space and K is fibered [19, 33, 39]. By the work ofMcCoy [30, Theorem 1.1], g ( K ) = g ( T r,s ) and ∆ K = ∆ T r,s .If K is hyperbolic, our result follows from Theorem 1.1.If K is a torus knot, since ∆ K = ∆ T r,s , we must have K = T r,s .If K is a satellite knot, let R ⊂ S \ K be an “innermost” incompressibletorus. Let V be the solid torus bounded by R in S , and let L be the core of V .Since S p/q ( K ) is irreducible and atoroidal, using [15], V p/q ( K ) must be a solidtorus. In this case, K is a 0–bridge or 1–bridge braid in V with winding number7 − − · · · − X . There are n − − w >
1, and gcd( p, w ) = 1 for homological reasons. The simple loop with slope p/ ( qw ) on R is null-homologous in V p/q ( K ), so S p/q ( K ) = S p/ ( qw ) ( L ). Since R is innermost, L is not a satellite knot, so L is either hyperbolic or a torusknot.If L is hypebolic, it follows from [28, Theorem 1.2] that w ≤
8, so w = 2.Hence K is a (2 h + 1 , L and p/q = 4 h + 2 + 1 /n for some n ∈ Z \ { } .Since pq ≥ g ( K ) + 4, we have4 h + 2 + 1 n ≥ g ( K ) + 4 = 4(2 g ( L ) + h ) + 4 , hence 2 + n ≥ g ( L ) + 4, which is not possible.If L is a torus knot, we can get a contradiction by the same argument as inthe proof of [34, Proposition 2.5]. Proof of Corollary 1.6.
Let p = 14 or 17, and T be T , or T , . It follows from[24] that if S p ( K ) ∼ = S p ( T ), then ∆ K = ∆ T . See also [35, Known Facts 1.2 (3)].Hence Theorem 1.1 rules out the possibility that K is hyperbolic. The finitesurgeries on satellite knots have been classified in [6, Corollary 1.4] and [4,Theorem 7], thus we can check K cannot be satellite. So K is a torus knot.Since ∆ K = ∆ T , we must have K = T . (4 n + 1) –surgery on T n +1 , T = T n +1 , and let K be a knot satisfying S n +1 ( K ) ∼ = S n +1 ( T ).The following lemma is a small part of Greene’s lens space realization the-orem [22]. Since we do not need the full strength of Greene’s theorem, we willgive a quick proof here. Lemma 4.1.
Either ∆ K = ∆ T , or n = 5 and ∆ K = ∆ T , .Proof. We know Y = S n +1 ( T ) ∼ = L (4 n + 1 ,
4) = − L (4 n + 1 , n ). (Note that ourconvention of the orientation of L ( p, q ) is opposite to the convention in [22].)Since 4 n + 1 n = [5 , , . . . , − := 5 − − −
1. . . − , n −
1) copies of 2 in the expression, we conclude that Y is theboundary of a negatively plumbed 4–manifold X with plumbing diagram givenin Figure 2. Since X is sharp, it follows from Theorem 2.8 that the lattice Λ isa changemaker lattice.Now Λ has a basis consisting of one vector with norm 5 and ( n −
1) vectorswith norm 2. Any norm 2 vector in Z n +1 must be of the form e i ± e j , where e , e , . . . , e n is the usual orthonormal basis. Since Λ = ( σ ) ⊥ , the two corre-sponding coordinates of σ must be equal. Thus we conclude that n coordinatesof σ are equal. Since σ = 1 (see the proof of [22, Theorem 1.6]), σ must be ofthe form (1 , a, a, . . . , a ) or (1 , , , . . . , , b )for some a, b ≥
1. Since Λ = ( σ ) ⊥ contains a vector with norm 5, we have a = 2and b ∈ { , } .If σ = (1 , , , . . . , K = ∆ T . (It is easierto see g ( K ) = n using (2).)If b = 2, the norm of σ is n + 4. By Theorem 2.8, the norm of σ is 4 n + 1,so n = 1. This falls into the case we just considered.If b = 4, the norm of σ is n + 16. Since the norm of σ is 4 n + 1, n = 5. Bythe computation in [36, Section 10.3], ∆ K = ∆ T , . Proof of Theorem 1.7.
By Lemma 4.1, either ∆ K = ∆ T , or n = 5 and ∆ K =∆ T , . In the latter case, K has the same knot Floer homology as T , by [39].Now we consider the case ∆ K = ∆ T . We have g ( K ) = g ( T ).If K is hyperbolic, we get a contradiction by Theorem 1.1.If K is a torus knot, since ∆ K = ∆ T , we get K = T .If K is a satellite knot, using the classification of lens space surgeries onsatellite knots [5, 46, 47], we see that K is the (2 uv + 1 , T u,v and4 n + 1 = 4 uv + 1. Then g ( K ) = ( u − v −
1) + uv > uv = n, a contradiction. Appendix: A criterion for veering
In this appendix, we assume the readers are familiar with the basic notionsof knot Floer homology [37, 43]. We will prove the following theorem.
Theorem A.1.
Let Y be a closed, oriented –manifold, K ⊂ Y be a fiberedknot with fiber F and monodromy φ . If rank \ HF K ( Y, K, [ F ] , g ( F ) −
1) = 1 , then φ is either right-veering or left-veering. K be a null-homologous knot in a closed, oriented 3–manifold Y , andlet F be a Seifert surface. There is a chain map ∂ z : \ CF K ( Y, K, [ F ] , i ) → \ CF K ( Y, K, [ F ] , i − n z = 1. Similarly, there is a chainmap ∂ w : \ CF K ( Y, K, [ F ] , i − → \ CF K ( Y, K, [ F ] , i ) . The following theorem is contained in the proof of [2, Theorem 1.1].
Theorem A.2.
Let K be a fibered knot in a closed, oriented –manifold Y . Let F be a Seifert surface, and let φ : F → F be the monodromy of the correspondingopen book. If φ is not left-veering, then the induced map ( ∂ z ) ∗ : \ HF K ( Y, K, [ F ] , − g ( F )) → \ HF K ( Y, K, [ F ] , − g ( F )) is nonzero. Similarly, if φ is not right-veering, then the induced map ( ∂ w ) ∗ : \ HF K ( Y, K, [ F ] , − g ( F )) → \ HF K ( Y, K, [ F ] , − g ( F )) is nonzero.Proof of Theorem A.1. We will use Q –coefficients for Heegaard Floer homology.Assume that φ is neither right-veering nor left-veering. Let C = CF K ∞ ( Y, K, [ F ])be the Z –filtered knot Floer chain complex. By [43, Lemma 4.5], C is filteredchain homotopy equivalent to a chain complex C ′ with C ′ { ( i, j ) } ∼ = \ HF K ( Y, K, [ F ] , j − i ) . Let a be a generator of C ′ { (0 , − g ( F )) } , consider the component b of ∂ a in C ′ { ( − , − − g ( F )) } . On one hand, since ∂ = 0, b = 0. On the other hand, b isjust ( ∂ z ) ∗ ◦ ( ∂ w ) ∗ ( a ). By Theorem A.2, both ( ∂ z ) ∗ and ( ∂ w ) ∗ are isomorphisms,so b = 0, a contradiction.It is well-known that \ HF K ( Y, K, [ F ] , g ( F )), the topmost term in knot Floerhomology, contains a lot of information about the topology of the knot comple-ment [19, 33, 38]. It is natural to ask what topological information is containedin other terms of \ HF K ( Y, K, [ F ] , g ( F )). Baldwin and Vela-Vick’s work [2] andour Theorem A.1 gave some partial answers to this question. References [1] Kenneth L. Baker,
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