The classification of quasi-alternating Montesinos links

# The classification of quasi-alternating Montesinos links

## Abstract.

In this note, we complete the classification of quasi-alternating Montesinos links. We show that the quasi-alternating Montesinos links are precisely those identified independently by Qazaqzeh-Chbili-Qublan and Champanerkar-Ording. A consequence of our proof is that a Montesinos link is quasi-alternating if and only if its double branched cover is an L-space, and bounds both a positive definite and a negative definite -manifold with vanishing first homology.

## 1. Introduction

Quasi-alternating links were defined by Ozsváth-Szabó [OS05, Definition 3.1] as a natural generalisation of the class of alternating links.

###### Definition 1.

The set of quasi-alternating links is the smallest set of links satisfying the following:

• The unknot U belongs to .

• If is a link with a diagram containing a crossing such that

1. both smoothings and of the link at the crossing , as in Figure 1, belong to ,

2. , and

3. ,

then is in . The crossing is called a quasi-alternating crossing.

Ozsváth-Szabó showed that the class of non-split alternating links is contained in [OS05, Lemma 3.2]. Moreover, quasi-alternating links share a number of properties with alternating links, we list a few of these. For a quasi-alternating link :

1. is homologically thin for both Khovanov homology and knot Floer homology [MO08].

2. The double branched cover of is an L-space [OS05, Proposition 3.3].

3. The -manifold bounds a smooth negative definite 4-manifold with [OS05, Proof of Lemma 3.6].

For some further properties see [LO15], [QC15], [Ter15] and [ORS13, Remark after Proposition 5.2].

Due to their recursive definition, it is difficult in general to determine whether or not a link is quasi-alternating. For example, there still remain examples of -crossing knots with unknown quasi-alternating status [Jab14]. Champanerkar-Kofman [CK09] showed that the quasi-alternating property is preserved by replacing a quasi-alternating crossing with an alternating rational tangle. They used this to determine an infinite family of quasi-alternating pretzel links, which Greene later showed is the complete set of quasi-alternating pretzel links [Gre10].

Qazaqzeh-Chbili-Qublan [QCQ15] and Champanerkar-Ording [CO15] independently generalised the sufficient conditions on pretzel links to obtain an infinite family of quasi-alternating Montesinos links. This family includes all examples of quasi-alternating Montesinos links found by Widmer [Wid09]. Furthermore, it was conjectured by Qazaqzeh-Chbili-Qublan that this family is the complete set of quasi-alternating Montesinos links. We mention that Watson [Wat11] gave an iterative surgical construction for constructing all quasi-alternating Montesinos links.

Some necessary conditions to be quasi-alternating in terms of the rational parameters of a Montesinos link were obtained in [QCQ15] and [CO15] based on the fact that a quasi-alternating link is homologically thin. Further conditions are described in [CO15] coming from the fact that the double branched cover of a quasi-alternating link is an L-space. Some additional restrictions were found in [QC15].

Our main result is the following theorem which states that the quasi-alternating Montesinos links are precisely those found by Qazaqzeh-Chbili-Qublan [QCQ15] and Champanerkar-Ording [CO15]: {restatable}thmmainthm Let be a Montesinos link in standard form, that is, where and are coprime for all . Then is quasi-alternating if and only if

1. , or

2. and for some with , or

3. , or

4. and for some with .

As a corollary of our proof we obtain the following characterisation of the Montesinos links which are quasi-alternating in terms of their double branched covers : {restatable}corolmaincorol A Montesinos link is quasi-alternating if and only if

1. is an L-space, and

2. there exist a smooth negative definite -manifold and a smooth positive definite -manifold with and for .

Note that in Corollary 1 and throughout, we assume all homology groups have coefficients.

In light of this corollary, Theorem 1 can also be seen as a classification of the L-space Seifert fibered spaces over which bound both positive and negative definite -manifolds with vanishing first homology. To what extent Corollary 1 generalises to non-Montesinos links remains an interesting question.

This work also gives a classification of the Seifert fibered space formal L-spaces. The notion of a formal L-space was defined by Greene and Levine [GL16] as a 3-manifold analogue of quasi-alternating links. In fact, the double branched cover of a quasi-alternating link is an example of a formal L-space. In [LS17], Lidman and Sivek classified the quasi-alternating links of determinant at most . In fact, they show that the formal L-spaces with are precisely the double branched covers of quasi-alternating links with determinant at most . In this same direction, as a consequence of Corollary 1, we have the following.

{restatable}

corolformallspace A Seifert fibered space over is a formal L-space if and only if it is the double branched cover of a quasi-alternating link.

Corollary 1 also seems significant given the recent independent characterisations of alternating knots by Greene [Gre17] and Howie [How17]. A non-split link is alternating if and only if it bounds negative definite and positive definite spanning surfaces (which are the checkerboard surfaces). The double branched cover of over such a surface is a definite -manifold of the appropriate sign. Generalising this, a quasi-alternating link has the property that it bounds a pair of surfaces in with double branched covers a positive definite and a negative definite -manifold (these surfaces cannot be embedded in in general). Corollary 1 shows that among Montesinos links with double branched covers which are L-spaces, this property characterises those which are quasi-alternating.

Our approach to proving Theorem 1 follows that of Greene [Gre10] on the determination of quasi-alternating pretzel links. One of Greene’s main strategies is as follows. Suppose is a quasi-alternating Montesinos link such that is the oriented boundary of the standard negative definite plumbing . Since the property of being quasi-alternating is closed under reflection, by property (iii) above, bounds a negative definite -manifold with . By Donaldson’s theorem [Don87], the smooth closed negative definite -manifold has diagonalisable intersection form. Hence, is an embedding of the intersection lattice of into the standard negative diagonal lattice. Moreover, using that is torsion free, it is shown that if is a matrix representing the lattice embedding then must be surjective.

When is a pretzel link of a certain form, Greene analyses the possible embeddings of the intersection lattice of into a negative diagonal lattice and shows that the aforementioned surjectivity condition cannot hold, and hence the link cannot be quasi-alternating. Our main contribution is to argue for more general Montesinos links that there is no lattice embedding for which is surjective. Key to our argument are some results on lattice embeddings by Lecuona-Lisca [LL11]. The condition we obtain combined with an obstruction based on being an L-space leads to the precise necessary conditions to complete the determination of quasi-alternating Montesinos links.

## 2. Preliminaries

We briefly recall some material on Montesinos links and plumbings. See [CO15] or [BZH14] for further detail on Montesinos links, and [NR78] for more on plumbings. The Montesinos link , where with and coprime integers, and is an integer, is given by the diagram in Figure 2. In the figure, each box labelled represents the corresponding rational tangle. The rational tangle is shown in Figure 3. Introducing an additional positive (resp. negative) half-twist to the bottom of an rational tangle produces a rational tangle represented by (resp. ), see Figure 3. Rotating (in either direction) a rational tangle represented by by degrees produces the rational tangle represented by . The rational tangle represented by any can be obtained from the rational tangle by a sequence of these two operations. See [Cro04] for a more thorough treatment of rational links. Note however that an rational tangle with our conventions corresponds to a rational tangle in [Cro04].

We also note that with our conventions for a Montesinos link , the integer has opposite sign to that used by Champanerkar-Ording [CO15], and agrees with that of Qazaqzeh-Chbili-Qublan [QCQ15] and Greene [Gre10].

The Montesinos link is isotopic to where , and is also isotopic to , where . Hence, a Montesinos link is isotopic to one in standard form, that is, of the form where for all .

Let where for all . Note that any Montesinos link can be put into this form. For each , there is a unique continued fraction expansion

 ti=[ai1,…,aihi]:=ai1−1ai2−1⋱aihi−1−1aihi,

where and for all .

The double branched cover of is the oriented boundary of the -dimensional plumbing of -bundles over described by the weighted star-shaped graph shown in Figure 4. We call the standard star-shaped plumbing graph for . The th leg of corresponding to is the linear subgraph generated by the vertices labelled with weights . The degree vertex labelled with weight is called the central vertex. Denote the vertices of by . The zero-sections of the -bundles over corresponding to each of in the plumbing together form a natural spherical basis for . With respect to this basis, the intersection form of is given by the weighted adjacency matrix with entries , given by

 Qij=⎧⎨⎩w(vi),if i=j1,if vi and vj are connected by an edge0,otherwise,

where is the weight of vertex . We call the intersection lattice of (or of ).

## 3. Results

Equivalent sufficient conditions for a Montesinos link to be quasi-alternating were given in [CO15, Theorem 5.3] and [QCQ15, Theorem 3.5]. The goal of this section is to prove Theorem 1 which states that these sufficient conditions for a Montesinos link to be quasi-alternating are also necessary conditions.

###### Lemma 1.

Let , , be a Montesinos link in standard form, i.e. where and are coprime for all . Suppose that and (in particular ). Then is not an L-space, and therefore is not quasi-alternating.

###### Proof.

The reflection of is given by . The space is the oriented boundary of a plumbing corresponding to the standard star-shaped plumbing graph for . Since , by [NR78, Theorem 5.2], has negative definite intersection form.

Since is negative definite and is almost-rational, by [Ném05, Theorem 6.3] we have that is an L-space if and only if is a rational surface singularity (more generally, see [Ném15]). Note that is almost-rational since by sufficiently decreasing the weight of the central vertex we obtain a plumbing graph satisfying for all vertices , where denotes the weight of , and such a graph is rational (for details see [Ném05, Example 8.2(3)]).

Laufer’s algorithm [Lau72, Section 4] can be used to determine whether the negative definite plumbing is a rational surface singularity as follows. Let be the vertices of and for , let be the spherical class naturally associated to . The algorithm is as follows (see [Sti08, Section 3] for a similar formulation).

1. Let .

2. In the th step, consider the pairings , for . Note that these pairings may be evaluated using the adjacency matrix . If for some the pairing is at least then the algorithm stops and is not a rational surface singularity. If for some , the pairing is equal to , then set and go to the next step. Otherwise all pairings are non-positive, the algorithm stops and is a rational surface singularity.

Applying Laufer’s algorithm to , we claim that the algorithm terminates at the step. To see this, note that for the central vertex of , (each vertex adjacent to contributes , the central vertex contributes ). By assumption so . Hence, the algorithm terminates, we conclude that is not a rational surface singularity and hence is not an L-space. Therefore is not an L-space. ∎

The following lemma will provide an obstruction to a Montesinos link being quasi-alternating.

###### Lemma 2 ([Gre10, Lemma 2.1]).

Suppose that and are a pair of 4-manifolds, is a rational homology sphere, and is torsion-free. Express the map with respect to a pair of bases by the matrix . This map is an inclusion, and is surjective. In particular, if some rows of contain all the non-zero entries of some of its columns, then the induced minor has determinant .

The following two technical lemmas will be useful when we apply the obstruction to being quasi-alternating based on Lemma 2.

###### Lemma 3 ([Ll11, Lemma 3.1]).

Suppose and where . Consider a weighted linear graph having two connected components, and , where consists of vertices with weights and of vertices with weights . Moreover, suppose that there is an embedding of the lattice into , with basis . For a subset of vertices of , define

 US={ei|ei⋅v≠0 for some v∈S}.

Suppose further that and . Then and .

###### Lemma 4 ([Ll11, Lemma 3.2]).

Let and be such that Then there exists and such that and satisfy .

\mainthm

*

###### Proof.

If one of the conditions (1)-(4) is satisfied then is quasi-alternating by either of [CO15, Theorem 5.3] or [QCQ15, Theorem 3.5], thus it suffices to show that if none of the conditions are satisfied then is not quasi-alternating. Thus, assume none of the conditions are satisfied, in particular .

By [Sav02, Section 1.2.3] (see also [CO15, Proposition 4.1]), we have that

 det(L)=∣∣ ∣∣α1…αp(e−p∑i=1βiαi)∣∣ ∣∣.

If , since none of the conditions are satisfied we must have and . Hence, , and so is not quasi-alternating (in fact must be the two component unlink). For the remainder of the argument we assume that , and , that is, .

First consider the case . The reflection of is given by

 ¯¯¯¯L=M(−e,−α1β1,…,−αpβp)=M(p−e,α1α1−β1,…,αpαp−βp),

where the latter is written in standard form and . Moreover, we see that a reflection reverses the sign of and thus by a reflection if necessary we may assume that . Then by Lemma 1, is not an L-space, so is not quasi-alternating.

It remains to consider the cases and . By a reflection if necessary we may assume that . Note that conditions (2) and (4) are equivalent under a reflection. We assume that condition (2) is not satisfied. We need to prove that this implies that is not quasi-alternating. If then by Lemma 1, is not an L-space, and therefore is not quasi-alternating.

Otherwise . We have that

 L=M(1;α1β1,…,αpβp)=M(1−p;α1β1−α1,…,αpβp−αp),

where for all .

The double branched cover of is therefore the boundary of a plumbing -manifold on the standard star-shaped planar graph with central vertex of weight and legs corresponding to the fractions , . Our assumption that implies that is negative definite [NR78, Theorem 5.2]. Suppose for the sake of contradiction that is quasi-alternating. Then is quasi-alternating and bounds a negative definite -manifold with [OS05, Proof of Lemma 3.6]. By Donaldson’s theorem [Don87], the smooth closed negative definite -manifold has diagonalisable intersection form. Thus, the map induced by the inclusion map is an embedding of the intersection lattice of into the standard negative diagonal lattice for some . Denote by a basis for .

We use the lattice embedding to identify elements of with their image in . For convenience, we will not distinguish between a vertex of and the vector it corresponds to in the lattice. The central vertex of has weight , and so for at most values of . Thus, by applying an automorphism if necessary, we may assume that pairs non-trivially with precisely where . Since there are legs, by the pigeonhole principle there must exist some , where , and two distinct vertices adjacent to with and . Without loss of generality we assume that and that for , the vertex belongs to the th leg of , i.e. corresponding to the fraction .

Since we are assuming condition (2) does not hold, we have that for all with . In particular, we have . Rearranging this gives . Note that the two legs correspond to the fractions and , where , and where our notation is as in Section 2. Thus, we have that . Since , by Lemma 4 there exist and such that and with .

Let be the union of the linear graph containing the first vertices of the first leg (where we count vertices in a leg starting away from the central vertex), and the linear graph containing the first vertices of the second leg. By restricting our embedding of , we have an embedding of the sublattice corresponding to into . The image of this embedding is contained in a sublattice of spanned by . Hence consists of elements (see Lemma 3 for definition of ). Let be the two vertices of adjacent to the central vertex in . By our choice of the two legs of which contain the vertices of , we know that for some . This shows that the hypothesis of Lemma 3 are satisfied, hence we conclude that .

Let be the matrix representing the embedding into . Then the columns of corresponding to the vertices of are supported in rows of corresponding to the -dimensional sublattice of . Denote this minor by . Then is a matrix for the intersection form of the plumbing corresponding to . Hence is a presentation matrix for where is the boundary of the (disconnected) plumbing corresponding to . The -manifold is the disjoint union of two lens spaces, each given by surgery on the unknot with framings and respectively. Therefore contradicting Lemma 2. Thus, is not quasi-alternating. ∎

\maincorol

*

###### Proof.

This is a corollary of the proof of Theorem 1. Suppose first that is quasi-alternating. By [OS05, Proposition 3.3], is an L-space. Furthermore, must bound a negative definite -manifold with [OS05, Proof of Lemma 3.6]. Applying this to the reflection of which is also quasi-alternating, we get that also bounds a positive definite -manifold with . For the converse, note that these two necessary conditions are the only conditions used to obstruct a Montesinos link from being quasi-alternating in the proof of Theorem 1. ∎

As a consequence, we obtain a classification of the Seifert fibered spaces which are formal L-spaces. Before stating it, we recall the definition of a formal L-space. We say that a triple of closed, oriented -manifolds form a triad if there is a -manifold with torus boundary, and three oriented curves at pairwise distance , such that is the result of Dehn filling along , for .

###### Definition 2.

The set of formal L-spaces is the smallest set of rational homology -spheres such that

1. , and

2. if is a triad with and

 |H1(Y)|=|H1(Y0)|+|H1(Y1)|,

then .

\formallspace

*

###### Proof.

Let be a quasi-alternating Montesinos link. Then the double branched cover of is a Seifert fibered space over . Ozsváth and Szabó show that the double branched cover of a quasi-alternating link is an L-space [OS05, Proposition 3.3]. Their proof in fact shows that the double branched cover of a quasi-alternating link is a formal L-space. Hence is a formal L-space Seifert fibered space over .

Now let be a formal L-space Seifert fibered space over . Then is the double branched cover of a Montesinos link . Ozsváth and Szabó’s in [OS05, Proof of Lemma 3.6] show that the double branched cover of a quasi-alternating link bounds both a positive definite, and a negative definite -manifold with vanishing first homology. However, their proof in fact shows this for all formal L-spaces. Hence is a formal L-space bounding positive and negative definite -manifolds with vanishing first homology. Thus, Corollary 1 implies that is quasi-alternating. ∎

## Acknowledgements

I would like to thank Cameron Gordon for his support and helpful conversations, and Duncan McCoy for his suggestions and many helpful comments. I would also like to thank the referee for useful feedback.

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