Quasi-alternating links with small determinant
Quasi-alternating links of determinant 1, 2, 3, and 5 were previously classified by Greene and Teragaito, who showed that the only such links are two-bridge. In this paper, we extend this result by showing that all quasi-alternating links of determinant at most 7 are connected sums of two-bridge links, which is optimal since there are quasi-alternating links not of this form for all larger determinants. We achieve this by studying their branched double covers and characterizing distance-one surgeries between lens spaces of small order, leading to a classification of formal L-spaces with order at most 7.
Quasi-alternating links are a natural generalization of non-split alternating links which have received a considerable amount of attention over the past decade. First introduced by Ozsváth-Szabó in [OS05], these links provide a more general family of links whose Khovanov homology and knot Floer homology are particularly simple – they are homologically thin [MO08] – and they have exhibited a number of other behaviors found in alternating links. This can also be translated into topological applications: for example, any branched double cover of a quasi-alternating link is an example of a manifold which cannot admit a co-orientable taut foliation [OS05].
For many invariants such as the Alexander polynomial, there are only finitely many alternating knots which attain a fixed value, and this has had applications to Dehn surgery questions (most recently, [Gai14, LP14]). For quasi-alternating links, it is harder to obtain such finiteness results. For instance, it is still unknown if there are only finitely many quasi-alternating links of any fixed determinant.
In [Gre10], Greene classifies quasi-alternating links with determinant one, two, and three as the unknot, Hopf link, and the two trefoils respectively. Teragaito [Ter15, Theorem 1.9] completes the classification of quasi-alternating links with determinant 5 by a different method – they are the torus knots and the figure eight – and points out that the classification of quasi-alternating links of determinant 4 is still unknown despite partial results in this case [Ter14], but he conjectures that the only such links should be the torus links and the connected sum of two Hopf links. The goal of this paper is to verify this and to also classify quasi-alternating links with determinants up to 7 as well. Specifically, we prove:
Let be a quasi-alternating link with determinant at most 7. Then is either two-bridge or a connected sum of two-bridge links.
In particular, is either the unknot, the figure-eight knot, the torus link with , a connected sum of two Hopf links, a connected sum of a trefoil with a Hopf link, or either the knot or its mirror. This answers a question of Teragaito [Ter15, Conjecture 1.10]:
If is a quasi-alternating link which is not alternating, then .
Theorem 1.1 also implies the following about positive knots.
Let be a nontrivial positive knot of genus at most 2 and determinant at most 7. Then is either , , or .
The main idea of the proof is that used originally by Greene, and by Teragaito [Ter14] in the determinant 4 case, which is to lift to the branched double cover and rephrase the problem in terms of Dehn surgery. Greene and Levine [GL14] define a notion of formal L-space (see Definition 2.2) which is meant to be a 3-manifold analogue of quasi-alternating links; and indeed the branched double cover of any quasi-alternating link is a formal L-space. Theorem 1.1 will be a consequence of the following classification result.
If is a formal L-space with , then is a connected sum of lens spaces.
In fact, Teragaito nearly completes the determinant-4 classification of Theorem 1.1 in [Ter14, Lemma 2.3]; he could obtain the desired conclusion if he knew that non-trivial knots in cannot have non-trivial distance-one surgeries. The key technical result which allows us to extend Teragaito’s work is a recent theorem of Gainullin [Gai15] giving a Dehn surgery characterization of the unknot for nullhomologous knots in L-spaces, which extends a result of Kronheimer-Mrowka-Ozsváth-Szabó for knots in [KMOS07]. We remark that while the case of in Theorem 1.4 follows from Greene’s work, Teragaito’s classification of determinant 5 quasi-alternating links does not determine the order 5 formal L-spaces.
One could likely use the arguments in this paper to extend the classification of quasi-alternating links to slightly larger determinants. However, the obstruction to continuing this process for all values of the determinant is that one needs the classification of lens space surgeries on knots (the Berge conjecture) as well as a complete list of links whose branched double cover gives a fixed manifold.
Moreover, for any there are quasi-alternating links of determinant which are not connected sums of two-bridge links. Indeed, if then the -pretzel link is quasi-alternating [CK09, Theorem 3.2] but not alternating. One can show that its branched double cover is -surgery on the right handed trefoil, which implies that . Note that this branched double cover is a formal L-space. This is because is the lens space , the branched double cover of a two-bridge link, hence quasi-alternating, and is a formal L-space whenever is.
Here, and throughout the rest of this paper, we use the convention that is -surgery on the unknot in . We will also write .
In Section 2 we recall the definition and properties of quasi-alternating links, show that Theorem 1.1 follows from Theorem 1.4, and review other related material necessary for the setup. In Sections 3 and 4 we prove some results characterizing when knot complements in lens spaces of small order can have distance-one fillings which are also lens spaces of small order. Finally, we use these results in Section 5 to prove Theorem 1.4.
We would like to thank Fyodor Gainullin for helpful discussions. The first author was partially supported by NSF RTG grant DMS-1148490. The second author was supported by NSF grants DMS-1204387 and DMS-1506157.
2. Quasi-alternating links, branched double covers, and surgery
We begin by recalling the definitions of quasi-alternating links and formal L-spaces.
Definition 2.1 ([Os05]).
The set of quasi-alternating links is the smallest set of links in containing the unknot such that for any link , if admits a diagram with a crossing whose two resolutions satisfy
In particular, all non-split alternating links are quasi-alternating [OS05], and is also closed under taking mirrors and connected sums. (This last claim follows for by induction on : if then , and given resolutions of as above we have by hypothesis, so as well.)
We say a collection of closed, oriented 3-manifolds forms a triad if there is a 3-manifold with torus boundary and a collection of oriented curves at pairwise distance 1 such that each is the result of Dehn filling along . This is precisely the condition under which the Heegaard Floer homologies of the (in some order) fit into a surgery exact triangle. We will define to be if and otherwise; note that if is a link then its branched double cover satisfies .
Definition 2.2 ([Gl14, Section 7]).
The set of formal L-spaces is the smallest set of rational homology 3-spheres containing such that whenever is a triad with and
we have as well.
This definition can be interpreted as a 3-manifold analogue of the notion of a quasi-alternating link. Indeed, given any triple of links as in Definition 2.1, the branched double covers form a triad, and , so the branched double cover of any quasi-alternating link is a formal L-space. It is easy to see that contains all lens spaces and is closed under orientation reversal and taking connected sums.
We recall that we are interested in classifying quasi-alternating links and formal L-spaces with small determinant. This classification has been carried out for determinant at most 3 by work of Greene.
Theorem 2.3 ([Gre10]).
If is a formal L-space with equal to 1, 2, or 3, then is , , or respectively.
In order to deduce the classification of quasi-alternating links from the classification of formal L-spaces, we appeal to the following.
Theorem 2.4 (Hodgson-Rubinstein [Hr85]).
If is a link whose branched double cover is the lens space , then is the two-bridge link with continued fraction equal to .
Theorem 2.5 (Kim-Tollefson [Kt80]).
If the branched double cover of a non-split link is a connected sum , with each prime, then is a connected sum of links such that and .
Proof of Theorem 1.1.
Recall that if is a quasi-alternating link, then is a formal L-space; if , then is a connected sum of lens spaces by Theorem 1.4. It follows that is a connected sum of two-bridge links, and these links are determined uniquely by the lens space summands, so Theorem 1.1 follows immediately from Theorem 1.4. ∎
Thus, the remainder of the paper is devoted to proving Theorem 1.4.
2.1. Some general facts about surgery
The following lemma will be useful in the proof of Theorem 1.4. We first recall that a knot in a rational homology sphere is primitive if it generates . If denotes the exterior of , then primitivity implies that and there exists a curve on which represents the generator of . Further, we have that and the rational longitude form a basis for , and so . (Recall that the rational longitude is the unique slope on the boundary of a rational homology solid torus which is torsion in .) Given an arbitrary slope , we have .
Let be a primitive knot in a rational homology sphere . If a non-trivial filling on the exterior results in a rational homology sphere , possibly homeomorphic to , then the distance from this filling slope to the trivial slope is a multiple of . In particular, if and are not relatively prime, then such a filling cannot have distance one from the trivial slope.
Let and be slopes on for which Dehn filling yields and respectively, and let and . Then we can write and for some integers and , since and . Then , which is clearly a multiple of as claimed. ∎
In particular, if represents a core of a genus one Heegaard splitting of , then is primitive. It follows from Lemma 2.6 that no fillings which are distance one from the trivial filling can yield for any .
We will occasionally make use of the Casson-Walker invariant [BL90, Wal92] in order to study manifolds arising from several surgeries on the same knot. This invariant agrees with the usual -valued Casson invariant if is a homology sphere, and it satisfies a surgery formula for a knot in a homology sphere :
Here , where we normalize the Alexander polynomial so that and .
The Casson-Walker invariant is related to the Heegaard Floer -invariants [OS03] by the following formula, which was first proved by Ozsváth–Szabó [OS03, Theorem 1.3] for homology spheres and then generalized to all rational homology spheres by Rustamov [Rus04, Theorem 3.3].
Another useful tool for us will be the linking form. For notation, if is cyclic of order , then we will say that has linking form if there exists a generator with self-linking . We note for cyclic groups, the two forms and are equivalent if and only if for a unit . In this notation, if is a knot in a homology sphere , then the linking form of is .
2.2. Heegaard Floer homology and surgeries
If is a torsion structure on , the Heegaard Floer invariants () admit an absolute -grading [OS06]. This has already appeared implicitly in the statement of Theorem 2.8: given a rational homology sphere , the -invariants are defined in [OS03] as the lowest grading of a nonzero element such that for all . In this subsection we will review some properties of surgeries and their relationship to gradings in Heegaard Floer homology.
Theorem 2.9 (Ozsváth-Szabó [Os06]).
The absolute grading on has the following properties:
If is a cobordism from to , with torsion, then the induced map changes grading by
The natural map preserves the absolute grading.
The -action on has degree .
The cobordism maps on are particularly simple. If , then the map is zero for all [OS06, Lemma 8.2]. On the other hand, if is a 2-handle cobordism with between rational homology spheres and , and restricts to as for , then
is an isomorphism [OS03, Proposition 9.4] between -modules of the form , so it is determined completely by its grading, which in this case simplifies to . Moreover, in the case , the elements of which determine and are both in by definition, so we must have
Let be a 2-handle cobordism between rational homology spheres and . If and is an L-space, then the map is zero. If instead and is an L-space, then is surjective.
Write , . The cobordism maps fit into a commutative diagram
If and is an L-space, then and so , but is surjective so we must have . On the other hand, if and is an L-space, then is an isomorphism and is surjective, so their composition is surjective, hence is surjective as well. ∎
The cobordism maps induced by various surgeries on a knot (namely, any three which form a triad) fit into exact triangles; the surgery exact triangle first appeared in [OS04], but we cite the version from [OS11].
Theorem 2.11 (Ozsváth-Szabó [Os11, Theorem 6.2]).
Let be a rationally null-homologous knot with framing and meridian . Then there is a long exact sequence
in which the maps are the cobordism maps corresponding to attaching 2-handles.
The maps in the exact triangle are described via holomorphic triangle counts in [OS04, OS11], but the claim that they are cobordism maps follows from the fact that 2-handle cobordism maps are defined in [OS06] using precisely these counts. Each cobordism comes from attaching a 2-handle to a meridian of the core of the previous surgery.
The signature of each 2-handle cobordism in the exact triangle can be computed according to [KM07, Lemma 42.3.1]: let be the exterior of , and a primitive oriented curve which generates . If is the 2-handle cobordism from the Dehn filling of along to the Dehn filling along , where and are oriented so that , then has signature (respectively ) if and have the same sign (respectively opposite signs). (If either of these is zero then .) If all three manifolds in the triangle are rational homology spheres, it follows that two of the cobordisms are negative definite and one is positive definite.
For example, if we let and let be the image of an -framed meridian of , then the surgery triangle corresponding to has the form
We can verify that the cobordism from to is positive definite for . Indeed, if and are the meridian and longitude of in its exterior , then the boundary orientation of gives . The manifolds and are Dehn fillings along and respectively, satisfying . We have and , and since these have the same sign.
Finally, we use Dehn surgery to verify the following property of the -invariants.
If , then for all . Further, if the linking form of is equivalent to that of , then the set of -invariants of agrees mod 2 with that of .
We first claim that for all . By [Tan09, Theorem 4], we can write
for some , where is a Dedekind sum. Since when [RG72], we have the desired claim.
Now, fix with and suppose the linking form of is equivalent to . Then it follows from [CGO01, Corollary 3.9 and Proposition 3.17] that we can obtain from by a sequence of -surgeries on nullhomologous knots. Therefore, it suffices to show that -surgery on a nullhomologous knot in a rational homology sphere preserves the set of -invariants mod 2. Here we will implicitly use the canonical identification between and .
If is obtained by -surgery on , then we consider the corresponding 2-handle cobordism from to which is negative definite. By (2.2), it suffices to show that for each , the grading shift is an even integer. We have
for some integer , and since , it follows that , where . On the other hand, if is obtained from by -surgery on a knot, then is obtained from by -surgery on the dual knot, and we can apply the same argument. Since we have shown that the -invariants of are in , we must have the same for . ∎
2.3. Lens space surgeries and L-spaces
Greene states Theorem 2.3 in [Gre10] as a classification of quasi-alternating links of determinant up to 3, namely that they are the unknot, the Hopf link, or a trefoil; but his proof, which passes through branched double covers, actually establishes Theorem 2.3 as stated here. Theorem 2.4 then yields the classification for the branch sets. We recall the argument here in order to suggest how our own classification will proceed and to introduce some necessary results.
The only formal L-space with determinant 1 is by definition. To classify formal L-spaces with , Greene observes that there must be a triad with , and hence ; thus both and result from nontrivial surgeries on a knot , and only the unknot in has a nontrivial surgery. Therefore, is the unknot and is therefore . In order to classify with , we must likewise have a triad , so both and arise as surgeries on the same knot ; but the only knot in with an surgery is the unknot, so must be a lens space of order three.
The key input needed for the above argument is an understanding of which knots in have lens space surgeries. While this is a difficult question in general, it is understood for lens spaces of small order.
Theorem 2.13 ([Kmos07, Corollary 8.4]).
Suppose that results in a lens space for . Then is the unknot or a trefoil. In particular, if then is the unknot.
Similarly, we will occasionally need to understand when knots in Heegaard Floer L-spaces have lens space surgeries. We will use a recent result of Gainullin [Gai15], who proved the following Dehn surgery characterization of the unknot, generalizing the main result of [KMOS07] (see also [OS11, Corollary 1.3]).
Theorem 2.14 ([Gai15, Theorem 8.2]).
Let be a nullhomologous knot in an L-space . If and are isomorphic as absolutely-graded -modules, then is the unknot.
It follows that nullhomologous knots in L-spaces are determined by their complements. In our applications of Theorem 2.14, we will have the stronger assumption that and are orientation-preserving homeomorphic, except in the case when is a homology sphere L-space. We remark that in the case of a homology sphere, the result can be proved exactly as in [OS11, Corollary 1.3]. (Here, the only change necessary from the case is to use [Ni06], which shows that knot Floer homology detects the unknot in homology spheres.)
Under some additional mild conditions, Theorem 2.14 yields two stronger results.
Let be a knot in an L-space with prime. If a non-trivial surgery on which is distance one from the trivial surgery gives , then is the unknot.
If is nullhomologous, then the result follows from Theorem 2.14. On the other hand, if is not nullhomologous, we see that is primitive, because is prime. Therefore, it follows from Lemma 2.6 that the distance between the trivial and non-trivial surgeries is a multiple of , which is a contradiction. ∎
Let be an invisible 3-manifold, i.e. a homology sphere L-space with . If there is a knot and a rational number satisfying , then and is the unknot.
3. Primitive knot complements with several lens space fillings
In this section we will prove the following theorem, which we will use to study when two lens spaces and can belong to a triad.
Let be a homology with torus boundary, and suppose there are a pair of slopes with and a positive integer such that Dehn filling along and produces the lens spaces and respectively. Then is a solid torus.
Let be a closed, oriented curve such that generates , and declare a meridian to be any oriented curve such that . For convenience, given a simple, closed curve we will also use to denote its homology class in and its induced slope. More generally, the notation for a slope will now always be used to refer to the rational longitude.
Given satisfying the hypotheses of Theorem 3.1, there is a meridian and a sign such that and .
Fix an initial choice of meridian, so that and generate as an abelian group. Then Dehn filling along produces a closed 3-manifold with first homology , so for some orientations of and integers we can write
Letting and , it follows that
and by definition we have , which is by assumption. ∎
If is a homology sphere and a knot such that and for some positive integer , then .
The two sides of have linking forms and , so these are equivalent and thus is a square mod . It follows that cannot be a multiple of 4 or of any prime , since is not a square modulo any of these numbers. Moreover, cannot be , since it is a product of primes which are either or 2, so must be either or modulo 4. Since it is not a multiple of 3, it must also be either or modulo 3. But the same holds true for , since the linking forms and are also equivalent, and so the only way this can be possible is if and modulo both 3 and 4. In particular, as claimed. ∎
Given and likewise , (2.1) now implies that
and thus .
Let be a homology sphere with , and suppose for some knot and integer that and is an L-space. Then is invisible, i.e. an L-space homology sphere with correction term , and in fact .
Writing for convenience, we use the surgery exact triangle
where and are the corresponding 2-handle cobordisms from to and from to respectively. The latter two groups in the triangle have the form
and , with the subscripts denoting the grading of the bottom-most element in each tower . In particular we have
computed as in [OS03]. We also know that , with an even integer and for . Our goal is to show that and .
The map decomposes into summands for each . If we let be the result of capping off a Seifert surface for then is generated by , which has self-intersection , and we can label the structures on as () such that . Then the structure is determined by . Now if is the cocore of the 2-handle, then we must have . The natural map sends to , so . It follows from Theorem 2.9 that
If is a homogeneous element of even grading, then has grading . On the other hand, each element of the codomain has grading congruent to
and so must be zero. Thus .
Next, since every element of is in for all , the same is true for . By exactness the latter is equal to , hence all elements of with even grading are in . It follows that is supported entirely in odd gradings, and so by Theorem 2.8 and the assumption that , we have
Now suppose that either is not an L-space or . Then we have shown that , and so is not the boundary of any smooth, negative definite 4-manifold by [OS03, Corollary 9.8]. If we take the positive definite 2-handle cobordism from to , turn it upside down, and reverse its orientation, then we get a negative definite 2-handle cobordism from to , and is the boundary of a negative definite linear plumbing of disk bundles over , so the composition is negative definite with . This is a contradiction, so must in fact be an L-space with , and we have . ∎
Proof of Theorem 3.1.
Using Lemmas 3.2 and 3.3, we have a homology sphere obtained from by Dehn filling with core so that and . It follows from (3.1) that , and so Proposition 3.4 ensures that is an invisible manifold. Since is invisible and , it follows from Corollary 2.16 that and is the unknot, and so the exterior of is a solid torus as desired. ∎
4. Knots in with distance-one lens space fillings of order 5
In this section we address the question of when and a lens space of the form can belong to a triad.
Let be a manifold with torus boundary for which a pair of distance-one Dehn fillings produce and a lens space . Then is a solid torus.
We begin by determining which lens spaces of order 5 can occur.
Let and satisfy the hypotheses of Theorem 4.1. Then admits a Dehn filling with core such that is a homology sphere, , and where is either 2 or 3. Moreover, every 3-manifold of the form has Casson-Walker invariant equal to .
We know that is a homology , since otherwise the core of the filling is nullhomologous and so any other filling would have homology of even order. We identify curves such that generates the kernel of the natural map and . For some odd integer , Dehn filling along the curve produces the filling. If we write and set then we have and . Thus Dehn filling along produces the desired homology sphere and core such that . The filling of is at distance one from by assumption, so it must be along where is either 2 or 3, i.e. . Since , we would like to see that is not homeomorphic to . However, since or , the linking form rules this out, and we can take .
For the second claim, we apply the Casson-Walker surgery formula (2.1) to and :
The left sides of both equations are zero, and we obtain . The surgery formula then says that for all . ∎
We claim that it suffices to consider the case and (i.e. ) for now. Let’s see how this implies the remaining case of Proposition 4.2, i.e. . Following the proof, we could instead take and define to be the homology sphere attained by filling along with core . We have , so . Then , so that . If we reverse orientation, then we get and . Thus, the case would imply that the exterior of in , which is orientation-reversing homeomorphic to , is a solid torus, completing the proof.
With the preceding understood, we now suppose for the remainder of this section that and . In what follows we will write for convenience.
For all , there is a short exact sequence
in which the two non-trivial maps are induced by negative definite 2-handle cobordisms.
These invariants fit into the surgery exact triangle (2.3), so it will suffice to show that if is the 2-handle cobordism from to , corresponding to attaching a handle along an -framed meridian of (after a 2-framed surgery on ), then the induced map is zero. By the discussion above Lemma 2.12, we see that is positive definite for . Since is an L-space, for all by Lemma 2.10. ∎
If and , then is an L-space.
The case of Lemma 4.3 produces a short exact sequence