The classification of quasialternating Montesinos links
Abstract.
In this note, we complete the classification of quasialternating Montesinos links. We show that the quasialternating Montesinos links are precisely those identified independently by QazaqzehChbiliQublan and ChampanerkarOrding. A consequence of our proof is that a Montesinos link is quasialternating if and only if its double branched cover is an Lspace, and bounds both a positive definite and a negative definite manifold with vanishing first homology.
1. Introduction
Quasialternating links were defined by OzsváthSzabó [OS05, Definition 3.1] as a natural generalisation of the class of alternating links.
Definition 1.
The set of quasialternating 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

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

, and

,
then is in . The crossing is called a quasialternating crossing.

OzsváthSzabó showed that the class of nonsplit alternating links is contained in [OS05, Lemma 3.2]. Moreover, quasialternating links share a number of properties with alternating links, we list a few of these. For a quasialternating link :

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

The double branched cover of is an Lspace [OS05, Proposition 3.3].

The manifold bounds a smooth negative definite 4manifold 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 quasialternating. For example, there still remain examples of crossing knots with unknown quasialternating status [Jab14]. ChampanerkarKofman [CK09] showed that the quasialternating property is preserved by replacing a quasialternating crossing with an alternating rational tangle. They used this to determine an infinite family of quasialternating pretzel links, which Greene later showed is the complete set of quasialternating pretzel links [Gre10].
QazaqzehChbiliQublan [QCQ15] and ChampanerkarOrding [CO15] independently generalised the sufficient conditions on pretzel links to obtain an infinite family of quasialternating Montesinos links. This family includes all examples of quasialternating Montesinos links found by Widmer [Wid09]. Furthermore, it was conjectured by QazaqzehChbiliQublan that this family is the complete set of quasialternating Montesinos links. We mention that Watson [Wat11] gave an iterative surgical construction for constructing all quasialternating Montesinos links.
Some necessary conditions to be quasialternating in terms of the rational parameters of a Montesinos link were obtained in [QCQ15] and [CO15] based on the fact that a quasialternating link is homologically thin. Further conditions are described in [CO15] coming from the fact that the double branched cover of a quasialternating link is an Lspace. Some additional restrictions were found in [QC15].
Our main result is the following theorem which states that the quasialternating Montesinos links are precisely those found by QazaqzehChbiliQublan [QCQ15] and ChampanerkarOrding [CO15]: {restatable}thmmainthm Let be a Montesinos link in standard form, that is, where and are coprime for all . Then is quasialternating if and only if

, or

and for some with , or

, or

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

is an Lspace, and

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 Lspace Seifert fibered spaces over which bound both positive and negative definite manifolds with vanishing first homology. To what extent Corollary 1 generalises to nonMontesinos links remains an interesting question.
This work also gives a classification of the Seifert fibered space formal Lspaces. The notion of a formal Lspace was defined by Greene and Levine [GL16] as a 3manifold analogue of quasialternating links. In fact, the double branched cover of a quasialternating link is an example of a formal Lspace. In [LS17], Lidman and Sivek classified the quasialternating links of determinant at most . In fact, they show that the formal Lspaces with are precisely the double branched covers of quasialternating links with determinant at most . In this same direction, as a consequence of Corollary 1, we have the following.
corolformallspace A Seifert fibered space over is a formal Lspace if and only if it is the double branched cover of a quasialternating link.
Corollary 1 also seems significant given the recent independent characterisations of alternating knots by Greene [Gre17] and Howie [How17]. A nonsplit 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 quasialternating 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 Lspaces, this property characterises those which are quasialternating.
Our approach to proving Theorem 1 follows that of Greene [Gre10] on the determination of quasialternating pretzel links. One of Greene’s main strategies is as follows. Suppose is a quasialternating Montesinos link such that is the oriented boundary of the standard negative definite plumbing . Since the property of being quasialternating 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 quasialternating. 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 LecuonaLisca [LL11]. The condition we obtain combined with an obstruction based on being an Lspace leads to the precise necessary conditions to complete the determination of quasialternating 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) halftwist 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 ChampanerkarOrding [CO15], and agrees with that of QazaqzehChbiliQublan [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
where and for all .
The double branched cover of is the oriented boundary of the dimensional plumbing of bundles over described by the weighted starshaped graph shown in Figure 4. We call the standard starshaped 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 zerosections 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
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 quasialternating 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 quasialternating 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 Lspace, and therefore is not quasialternating.
Proof.
The reflection of is given by . The space is the oriented boundary of a plumbing corresponding to the standard starshaped plumbing graph for . Since , by [NR78, Theorem 5.2], has negative definite intersection form.
Since is negative definite and is almostrational, by [Ném05, Theorem 6.3] we have that is an Lspace if and only if is a rational surface singularity (more generally, see [Ném15]). Note that is almostrational 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).

Let .

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 nonpositive, 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 Lspace. Therefore is not an Lspace. ∎
The following lemma will provide an obstruction to a Montesinos link being quasialternating.
Lemma 2 ([Gre10, Lemma 2.1]).
Suppose that and are a pair of 4manifolds, is a rational homology sphere, and is torsionfree. 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 nonzero 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 quasialternating 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
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 .
*
Proof.
If one of the conditions (1)(4) is satisfied then is quasialternating 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 quasialternating. 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
If , since none of the conditions are satisfied we must have and . Hence, , and so is not quasialternating (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
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 Lspace, so is not quasialternating.
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 quasialternating. If then by Lemma 1, is not an Lspace, and therefore is not quasialternating.
Otherwise . We have that
where for all .
The double branched cover of is therefore the boundary of a plumbing manifold on the standard starshaped 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 quasialternating. Then is quasialternating 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 nontrivially 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 quasialternating. ∎
*
Proof.
This is a corollary of the proof of Theorem 1. Suppose first that is quasialternating. By [OS05, Proposition 3.3], is an Lspace. Furthermore, must bound a negative definite manifold with [OS05, Proof of Lemma 3.6]. Applying this to the reflection of which is also quasialternating, 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 quasialternating in the proof of Theorem 1. ∎
As a consequence, we obtain a classification of the Seifert fibered spaces which are formal Lspaces. Before stating it, we recall the definition of a formal Lspace. 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 Lspaces is the smallest set of rational homology spheres such that

, and

if is a triad with and
then .
*
Proof.
Let be a quasialternating 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 quasialternating link is an Lspace [OS05, Proposition 3.3]. Their proof in fact shows that the double branched cover of a quasialternating link is a formal Lspace. Hence is a formal Lspace Seifert fibered space over .
Now let be a formal Lspace 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 quasialternating 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 Lspaces. Hence is a formal Lspace bounding positive and negative definite manifolds with vanishing first homology. Thus, Corollary 1 implies that is quasialternating. ∎
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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