The classification of quasi-alternating Montesinos links

TL;DR

This paper completes the classification of quasi-alternating Montesinos links, showing they match previously identified classes and linking their properties to topological invariants like L-spaces and definite 4-manifolds.

Contribution

It finalizes the classification of quasi-alternating Montesinos links and establishes a topological characterization involving double branched covers and 4-manifold bounds.

Findings

Quasi-alternating Montesinos links are exactly those previously identified.

A Montesinos link is quasi-alternating if its double branched cover is an L-space.

Such links bound both positive and negative definite 4-manifolds with trivial first homology.

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 $L$ 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 4-manifold with vanishing first homology.

Equations

$$t_i=[a_1^i,\dots ,a_{h_i}^i]:=a_1^i-\frac{1}{a_2^i-\frac{1}{\begin{array}{rl}\ddots & \\ & a_{h_i-1}^i-\frac{1}{a_{h_i}^i}\end{array}}},$$

$$Q_{ij}=\{\begin{array}{ll}\text{w}(v_i),& \text{mbox}ifi=j\\ 1,& \text{mbox}if{v}_i\text{mbox}and{v}_j\text{mbox}areconnectedbyanedge\\ 0,& \text{mbox}otherwise\end{array},$$

$$U_S=\{e_i|e_i\cdot v\ne 0\text{mbox}forsomev\in S\}.$$

$$\text{mbox}det(L)=\left|{\alpha }_1\dots {\alpha }_p\left(e-\sum _{i=1}^p\frac{{\beta }_i}{{\alpha }_i}\right)\right|.$$

$$\overline{L}=M\left(-e,-\frac{{\alpha }_1}{{\beta }_1},\dots ,-\frac{{\alpha }_p}{{\beta }_p}\right)=M\left(p-e,\frac{{\alpha }_1}{{\alpha }_1-{\beta }_1},\dots ,\frac{{\alpha }_p}{{\alpha }_p-{\beta }_p}\right),$$

$$L=M\left(1;\frac{{\alpha }_1}{{\beta }_1},\dots ,\frac{{\alpha }_p}{{\beta }_p}\right)=M\left(1-p;\frac{{\alpha }_1}{{\beta }_1-{\alpha }_1},\dots ,\frac{{\alpha }_p}{{\beta }_p-{\alpha }_p}\right),$$

$$|H_1(Y)|=|H_1(Y_0)|+|H_1(Y_1)|,$$

Full text

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 $\mathcal{Q}$ of quasi-alternating links is the smallest set of links satisfying the following:

• •

The unknot U belongs to $\mathcal{Q}$.

• •

If $L$ is a link with a diagram containing a crossing $c$ such that

1. (1)

both smoothings $L_{0}$ and $L_{1}$ of the link $L$ at the crossing $c$, as in Figure 1, belong to $\mathcal{Q}$, 2. (2)

$\mbox{det}(L_{0}),\mbox{det}(L_{1})\geq 1$, and 3. (3)

$\mbox{det}(L)=\mbox{det}(L_{0})+\mbox{det}(L_{1})$,

then $L$ is in $\mathcal{Q}$. The crossing $c$ is called a quasi-alternating crossing.

Ozsváth-Szabó showed that the class of non-split alternating links is contained in $\mathcal{Q}$ [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 $L$:

1. (i)

$L$ is homologically thin for both Khovanov homology and knot Floer homology [MO08]. 2. (ii)

The double branched cover $\Sigma(L)$ of $L$ is an L-space [OS05, Proposition 3.3]. 3. (iii)

The $3$-manifold $\Sigma(L)$ bounds a smooth negative definite 4-manifold $W$ with $H_{1}(W)=0$ [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 $12$-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]:

Theorem 1**.**

Let $L=M(e;t_{1},\ldots,t_{p})$ be a Montesinos link in standard form, that is, where $t_{i}=\frac{\alpha_{i}}{\beta_{i}}>1$ and $\alpha_{i},\beta_{i}>0$ are coprime for all $i=1,\ldots,p$. Then $L$ is quasi-alternating if and only if

1. (1)

$e<1$, or 2. (2)

$e=1$* and $\frac{\alpha_{i}}{\alpha_{i}-\beta_{i}}>\frac{\alpha_{j}}{\beta_{j}}$ for some $i,j$ with $i\neq j$, or* 3. (3)

$e>p-1$, or 4. (4)

$e=p-1$* and $\frac{\alpha_{i}}{\alpha_{i}-\beta_{i}}<\frac{\alpha_{j}}{\beta_{j}}$ for some $i,j$ with $i\neq j$.*

As a corollary of our proof we obtain the following characterisation of the Montesinos links $L$ which are quasi-alternating in terms of their double branched covers $\Sigma(L)$:

Corollary 1**.**

A Montesinos link $L$ is quasi-alternating if and only if

1. (1)

$\Sigma(L)$* is an L-space, and* 2. (2)

there exist a smooth negative definite $4$-manifold $W_{1}$ and a smooth positive definite $4$-manifold $W_{2}$ with $\partial W_{i}=\Sigma(L)$ and $H_{1}(W_{i})=0$ for $i=1,2$.

Note that in Corollary 1 and throughout, we assume all homology groups have $\mathbb{Z}$ coefficients.

In light of this corollary, Theorem 1 can also be seen as a classification of the L-space Seifert fibered spaces over $S^{2}$ which bound both positive and negative definite $4$-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 $7$. In fact, they show that the formal L-spaces $M^{3}$ with $|H_{1}(M)|\leq 7$ are precisely the double branched covers of quasi-alternating links with determinant at most $7$. In this same direction, as a consequence of Corollary 1, we have the following.

Corollary 2**.**

A Seifert fibered space over $S^{2}$ 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 $B^{4}$ over such a surface is a definite $4$-manifold of the appropriate sign. Generalising this, a quasi-alternating link has the property that it bounds a pair of surfaces in $B^{4}$ with double branched covers a positive definite and a negative definite $4$-manifold (these surfaces cannot be embedded in $S^{3}$ 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 $L$ is a quasi-alternating Montesinos link such that $\Sigma(L)$ is the oriented boundary of the standard negative definite plumbing $X^{4}$. Since the property of being quasi-alternating is closed under reflection, by property (iii) above, $-\Sigma(L)=\Sigma(\overline{L})$ bounds a negative definite $4$-manifold $W$ with $H_{1}(W)=0$. By Donaldson’s theorem [Don87], the smooth closed negative definite $4$-manifold $X\cup W$ has diagonalisable intersection form. Hence, $H_{2}(X)/\mbox{Tors}\hookrightarrow H_{2}(X\cup W)/\mbox{Tors}$ is an embedding of the intersection lattice of $X$ into the standard negative diagonal lattice. Moreover, using that $H_{1}(W)$ is torsion free, it is shown that if $A$ is a matrix representing the lattice embedding then $A^{T}$ must be surjective.

When $L$ is a pretzel link of a certain form, Greene analyses the possible embeddings of the intersection lattice of $X$ 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 $L$ that there is no lattice embedding for which $A^{T}$ 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 $\Sigma(L)$ 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 $M(e;t_{1},\ldots,t_{p})$, where $t_{i}=\frac{\alpha_{i}}{\beta_{i}}\in\mathbb{Q}$ with $\alpha_{i}>1$ and $\beta_{i}$ coprime integers, and $e$ is an integer, is given by the diagram in Figure 2. In the figure, each box labelled $t_{i}$ represents the corresponding rational tangle. The [math] rational tangle is shown in Figure 3. Introducing an additional positive (resp. negative) half-twist to the bottom of an $a/b$ rational tangle produces a rational tangle represented by $a/b+1$ (resp. $a/b-1$), see Figure 3. Rotating (in either direction) a rational tangle represented by $t\in\mathbb{Q}\cup\{1/0\}$ by $90$ degrees produces the rational tangle represented by $-1/t$. The rational tangle represented by any $a/b\in\mathbb{Q}\cup\{1/0\}$ can be obtained from the [math] rational tangle by a sequence of these two operations. See [Cro04] for a more thorough treatment of rational links. Note however that an $a/b$ rational tangle with our conventions corresponds to a $b/a$ rational tangle in [Cro04].

We also note that with our conventions for a Montesinos link $M(e;t_{1},\ldots,t_{p})$, the integer $e$ 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 $M(e;t_{1},\ldots,t_{p})$ is isotopic to $M(e+1;t_{1},\ldots,t_{i-1},t_{i}^{\prime},t_{i+1},\ldots,t_{p})$ where $t_{i}^{\prime}=\frac{\alpha_{i}}{\beta_{i}+\alpha_{i}}$, and is also isotopic to $M(e-1;t_{1},\ldots,t_{i-1},t_{i}^{\prime},t_{i+1},\ldots,t_{p})$, where $t_{i}^{\prime}=\frac{\alpha_{i}}{\beta_{i}-\alpha_{i}}$. Hence, a Montesinos link is isotopic to one in standard form, that is, of the form $M(e;t_{1},\ldots,t_{p})$ where $t_{i}>1$ for all $i$.

Let $L=M(e;t_{1},\ldots,t_{p})$ where $t_{i}<-1$ for all $i$. Note that any Montesinos link can be put into this form. For each $i$, there is a unique continued fraction expansion

[TABLE]

where $h_{i}\geq 1$ and $a_{j}^{i}\leq-2$ for all $j\in\{1,\ldots,h_{i}\}$.

The double branched cover $\Sigma(L)$ of $L$ is the oriented boundary of the $4$-dimensional plumbing $X_{\Gamma}$ of $D^{2}$-bundles over $S^{2}$ described by the weighted star-shaped graph $\Gamma$ shown in Figure 4. We call $\Gamma$ the standard star-shaped plumbing graph for $L$. The $i$th leg of $\Gamma$ corresponding to $t_{i}$ is the linear subgraph generated by the vertices labelled with weights $a_{1}^{i},\ldots,a_{h_{i}}^{i}$. The degree $p$ vertex labelled with weight $e$ is called the central vertex. Denote the vertices of $\Gamma$ by $v_{1},v_{2},\ldots,v_{k}$. The zero-sections of the $D^{2}$-bundles over $S^{2}$ corresponding to each of $v_{1},\ldots,v_{k}$ in the plumbing together form a natural spherical basis for $H_{2}(X_{\Gamma})$. With respect to this basis, the intersection form of $X_{\Gamma}$ is given by the weighted adjacency matrix $Q_{\Gamma}$ with entries $Q_{ij}$, $1\leq i,j\leq k$ given by

[TABLE]

where $\text{w}(v_{i})$ is the weight of vertex $v_{i}$. We call $(\mathbb{Z}^{k},Q_{\Gamma})$ the intersection lattice of $X_{\Gamma}$ (or of $\Gamma$).

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 $L=M(e;t_{1},\ldots,t_{p})$, $p\geq 3$, be a Montesinos link in standard form, i.e. where $t_{i}=\frac{\alpha_{i}}{\beta_{i}}>1$ and $\alpha_{i},\beta_{i}>0$ are coprime for all $i$. Suppose that $e\leq p-2$ and $e-\sum_{i=1}^{p}\frac{1}{t_{i}}>0$ (in particular $e\geq 1$). Then $\Sigma(L)$ is not an L-space, and therefore $L$ is not quasi-alternating.

Proof.

The reflection of $L$ is given by $\overline{L}=M(e^{\prime};t_{1}^{\prime},\ldots,t_{p}^{\prime})=M(-e;-t_{1},\ldots,-t_{p})$. The space $\Sigma(\overline{L})$ is the oriented boundary of a plumbing $X_{\Gamma}$ corresponding to the standard star-shaped plumbing graph $\Gamma$ for $\overline{L}$. Since $e^{\prime}-\sum_{i=1}^{p}\frac{1}{t_{i}^{\prime}}=-\left(e-\sum_{i=1}^{p}\frac{1}{t_{i}}\right)<0$, by [NR78, Theorem 5.2], $X_{\Gamma}$ has negative definite intersection form.

Since $X_{\Gamma}$ is negative definite and $\Gamma$ is almost-rational, by [Ném05, Theorem 6.3] we have that $\Sigma(\overline{L})$ is an L-space if and only if $X_{\Gamma}$ is a rational surface singularity (more generally, see [Ném15]). Note that $\Gamma$ is almost-rational since by sufficiently decreasing the weight of the central vertex we obtain a plumbing graph satisfying $-\text{w}(v)\geq\text{deg}(v)$ for all vertices $v$, where $\text{w}(v)$ denotes the weight of $v$, 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 $X_{\Gamma}$ is a rational surface singularity as follows. Let $v_{1},\ldots,v_{k}$ be the vertices of $\Gamma$ and for $i\in\{1,\ldots,k\}$, let $[\Sigma_{v_{i}}]\in H_{2}(X_{\Gamma})$ be the spherical class naturally associated to $v_{i}$. The algorithm is as follows (see [Sti08, Section 3] for a similar formulation).

1. (1)

Let $K_{0}=\sum_{i=1}^{k}[\Sigma_{v_{i}}]\in H_{2}(X_{\Gamma})$. 2. (2)

In the $i$th step, consider the pairings $\langle PD[K_{i}],[\Sigma_{v_{j}}]\rangle$, for $j\in\{1,\ldots,k\}$. Note that these pairings may be evaluated using the adjacency matrix $Q$. If for some $j$ the pairing is at least $2$ then the algorithm stops and $X_{\Gamma}$ is not a rational surface singularity. If for some $j$, the pairing is equal to $1$, then set $K_{i+1}=K_{i}+[\Sigma_{v_{j}}]$ and go to the next step. Otherwise all pairings are non-positive, the algorithm stops and $X_{\Gamma}$ is a rational surface singularity.

Applying Laufer’s algorithm to $X_{\Gamma}$, we claim that the algorithm terminates at the $0^{\mbox{th}}$ step. To see this, note that for $v$ the central vertex of $\Gamma$, $\langle PD[K_{0}],[\Sigma_{v}]\rangle=p-e$ (each vertex adjacent to $v$ contributes $1$, the central vertex contributes $-e$). By assumption $e\leq p-2$ so $\langle PD[K_{0}],[\Sigma_{v}]\rangle=p-e\geq 2$. Hence, the algorithm terminates, we conclude that $X_{\Gamma}$ is not a rational surface singularity and hence $\Sigma(\overline{L})$ is not an L-space. Therefore $\Sigma(L)$ 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 $X$ and $W$ are a pair of 4-manifolds, $\partial X=-\partial W=Y$ is a rational homology sphere, and $H_{1}(W)$ is torsion-free. Express the map $H_{2}(X)/\mbox{Tors}\rightarrow H_{2}(X\cup W)/\mbox{Tors}$ with respect to a pair of bases by the matrix $A$. This map is an inclusion, and $A^{T}$ is surjective. In particular, if some $k$ rows of $A$ contain all the non-zero entries of some $k$ of its columns, then the induced $k\times k$ minor has determinant $\pm 1$.

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 $-1/r=[a_{1},\ldots,a_{n}]$ and $-1/s=[b_{1},\ldots,b_{m}]$ where $r+s=1$. Consider a weighted linear graph $\Psi$ having two connected components, $\Psi_{1}$ and $\Psi_{2}$, where $\Psi_{1}$ consists of $n$ vertices $v_{1},\ldots,v_{n}$ with weights $a_{1},\ldots,a_{n}$ and $\Psi_{2}$ of $m$ vertices $w_{1},\ldots,w_{m}$ with weights $b_{1},\ldots,b_{m}$. Moreover, suppose that there is an embedding of the lattice $(\mathbb{Z}^{n+m},Q_{\Psi})$ into $(\mathbb{Z}^{k},-\mbox{Id})$, with basis $e_{1},\ldots,e_{k}$. For $S$ a subset of vertices of $\Psi$, define

[TABLE]

Suppose further that $e_{1}\in U_{v_{1}}\cap U_{w_{1}}$ and $U_{\Psi}=\{e_{1},\ldots,e_{k}\}$. Then $U_{\Psi_{1}}=U_{\Psi_{2}}$ and $k=n+m$.

Lemma 4** ([LL11, Lemma 3.2]).**

Let $-1/r=[a_{1},\ldots,a_{n}]$ and $-1/s=[b_{1},\ldots,b_{m}]$ be such that $r+s\geq 1.$ Then there exists $n_{0}\leq n$ and $m_{0}\leq m$ such that $-1/r_{0}=[a_{1},\ldots,a_{n_{0}}]$ and $-1/s_{0}=[b_{1},\ldots,b_{m_{0}}]$ satisfy $r_{0}+s_{0}=1$.

See 1

Proof.

If one of the conditions (1)-(4) is satisfied then $L$ 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 $L$ is not quasi-alternating. Thus, assume none of the conditions are satisfied, in particular $p\geq 2$.

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

[TABLE]

If $p=2$, since none of the conditions are satisfied we must have $e=1$ and $\frac{\alpha_{1}}{\alpha_{1}-\beta_{1}}=\frac{\alpha_{2}}{\beta_{2}}$. Hence, $\mbox{det}(L)=|\alpha_{1}\alpha_{2}(1-\frac{\beta_{1}}{\alpha_{1}}-\frac{\beta_{2}}{\alpha_{2}})|=0$, and so $L$ is not quasi-alternating (in fact $L$ must be the two component unlink). For the remainder of the argument we assume that $p\geq 3$, and $\mbox{det}(L)\neq 0$, that is, $e-\sum_{i=1}^{p}\frac{\beta_{i}}{\alpha_{i}}\neq 0$.

First consider the case $1<e<p-1$. The reflection of $L$ is given by

[TABLE]

where the latter is written in standard form and $1<p-e<p-1$. Moreover, we see that a reflection reverses the sign of $e-\sum_{i=1}^{p}\frac{\beta_{i}}{\alpha_{i}}$ and thus by a reflection if necessary we may assume that $e-\sum_{i=1}^{p}\frac{\beta_{i}}{\alpha_{i}}>0$. Then by Lemma 1, $\Sigma(L)$ is not an L-space, so $L$ is not quasi-alternating.

It remains to consider the cases $e=1$ and $e=p-1$. By a reflection if necessary we may assume that $e=1$. 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 $L$ is not quasi-alternating. If $e-\sum_{i=1}^{p}\frac{\beta_{i}}{\alpha_{i}}>0$ then by Lemma 1, $\Sigma(L)$ is not an L-space, and therefore $L$ is not quasi-alternating.

Otherwise $e-\sum_{i=1}^{p}\frac{\beta_{i}}{\alpha_{i}}<0$. We have that

[TABLE]

where $\frac{\alpha_{i}}{\beta_{i}-\alpha_{i}}<-1$ for all $i$.

The double branched cover $\Sigma(L)$ of $L$ is therefore the boundary of a plumbing $4$-manifold $X_{\Gamma}$ on the standard star-shaped planar graph $\Gamma$ with central vertex of weight $-(p-1)$ and legs corresponding to the fractions $\frac{\alpha_{i}}{\beta_{i}-\alpha_{i}}$, $i\in\{1,\ldots,p\}$. Our assumption that $e-\sum_{i=1}^{p}\frac{\beta_{i}}{\alpha_{i}}<0$ implies that $X_{\Gamma}$ is negative definite [NR78, Theorem 5.2]. Suppose for the sake of contradiction that $L$ is quasi-alternating. Then $\overline{L}$ is quasi-alternating and $-\Sigma(L)=\Sigma(\overline{L})$ bounds a negative definite $4$-manifold $W$ with $H_{1}(W)=0$ [OS05, Proof of Lemma 3.6]. By Donaldson’s theorem [Don87], the smooth closed negative definite $4$-manifold $X_{\Gamma}\cup W$ has diagonalisable intersection form. Thus, the map $H_{2}(X_{\Gamma})/\mbox{Tors}\hookrightarrow H_{2}(X_{\Gamma}\cup W)/\mbox{Tors}$ induced by the inclusion map is an embedding of the intersection lattice $(\mathbb{Z}^{k},Q_{\Gamma})$ of $X_{\Gamma}$ into the standard negative diagonal lattice $(\mathbb{Z}^{n},-\mbox{Id})$ for some $n$. Denote by $e_{1},\ldots,e_{n}$ a basis for $(\mathbb{Z}^{n},-\mbox{Id})$.

We use the lattice embedding to identify elements of $(\mathbb{Z}^{k},Q_{\Gamma})$ with their image in $(\mathbb{Z}^{n},-\mbox{Id})$. For convenience, we will not distinguish between a vertex of $\Gamma$ and the vector it corresponds to in the lattice. The central vertex $v$ of $\Gamma$ has weight $-(p-1)$, and so $v\cdot e_{i}\neq 0$ for at most $p-1$ values of $i\in\{1,\ldots,n\}$. Thus, by applying an automorphism if necessary, we may assume that $v$ pairs non-trivially with precisely $e_{1},\ldots,e_{m}$ where $m\leq p-1$. Since there are $p$ legs, by the pigeonhole principle there must exist some $e_{j}$, where $j\in\{1,\ldots,m\}$, and two distinct vertices $v_{1},v_{2}$ adjacent to $v$ with $v_{1}\cdot e_{j}\neq 0$ and $v_{2}\cdot e_{j}\neq 0$. Without loss of generality we assume that $j=1$ and that for $i\in\{1,2\}$, the vertex $v_{i}$ belongs to the $i$th leg of $\Gamma$, i.e. corresponding to the fraction $\frac{\alpha_{i}}{\beta_{i}-\alpha_{i}}$.

Since we are assuming condition (2) does not hold, we have that $\frac{\alpha_{i}}{\alpha_{i}-\beta_{i}}\leq\frac{\alpha_{j}}{\beta_{j}}$ for all $i,j$ with $i\neq j$. In particular, we have $\frac{\alpha_{1}}{\alpha_{1}-\beta_{1}}\leq\frac{\alpha_{2}}{\beta_{2}}$. Rearranging this gives $\frac{\beta_{1}}{\alpha_{1}}+\frac{\beta_{2}}{\alpha_{2}}\leq 1$. Note that the two legs correspond to the fractions $-1/r:=-\frac{\alpha_{1}}{\alpha_{1}-\beta_{1}}=[a_{1}^{1},\ldots,a_{h_{1}}^{1}]$ and $-1/s:=-\frac{\alpha_{2}}{\alpha_{2}-\beta_{2}}=[a_{1}^{2},\ldots,a_{h_{2}}^{2}]$, where $r,s\in\mathbb{Q}$, and where our notation is as in Section 2. Thus, we have that $r+s=2-\frac{\beta_{1}}{\alpha_{1}}-\frac{\beta_{2}}{\alpha_{2}}\geq 1$. Since $r+s\geq 1$, by Lemma 4 there exist $h_{1}^{\prime}\leq h_{1}$ and $h_{2}^{\prime}\leq h_{2}$ such that $-1/r_{0}=[a_{1}^{1},\ldots,a_{h_{1}^{\prime}}^{1}]$ and $-1/s_{0}=[a_{1}^{2},\ldots,a_{h_{2}^{\prime}}^{2}]$ with $r_{0}+s_{0}=1$.

Let $\Psi$ be the union of the linear graph containing the first $h_{1}^{\prime}$ 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 $h_{2}^{\prime}$ vertices of the second leg. By restricting our embedding of $(\mathbb{Z}^{k},Q_{\Gamma})$, we have an embedding of the sublattice corresponding to $\Psi$ into $(\mathbb{Z}^{n},-\mbox{Id})$. The image of this embedding is contained in a sublattice $(\mathbb{Z}^{d},-\mbox{Id})$ of $(\mathbb{Z}^{n},-\mbox{Id})$ spanned by $\{e_{i}\in\mathbb{Z}^{n}\;|\;e_{i}\cdot v\neq 0\mbox{ for some vertex }v\mbox{ of }\Psi\}$. Hence $U_{\Psi}$ consists of $d$ elements (see Lemma 3 for definition of $U_{\Psi}$). Let $v_{1},w_{1}$ be the two vertices of $\Psi$ adjacent to the central vertex in $\Gamma$. By our choice of the two legs of $\Gamma$ which contain the vertices of $\Psi$, we know that $e_{j}\in U_{v_{1}}\cap U_{w_{1}}$ for some $j\in\{1,\ldots,n\}$. This shows that the hypothesis of Lemma 3 are satisfied, hence we conclude that $d=h_{1}^{\prime}+h_{2}^{\prime}$.

Let $A$ be the matrix representing the embedding $(\mathbb{Z}^{k},Q_{\Gamma})$ into $(\mathbb{Z}^{n},-\mbox{Id})$. Then the $h_{1}^{\prime}+h_{2}^{\prime}$ columns of $A$ corresponding to the vertices of $\Psi$ are supported in $d=h_{1}^{\prime}+h_{2}^{\prime}$ rows of $A$ corresponding to the $d$-dimensional sublattice of $(\mathbb{Z}^{n},-\mbox{Id})$. Denote this $d\times d$ minor by $B$. Then $-B^{T}B$ is a matrix for the intersection form of the plumbing corresponding to $\Psi$. Hence $-B^{T}B$ is a presentation matrix for $H_{1}(Y)$ where $Y$ is the boundary of the (disconnected) plumbing corresponding to $\Psi$. The $3$-manifold $Y$ is the disjoint union of two lens spaces, each given by surgery on the unknot with framings $-1/r_{0}<-1$ and $-1/s_{0}<-1$ respectively. Therefore $|\mbox{det}(B)|^{2}=|H_{1}(Y)|>1$ contradicting Lemma 2. Thus, $L$ is not quasi-alternating. ∎

See 1

Proof.

This is a corollary of the proof of Theorem 1. Suppose first that $L$ is quasi-alternating. By [OS05, Proposition 3.3], $\Sigma(L)$ is an L-space. Furthermore, $\Sigma(L)$ must bound a negative definite $4$-manifold $W_{1}$ with $H_{1}(W_{1})=0$ [OS05, Proof of Lemma 3.6]. Applying this to the reflection of $L$ which is also quasi-alternating, we get that $\Sigma(L)$ also bounds a positive definite $4$-manifold $W_{2}$ with $H_{1}(W_{2})=0$. 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 $(Y_{1},Y_{2},Y_{3})$ of closed, oriented $3$-manifolds form a triad if there is a $3$-manifold $M$ with torus boundary, and three oriented curves $\gamma_{1},\gamma_{2},\gamma_{3}\subset\partial M$ at pairwise distance $1$, such that $Y_{i}$ is the result of Dehn filling $M$ along $\gamma_{i}$, for $i=1,2,3$.

Definition 2**.**

The set $\mathcal{F}$ of formal L-spaces is the smallest set of rational homology $3$-spheres such that

1. (1)

$S^{3}\in\mathcal{F}$, and 2. (2)

if $(Y,Y_{0},Y_{1})$ is a triad with $Y_{0},Y_{1}\in\mathcal{F}$ and

[TABLE]

then $Y\in\mathcal{F}$.

See 2

Proof.

Let $L$ be a quasi-alternating Montesinos link. Then the double branched cover of $L$ is a Seifert fibered space over $S^{2}$. 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 $\Sigma(L)$ is a formal L-space Seifert fibered space over $S^{2}$.

Now let $M$ be a formal L-space Seifert fibered space over $S^{2}$. Then $M$ is the double branched cover of a Montesinos link $L$. 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 $4$-manifold with vanishing first homology. However, their proof in fact shows this for all formal L-spaces. Hence $M=\Sigma(L)$ is a formal L-space bounding positive and negative definite $4$-manifolds with vanishing first homology. Thus, Corollary 1 implies that $L$ 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.