Strongly quasipositive quasi-alternating links and Montesinos links

TL;DR

This paper characterizes strongly quasipositive quasi-alternating links and identifies new classes of such links among Montesinos links, establishing conditions under which these links are definite and strongly quasipositive.

Contribution

It provides a characterization of strongly quasipositive quasi-alternating links and detects new classes of strongly quasipositive Montesinos links based on their properties.

Findings

Strongly quasipositive quasi-alternating links are characterized by definiteness.

New classes of strongly quasipositive Montesinos links are identified.

Conditions involving quasi-alternating crossings and properties of the link components are established.

Abstract

The aim of this article is to give a characterization of strongly quasipositive quasi-alternating links and detect new classes of strongly quasipositive Montesinos links and non-strongly quasipositive Montesinos links. In this direction, we show that, if $L$ is an oriented quasi-alternating link with a quasi-alternating crossing $c$ such that $L_0$ is alternating (where $L_0$ has the induced orientation), then $L$ is definite if and only if it is strongly quasipositive (up to mirroring). We also show that if $L$ is an oriented quasi-alternating link with a quasi-alternating crossing $c$ such that $L_0$ is fibred or more generally has a unique minimal genus Seifert surface (where $L_0$ has the induced orientation), then $L$ is definite if and only if it is strongly quasipositive (up to mirroring).

Equations

$$[a_1,a_2,\cdots ,a_n]=\frac{1}{a_1-\frac{1}{a_2-\frac{1}{a_3-\cdots }}}.$$

$$g(D)=\frac{c(D)-s(D)+2-m}{2},$$

$$\det L=|\sum _{T\in Tr(D)}\prod _{e\in T\cap E}\chi (e)|$$

$$\begin{array}{rl}\det L& =|\sum _{T\in Tr(D)}\prod _{e\in T\cap E}\chi (e)|\\ & =|\sum _{T\in A}\prod _{e\in A\cap E}\chi (e)+\sum _{T\in Tr(D)\setminus A}\prod _{e\in T\cap E}\chi (e)|\\ & =|-card(A)+card(Tr(D)\setminus A)|\\ & <|card(A)+card(Tr(D)\setminus A)|\\ & <\det L_{\infty }^c+\det L_0^c=\det L\end{array}$$

$$\begin{array}{rl}\det L^{\prime }& =|\sum _{T\in Tr(L^{\prime })}\prod _{e\in T\cap E}\chi (e)|\\ & =|\sum _{T\in A}\prod _{e\in A\cap E}\chi (e)+\sum _{T\in A^{\prime }}\prod _{e\in A^{\prime }\cap E}\chi (e)+\sum _{T\in Tr(L^{\prime })\setminus \{A\cup A^{\prime }\}}\prod _{e\in T\cap E}\chi (e)|\\ & =|-card(A)-card(A^{\prime })+card(Tr(L^{\prime })\setminus \{A\cup A^{\prime }\})|\\ & =|-2\det L_{\infty }^{\prime }+\det L_{0,0}^{\prime }|\end{array}$$

$$\frac{\beta }{\alpha }=[c_1,c_2,\cdots ,c_m]=\frac{1}{c_1-\frac{1}{c_2-\frac{1}{c_3-\cdots }}}.$$

$$K^{\prime }=M(e;t_1,t_2,\cdots ,t_{i_0-1},t_{i_0+2},\cdots ,t_r)$$

$$\frac{q}{p}=[2c_2^{i_0}+2c_2^{i_0+1},2c_3^{i_0+1},\cdots ,2c_{m_{i_0+1}}^{i_0+1}]$$

$$\frac{q}{p}=[2c_2^{i_0+1}+2c_2^{i_0},2c_3^{i_0},\cdots ,2c_{m_{i_0}}^{i_0}].$$

$$g(L(\frac{q}{p}))<max(\frac{m_{i_0}}{2},\frac{m_{i_0+1}}{2}).$$

$$g_4(K)\le g(K^{\prime })+g(L(\frac{q}{p}))+1.$$

$$\begin{array}{rl}g(K)& =\frac{1}{2}(1+\sum _{i=1}^r{m}_i)\\ & =\frac{1}{2}(1+\sum _{i\ne \{i_0,i_0+1\}}{m}_i)+\frac{m_{i_0}}{2}+\frac{m_{i_0+1}}{2}\\ & =g(K^{\prime })+\frac{m_{i_0}}{2}+\frac{m_{i_0+1}}{2}\\ & =g(K^{\prime })+max(\frac{m_{i_0}}{2},\frac{m_{i_0+1}}{2})+1\\ & >g(K^{\prime })+g(L(\frac{q}{p}))+1\ge g_4(K).\end{array}$$

$$\begin{array}{rl}g(K)& =\frac{1}{2}(-1+\sum _{i=1}^r{m}_i)\\ & =\frac{1}{2}(-1+\sum _{i\ne \{i_0,i_0+1\}}{m}_i)+\frac{m_{i_0}}{2}+\frac{m_{i_0+1}}{2}\\ & =g(K^{\prime })+\frac{m_{i_0}}{2}+\frac{m_{i_0+1}}{2}\\ & =g(K^{\prime })+max(\frac{m_{i_0}}{2},\frac{m_{i_0+1}}{2})+1\\ & >g(K^{\prime })+g(L(\frac{q}{p}))+1\ge g_4(K).\end{array}$$

$$\begin{array}{rl}g(K)& =\frac{1}{2}(1+\sum _{i=1}^r{m}_i)-(p+1)\\ & =\frac{1}{2}(1+\sum _{i\ne \{i_0,i_0+1\}}{m}_i)-(p+1)+\frac{m_{i_0}}{2}+\frac{m_{i_0+1}}{2}\\ & \ge g(K^{\prime })+\frac{m_{i_0}}{2}+\frac{m_{i_0+1}}{2}\\ & \ge g(K^{\prime })+max(\frac{m_{i_0}}{2},\frac{m_{i_0+1}}{2})+1\\ & >g(K^{\prime })+g(L(\frac{q}{p}))+1\ge g_4(K).\end{array}$$

Full text

1. introduction

In this paper we assume, unless otherwise stated, that links are oriented and contained in $\mathbb{S}^{3}$. A Seifert surface $S$ is quasipositive if it is isotopic to a braided Seifert surface $S[b]$ for some positive bandword $b$ (see Rudolph [Ru5]). A link $L$ is strongly quasipositive if it has a quasipositive Seifert surface. Let $L$ be a link. Let $\chi(L)$ ($\chi_{4}(L)$) be the maximal Euler characteristic number of an oriented 2-manifold without closed components smoothly embedded in $\mathbb{S}^{3}$ (in $D^{4}$ respectively) with boundary $L$. Let $g(L)$, $g_{4}(L)$, and $g_{4}^{top}(L)$ be the $3$-sphere genus, smooth $4$-ball genus, and topologically (locally flat) $4$-ball genus of $L$, respectively. Let $\sigma$ be the signature of $L$.

Let $L$ be a link. We say that $L$ is definite if $g(L)=\frac{1}{2\texttimes}(|\sigma(L)|-(m-1))$, where $m$ is the number of components of $L$.

For an oriented quasi-alternating link $L$ with a quasi-alternating crossing $c$, the link $L_{0}$ is given as in Figure 3.

Theorem 1.1**.**

Let $L$ be an oriented quasi-alternating link with a quasi-alternating crossing $c$ such that $L_{0}$ is alternating. Then $L$ is definite if and only if it is strongly quasipositive $($up to mirroring$)$.

Theorem 1.2**.**

Let $L$ be an oriented quasi-alternating link with a quasi-alternating crossing $c$ such that $L_{0}$ is either,

1. (1)

fibred, or more generally 2. (2)

has a unique minimal genus Seifert surface.

Then $L$ is definite if and only if it is strongly quasipositive $($up to mirroring$)$.

Based in Theorem 1.1 and Theorem 1.2 we made the following conjecture:

Conjecture 1.3**.**

Let $L$ be an oriented quasi-alternating link. Then $L$ is definite if and only if it is strongly quasipositive $($up to mirroring$)$.

Proposition 1.4**.**

If $L$ is an oriented quasi-alternating link which is definite then $L_{0}$ is also definite.

Let $M(e;t_{1},\cdots,t_{r})$ be a Montesinos knot or link, where $t_{i}=\frac{\beta_{i}}{\alpha_{i}}=[c_{1}^{i},\cdots,c_{m_{i}}^{i}]$. We make the following assumptions below: $\alpha_{i}>1$ and $-\alpha_{i}<\beta_{i}<\alpha_{i}$, with gcd$(\alpha_{i},\beta_{i})=1$. We follow the convention that

[TABLE]

Proposition 1.5**.**

Let $L=M(e;t_{1},\cdots,t_{r})$ be a Montesinos link with $e$ even, $r\geq 3$, $t_{i}=\frac{\beta_{i}}{\alpha_{i}}=[c_{1}^{i},\cdots,c_{m_{i}}^{i}]$.

1. (1)

If $e\geq 0$ and $t_{i}=[2q_{1}^{i},2s_{1}^{i},2q_{2}^{i},2s_{2}^{i},\cdots,2q_{n_{i}}^{i},2s_{n_{i}}^{i}]$ where $q_{j}^{i}<0$, $s_{j}^{i}<0$ for any $i=1,\cdots,r$, $j=1,\cdots,n_{i}$, then $L$ is strongly quasipositive. 2. (2)

If $e\geq 0$ and $t_{i}=[2q_{1}^{i},2s_{1}^{i},2q_{2}^{i},2s_{2}^{i},\cdots,2q_{n_{i}}^{i}]$ where $q_{j}^{i}<0$, $s_{j}^{i}<0$ for any $i=1,\cdots,r$, $j=1,\cdots,n_{i}$ for $q_{j}^{i}$, and $j=1,\cdots,n_{i}-1$ for $s_{j}^{i}$ , then $L$ is strongly quasipositive. 3. (3)

*If $e\geq 0$ and there exists a unique $t_{i_{0}}=[2q_{1}^{i_{0}},2s_{1}^{i_{0}},2q_{2}^{i_{0}},2s_{2}^{i_{0}},\cdots,2q_{n_{i_{0}}}^{i_{0}}]$ where $q_{j}^{i_{0}}<0$, $s_{j}^{i_{0}}<0$ for any $j=1,\cdots,n_{i_{0}}$ for $q_{j}^{i_{0}}$, and $j=1,\cdots,n_{i_{0}}-1$ for $s_{j}^{i_{0}}$ and *

$t_{i}=[2q_{1}^{i},2s_{1}^{i},2q_{2}^{i},2s_{2}^{i},\cdots,2q_{n_{i}}^{i},2s_{n_{i}}^{i}]$* for any $i\neq i_{0}$ where $q_{j}^{i}<0$, $s_{j}^{i}<0$ for any, $j=1,\cdots,n_{i}$, then $L$ is strongly quasipositive.*

Proposition 1.6**.**

Let $K=M(e;t_{1},\cdots,t_{r})$ be a Montesinos knot with $e$ even, $r\geq 3$ and $t_{i}=\frac{\beta_{i}}{\alpha_{i}}$. Suppose that $\alpha_{1}$ is even and that, for any $i\geq 2$, $\alpha_{i}$ is odd and $\beta_{i}$ is even. Let $t_{i}=[2c_{1}^{i},2c_{2}^{i},\cdots,2c_{m_{i}}^{i}]$ be the even continued fraction of $t_{i}=\frac{\beta_{i}}{\alpha_{i}}$, for $i=1,\cdots,r$. Note that $m_{1}$ is odd and $m_{i}$ is even for $i>1$. If there exists $i_{0}$, $2\leq i_{0}\leq r$ such that $c_{1}^{i_{0}}=-c_{1}^{i_{0}+1}$, then $K$ is not strongly quasipositive.

The paper is organized as follows. In the second section we show Theorem 1.1, Theorem 1.2 and Proposition 1.4. In section 3 we prove Proposition 1.5 and Proposition 1.6.

Acknowledgments

I would like to thank my PhD supervisor Professor Steven Boyer for drawing my attention to the topic of the current paper and his consistent encouragement and support. I am also grateful as well to Kenneth Baker for his many helpful comments and encouragement.

2. strongly quasipositive quasi-alternating links

The concept of quasi-alternating links is due to (Ozsváth and Szabó, 2005).

The set $\mathcal{Q}$ of quasi-alternating links is the smallest set of links which satisfies the following properties:

1. (1)

the unknot is in $\mathcal{Q}$, 2. (2)

the set $\mathcal{Q}$ is closed under the following operation. Suppose $L$ is any link which admits a projection with a crossing with the following properties:

• •

both resolution $L_{0}$ and $L_{\infty}$ are in $\mathcal{Q}$,

• •

det$(L)$ = det$(L_{0})$ + det$(L_{\infty})$

then $L$ is in $\mathcal{Q}$.

Such a crossing $c$ is called quasi-alternating crossing.

We have the following concept, which is due to Thurston and taken from [ST]. The complexity $\chi^{-}(S)$ of an oriented surface $S$ is $-\chi(C)$, where $C$ is the union of all non-simply connected components of $S$ and $\chi(C)$ is the Euler characteristic. For $M$ a compact oriented 3-manifold and $N$ a (possibly empty) surface in $\partial M$, assign to any homology class $\alpha\in H_{2}(M,N,;\mathbb{Z})$ the minimum complexity $x(\alpha)$ of all oriented embedded surfaces whose fundamental class represents $\alpha$. The function $x:H_{2}(M,N,;\mathbb{Z})\longrightarrow\mathbb{Z}_{+}$ is called the Thurston norm. An oriented surface ($S,\partial S$)$\subset$($M,\partial M$) is taut if it is incompressible and $\chi^{-}(S)=x$($[S,\partial S])$ in $H_{2}(M,\nu(\partial S))$, where $\nu(\partial S)$ is a bicollar neighborhood of $\partial S$ in $\partial M$.

Lemma 2.1**.**

(Lemma in [ST])* A Seifert surface $S$ for a link $L$ is taut if and only if $\chi(S)=\chi(L)$.*

Corollary 2.2**.**

(Corollary 1.12 in [BBG])* If $L$ is a strongly quasipositive quasi-alternating link then $g_{4}^{top}(L)=g(L)=\frac{1}{2\texttimes}(|\sigma(L)|-(m-1))$, where $m$ is the number of components of $L$.*

Theorem 2.3**.**

(Lemma 3 in [MO])* Let $L$ be an oriented link. The following two conditions are equivalent, providing that the determinants of $L_{0}$ and $L_{\infty}$ are not zero*

1. (1)

${\rm det}(L_{+})={\rm det}(L_{0})+{\rm det}(L_{\infty})$** 2. (2)

$\sigma(L_{+})=\sigma(L_{0})-1$* and $\sigma(L_{+})-\sigma(L_{\infty})=-e$.*

A similar equivalence also holds for negative crossings:

1. (1)

${\rm det}(L_{-})={\rm det}(L_{0})+{\rm det}(L_{\infty})$** 2. (2)

$\sigma(L_{-})=\sigma(L_{0})+1$* and $\sigma(L_{-})-\sigma(L_{\infty})=-e$.*

where $e$ is the number of negative crossings in $L_{\infty}$ minus the number of negative crossings in $L_{0}$. Where also $L_{+}$ and $L_{-}$ are as in Figure 3, choose any orientation of $L$ and related orientation of $L_{0}$ and any orientation of $L_{\infty}$.

Theorem 2.4**.**

(Theorem 9.6 in [Li])* Let $L=L_{+}$ be a link. Then det $L_{+}$ $=$ det $L_{0}$ $+$ det $L_{\infty}$ if and only if det $L_{-}$ $=$ $|$det $L_{0}$ $-$ det $L_{\infty}|$, where $L_{+}$ and $L_{-}$ are related by a crossing change at a crossing.*

We say that a link is positive if it admits a diagram with only positive crossings. We say that a link is negative if it admits a diagram with only negative crossings. We say that a link is homogeneous if it has a diagram which can be written as a $\ast$-product of special alternating diagrams (see [Cro]). The genus of the Seifert surface obtained by applying the Seifert algorithm to a link diagram $D$ is given by

[TABLE]

where $c(D)$ is the number of crossings, $s(D)$ is the number of Seifert circles and $m$ is the number of components of $D$.

Theorem 2.5**.**

(Theorem in [Ru4])* A Murasugi sum is quasipositive iff the summands are quasipositive.*

Theorem 2.6**.**

(Theorem 4.1 in [LPS])* If $D$ is a diagram of a homogeneous link $L$ with $m$ components, then $|w(D)|\leq s(D)+2g(L)+m-1$, where $w(D)$ is the width and $s(D)$ is the number of Seifert circle. In particular, the equality holds if and only if $D$ is a positive diagram.*

Corollary 2.7**.**

If $D$ is a diagram of a homogeneous link $L$, then $g(L)\geq g(D)-c^{-}(D)$, where $c^{-}(D)$ is the number of negative crossings in $D$.

Theorem 2.8**.**

(Theorem in [Pr1])* Nontrivial positive links have negative signature.*

Theorem 2.9**.**

(Theorem 1.2 in [AbT])* A non-split link $L$ is positive if and only if it is homogeneous and strongly quasipositive.*

Theorem 2.10**.**

(Corollary [Cro])* The surface obtained by applying the Seifert algorithm to a positive diagram of a positive link is a minimal genus Seifert surface.*

Remark 2.11**.**

If $L$ is a definite link such that $\sigma(L)=0$ then $L$ is the trivial knot. Indeed, we have that $g(L)=\frac{1}{2\texttimes}(|\sigma(L)|-(m-1))=\frac{1}{2\texttimes}(0-(m-1))$ which implies that $g(L)=0$ and $m=1$.

2. Figure 7 is obtained by considering the Conway moves obtained by giving a half-twist to all the diagrams of Figure 4 (as in Figure 5)

Let $L$ and $L^{\prime}$ be links. The distant union of $L$ and $L^{\prime}$, written $L\sqcup L^{\prime}$, is obtained by first moving $L$ and $L^{\prime}$ so that they are separated by a plane, and then taking the union. An arbitrary link $L$ is isotopic to the distant union of its non-splittable sublinks. The number of such non-splittable sublinks is called the splitting number.

Since a positive Hopf link is positive then it strongly quasipositive by [Ru2].

Theorem 2.12**.**

(Theorem 1.4 in [ST])* Let $L_{+}$, $L_{-}$ and $L_{0}$ be three oriented links which are related by the Conway moves at a crossing. Then two of $\chi(L_{+})$, $\chi(L_{-})$ and $\chi(L_{0})-1$ are equal and are no larger than the third. The splitting numbers of the same pair of links are equal and are no larger than the third. Therefore, only one of the following three possibilities happens,*

1. (1)

$-\chi(L_{+})=-\chi(L_{-})\geq 1-\chi(L_{0})$; 2. (2)

$-\chi(L_{+})=1-\chi(L_{0})>-\chi(L_{-})$; 3. (3)

$-\chi(L_{-})=1-\chi(L_{0})>-\chi(L_{+})$.

Corollary 2.13**.**

Let $L_{+}$, $L_{-}$ and $L_{0}$ be three oriented, non-split links which are related by the Conway moves at a crossing. Then only one of the following three possibilities happens,

1. (1)

$2g(L_{+})+m-1=2g(L_{-})+m-1\geq 1+2g(L_{0})+m_{0}-1$; 2. (2)

$2g(L_{+})+m-1=1+2g(L_{0})+m_{0}-1>2g(L_{-})+m-1$; 3. (3)

$2g(L_{-})+m-1=1+2g(L_{0})+m_{0}-1>2g(L_{+})+m-1$**

where $m$ and $m_{0}$ are the number of components of $L_{\pm}$ and $L_{0}$, respectively.

Lemma 2.14**.**

(Proposition 3.1 in [ST])* When $-\chi(L_{+})=1-\chi(L_{0})>-\chi(L_{-}),$ there are taut Seifert surfaces $S^{\prime}$, $S$ for $L_{+}$ and $L_{0}$ respectively which appear as in Figure 6, i.e $S^{\prime}$ is obtained from $S$ by plumbing on a Hopf band: $($An analogous conclusion holds when $-\chi(L_{-})=1-\chi(L_{0})>-\chi(L_{+})$.$)$*

Proof of Proposition 1.4.

Assume that $L$ is a definite, oriented quasi-alternating link. Since, if $\sigma(L)=0$ then $L$ is the trivial knot, we can assume that $\sigma(L)\neq 0$. Therefore by Theorem 2.3 $\sigma(L)$ and $\sigma(L_{0})$ will have the same sign.

Since $L_{+}$, $L_{-}$ and $L_{0}$ are related by the Conway moves at a crossing, $m_{0}\in\{m,m-1,m+1\}$. We have that $2g(L_{0})\geq|\sigma(L_{0})|-(m_{0}-1)$. We have two cases,

First case: If $L=L_{+}$, then $\sigma(L)=\sigma(L_{0})-1$ by Theorem 2.3, and we have that

• •

$2g(L_{0})\geq 2g(L)$ if $\sigma(L)<0$ and $m_{0}\in\{m,m-1\}$.

• •

$2g(L_{0})\geq 2g(L)-2$ if $\sigma(L)<0$ and $m_{0}=m+1$.

• •

$2g(L_{0})>2g(L)$ if $\sigma(L)>0$ and $m_{0}\in\{m,m-1\}$.

• •

$2g(L_{0})\geq 2g(L)$ if $\sigma(L)>0$ and $m_{0}=m+1$.

Since $L$ and $L_{0}$ are quasi-alternating links then they are non-split links. Therefore by (1), (2), (3) of Corollary 2.13, Lemma 2.14 and the fact that $L$ is definite, we have that $m\neq m_{0}$ and,

$(\ast)$ \begin{cases}2g(L_{0})=2g(L), \hskip 31.2982pt if $m_{0}=m-1,\;L=L_{+}\;{\rm and}\;\sigma(L)<0$; $\\ 2g(L_{0})=2g(L)-2$,;;; ; if $m_{0}=m+1\;L=L_{+}\;{\rm and}\;\sigma(L)<0$. \end{cases}

In both cases $L_{0}$ is definite.

Second case: If $L=L_{-}$, then $\sigma(L)=\sigma(L_{0})+1$ by Theorem 2.3, and we have that

• •

$2g(L_{0})\geq 2g(L)$ if $\sigma(L)>0$ and $m_{0}\in\{m,m-1\}$.

• •

$2g(L_{0})\geq 2g(L)-2$ if $\sigma(L)>0$ and $m_{0}=m+1$.

• •

$2g(L_{0})>2g(L)$ if $\sigma(L)<0$ and $m_{0}\in\{m,m-1\}$.

• •

$2g(L_{0})\geq 2g(L)$ if $\sigma(L)<0$ and $m_{0}=m+1$.

Since $L$ and $L_{0}$ are quasi-alternating links then they are non-split links. Therefore by (1), (2), (3) of Corollary 2.13, Lemma 2.14 and the fact that $L$ is definite, we have that $m\neq m_{0}$ and,

$(\ast\ast)$ \begin{cases}2g(L_{0})=2g(L), \hskip 31.2982pt if $m_{0}=m-1,\;L=L_{-}\;{\rm and}\;\sigma(L)>0$; $\\ 2g(L_{0})=2g(L)-2$,;;; ; if $m_{0}=m+1\;L=L_{-}\;{\rm and}\;\sigma(L)>0$. \end{cases}

In both cases $L_{0}$ is definite.

Combining $(\ast)$ and $(\ast\ast)$ we have that,

\begin{cases}2g(L_{0})=2g(L), \hskip 31.2982pt if $m_{0}=m-1,\;L=L_{+}\;{\rm and}\;\sigma(L)<0$; $\\ 2g(L_{0})=2g(L)$, \hskip 31.2982pt if $m_{0}=m-1,\;L=L_{-}\;{\rm and}\;\sigma(L)>0$; $\\ 2g(L_{0})=2g(L)-2$,;;; ; if $m_{0}=m+1\;L=L_{+}\;{\rm and}\;\sigma(L)<0$; $\\ 2g(L_{0})=2g(L)-2$,;;; ; if $m_{0}=m+1\;L=L_{-}\;{\rm and}\;\sigma(L)>0$. \end{cases}

∎

Lemma 2.15**.**

(Claim 2 in [ST])* Assume that $-\chi(L_{+})=1-\chi(L_{0})$ and let $S_{0}$ be any Seifert surface of $L_{0}$ which is taut in $\mathbb{S}^{3}\setminus K$ $($as in Figure 7$)$. Then $S_{0}$ remains taut either in $\mathbb{S}^{3}$ or in the manifold obtained by doing [math]-surgery along $K$ (see Figure 7. In the latter case, the surface $S_{-}$ obtained by altering $S_{0}$ locally as in Figure 7 is a taut Seifert surface for $L_{-}$ in $\mathbb{S}^{3}$ and $\chi(S_{-})=\chi(S_{0})+1$. $($An analogous conclusion holds when $-\chi(L_{-})=1-\chi(L_{0})$.$)$*

We have the following theorem which was proved by Boileau-Boyer-Gordon in [BBG] using another method.

Theorem 2.16**.**

Let $L$ be a non-split alternating link with $m$ components, then $L$ is definite if and only if it is positive or negative.

Proof.

Let $L$ be an oriented, non-split alternating link. Choose a reduced alternating projection $D$ of $L$. Then each crossing of $D$ is a quasi-alternating crossing.

$\Leftarrow$) If the link $L$ is positive or negative then, since $L$ is quasi-alternating, it is definite by Corollary 2.2 and [Ru2].

$\Rightarrow$) Assume that the link $L$ is definite. We will show by induction on det $L=n$ that $L$ is positive or negative. Since, if $\sigma(L)=0$ then $L$ is the trivial knot, we can assume that $\sigma(L)\neq 0$. Therefore, det $L\geq 2$.

If $n=$ det $L=2$, then, by [LS], $L$ is the Hopf link. Therefore, $L$ is positive or negative. Now assume that $n={\rm det}$ $L>2$, and any definite, non-split alternating link $L^{\prime}$ with det $L^{\prime}<n$ is positive or negative. We show that $L$ is positive or negative.

Since $\sigma(L)\neq 0$, then $\sigma(L)$ and $\sigma(L_{0})$ have the same sign by Theorem 2.3. Since $L$ is definite then $L_{0}$ is also definite by Proposition 1.4. Therefore, $L_{0}$ is a positive or negative link by the induction hypothesis or it is the trivial knot. Without loss of generality assume that $L_{0}$ is a positive link. Therefore, $\sigma(L_{0})\leq 0$ by Theorem 2.8 and hence $\sigma(L)<0$. By the proof of Proposition 1.4, if $m$ and $m_{0}$ are the number of components of $L$ and $L_{0}$ respectively then we have that $m\neq m_{0}$ and

\begin{cases}2g(L_{0})=2g(L), \hskip 31.2982pt if $m_{0}=m-1,\;L=L_{+}\;{\rm and}\;\sigma(L)<0$; $\\ 2g(L_{0})=2g(L)$, \hskip 31.2982pt if $m_{0}=m-1,\;L=L_{-}\;{\rm and}\;\sigma(L)>0$; $\\ 2g(L_{0})=2g(L)-2$,;;; ; if $m_{0}=m+1\;L=L_{+}\;{\rm and}\;\sigma(L)<0$; $\\ 2g(L_{0})=2g(L)-2$,;;; ; if $m_{0}=m+1\;L=L_{-}\;{\rm and}\;\sigma(L)>0$. \end{cases}

Therefore, since each crossing of $D$ is a quasi-alternating crossing and $\sigma(L)<0$, each crossing of $D$ is a positive crossing.

∎

Theorem 2.17**.**

(Proposition 7.1 in [BBG])* Let $L$ be a non-split alternating link with $m$ components, then $L$ is definite if and only if it is special alternating.*

Theorem 2.18**.**

(Corollary 1.3 in [Gr])* Let $L$ be a special alternating link. Every minimal genus Seifert surface for $L$ is given by applying Seifert algorithm to some special alternating diagram for $L$.*

Proof of Theorem 1.1.

Let $L$ be an oriented quasi-alternating link with a quasi-alternating crossing $c$ such that $L_{0}$ is alternating.

$\Leftarrow$) If the link $L$ is strongly quasipositive then, since $L$ is quasi-alternating, it is definite by Corollary 2.2.

$\Rightarrow$) Assume that the link $L$ is definite. We will show by induction on det $L=n$ that $L$ is strongly quasipositive. Since, if $\sigma(L)=0$ then $L$ is the trivial knot, we can assume that $\sigma(L)\neq 0$. Therefore, det $L\geq 2$.

If $n=$ det $L=2$, then by [LS] $L$ is the Hopf link. Therefore, $L$ is strongly quasipositive. Now assume that $n={\rm det}$ $L>2$, and any definite quasi-alternating link $L^{\prime}$ with a quasi-alternating crossing $c^{\prime}$ such that $L_{0}^{\prime}$ is alternating with det $L^{\prime}<n$ is strongly quasipositive. We show that $L$ is strongly quasipositive.

Since $\sigma(L)\neq 0$, then $\sigma(L)$ and $\sigma(L_{0})$ have the same sign by Theorem 2.3. Since $L$ is definite then $L_{0}$ is also definite by Proposition 1.4. Therefore, since $L_{0}$ is alternating, it is strongly quasipositive by the induction hypothesis. By the proof of Proposition 1.4, if $m$ and $m_{0}$ are the number of components of $L$ and $L_{0}$, respectively, then we have that $m\neq m_{0}$ and

\begin{cases}2g(L_{0})=2g(L), \hskip 31.2982pt if $m_{0}=m-1,\;L=L_{+}\;{\rm and}\;\sigma(L)<0$; $\\ 2g(L_{0})=2g(L)$, \hskip 31.2982pt if $m_{0}=m-1,\;L=L_{-}\;{\rm and}\;\sigma(L)>0$; $\\ 2g(L_{0})=2g(L)-2$,;;; ; if $m_{0}=m+1\;L=L_{+}\;{\rm and}\;\sigma(L)<0$; $\\ 2g(L_{0})=2g(L)-2$,;;; ; if $m_{0}=m+1\;L=L_{-}\;{\rm and}\;\sigma(L)>0$. \end{cases}

Therefore, only the following three cases are possible

1. (1)

$-\chi(L_{+})=1-\chi(L_{0})>-\chi(L_{-})$ with $L=L_{+}$ and $\sigma(L)<0$; 2. (2)

$-\chi(L_{-})=1-\chi(L_{0})>-\chi(L_{+})$ with $L=L_{-}$ and $\sigma(L)>0$; 3. (3)

$-\chi(L_{+})=-\chi(L_{-})=1-\chi(L_{0})$ with $L=L_{+}$ and $\sigma(L)<0$.

We will study each of the cases (1), (2) and (3):

(1) Assume that $-\chi(L_{+})=1-\chi(L_{0})>-\chi(L_{-})$. Then $L=L_{+}$ and $\sigma(L)<0$. Therefore by Lemma 2.14, there are taut Seifert surfaces $S$, $S_{0}$ for $L_{+}$ and $L_{0}$, respectively, which appear as in Figure 6, i.e $S$ is obtained from $S_{0}$ by plumbing on a positive Hopf band. Since $L_{0}$ is definite and alternating, it is positive or negative by Theorem 2.16. Since $\sigma(L)<0$, then $\sigma(L_{0})\leq 0$. Therefore $L_{0}$ is positive. Since $L_{0}$ is positive and alternating then it is special alternating. Since $S_{0}$ is a minimal genus Seifert surface for $L_{0}$, by Theorem 2.18 $S_{0}$ is obtained by applying the Seifert algorithm to a special alternating diagram $D_{0}$ for $L_{0}$. Since $L_{0}$ is positive, then $S_{0}$ is quasipositive by [Ru2]. Therefore, by Theorem 2.5 $S$ is also quasipositive, and hence $L$ is strongly quasipositive.

(2) Assume that $-\chi(L_{-})=1-\chi(L_{0})>-\chi(L_{+})$. Then $L=L_{-}$ and $\sigma(L)>0$. In this case, let $L^{\prime}$ be the mirror image of $L$. Then, using the same argument as in case (1) for $L^{\prime}$, we have that $L^{\prime}$ is strongly quasipositive.

(3) Assume that $-\chi(L_{+})=-\chi(L_{-})=1-\chi(L_{0})$. In this case $L$ can be equal to $L_{+}$ or $L_{-}$. Assume that $L=L_{+}$. Then $\sigma(L)<0$. By Lemma 2.15, if $S_{0}$ is any Seifert surface of $L_{0}$ which is taut in $\mathbb{S}^{3}\setminus K$ (as in Figure 7), then we have two subcases:

• •

$S_{0}$ remains taut in the manifold obtained by doing [math]-surgery to $K$. In this case, the surface $S_{-}$ obtained by altering $S_{0}$ locally as in Figure 7 is a taut Seifert surface for $L_{-}$ in $\mathbb{S}^{3}$ and $\chi(S_{-})=\chi(S_{0})+1$. Since $-\chi(L_{+})=-\chi(L_{-})=1-\chi(L_{0})$ then $S_{0}$ cannot be taut in $\mathbb{S}^{3}$ because in that case the surface $S_{-}$ obtained by altering $S_{0}$ locally as in Figure 7 is a taut Seifert surface for $L_{-}$ in $\mathbb{S}^{3}$ and $\chi(L)=\chi(L_{-})=\chi(S_{-})=\chi(S_{0})+1$. Let $L_{2}$ be the link obtained by plumbing on a positive $T(2,4)$ torus link to the Seifert surface $S_{0}$. Then $|\sigma(L_{2})|=|\sigma(L_{0})-1|=|\sigma(L)|=2g(L)+m-1$ and by Theorem 2.12 either $\chi(L_{2})=\chi(L)=\chi(L_{-})$ or $\chi(L_{2})=\chi(L_{0})-1$ which implies that $\chi(L_{2})=\chi(L)=\chi(L_{-})=\chi(L_{0})-1$ and $g(L_{2})=g(L)=g(L_{-})$, continuing in this way we can plumb on a positive $T(2,2l)$ torus link and still get a link $L^{\prime}$ with same genus as $L$. Hence we can assume that $L$ is as in case (1) and then use the same argument.

• •

$S_{0}$ remains taut in $\mathbb{S}^{3}$. Since $-\chi(L)=1-\chi(L_{0})$, the Seifert surface $S$ for $L$ obtained from $S_{0}$ by plumbing on a positive Hopf band is taut. Therefore, since $L_{0}$ is strongly quasipositive and alternating, it is special alternating, and since $S_{0}$ is a minimal genus Seifert surface for $L_{0}$, by Theorem 2.18 $S_{0}$ is obtained by applying the Seifert algorithm to a special alternating diagram $D_{0}$ for $L_{0}$. Since $L_{0}$ is definite, then $S_{0}$ is quasipositive by [BBG] and [Ru2]. Therefore, by Theorem 2.5 $S$ is also quasipositive, and hence $L$ is strongly quasipositive.

The case where $L=L_{-}$ is done in the same way using (2).

This completes the proof that $L$ is strongly quasipositive. ∎

Theorem 2.19**.**

(Theorem 5.1 in [Ko])* Let $L_{1}$, $L_{2}$ be links with minimal genus Seifert surfaces $R_{1}$, $R_{2}$, respectively, and $R$ a Murasugi sum of $R_{1}$ and $R_{2}$. Then the minimal genus Seifert surfaces for $L=\partial R$ are unique if and only if one of $L_{1}$, $L_{2}$, say $L_{1}$, is fibred and the minimal genus Seifert surfaces for $L_{2}$ are unique.*

Proof of Theorem 1.2.

Since fibred links have unique minimal genus Seifert surfaces, it suffices to show this Theorem when $L_{0}$ has a unique minimal genus Seifert surface.

Let $L$ be an oriented quasi-alternating link with a quasi-alternating crossing $c$ such that $L_{0}$ has a unique minimal genus Seifert surface.

$\Leftarrow$) If the link $L$ is strongly quasipositive, then it is definite by Corollary 2.2.

$\Rightarrow$) Assume that the link $L$ is definite. We will show by induction on det $L=n$ that $L$ is strongly quasipositive. We can assume that $\sigma(L)\neq 0$. Therefore, det $L\geq 2$.

If $n=$ det $L=2$, then by [LS] $L$ is the Hopf link. Therefore, $L$ is strongly quasipositive. Now assume that $n={\rm det}$ $L>2$, and any definite quasi-alternating link $L^{\prime}$ with a quasi-alternating crossing $c^{\prime}$ such that $L_{0}^{\prime}$ has a unique minimal genus Seifert surface and with det $L^{\prime}<n$ is strongly quasipositive. We show that $L$ is strongly quasipositive.

Since $L$ is definite then $L_{0}$ is also definite by Proposition 1.4. Hence, if $\sigma(L_{0})=0$ then $L_{0}$ is the trivial knot. Therefore, we can assume that $\sigma(L_{0})\neq 0$. Let $c_{0}$ be a quasi-alternating crossing for $L_{0}$. By the proof of Proposition 1.4, if $m_{0}$ and $m_{00}$ are the number of components of $L_{0}$ and $L_{00}$, respectively, then we have that $m_{0}\neq m_{00}$ and

\begin{cases}2g(L_{00})=2g(L_{0}), \hskip 31.2982pt if $m_{00}=m_{0}-1,\;L_{0}=L_{0+}\;{\rm and}\;\sigma(L_{0})<0$; $\\ 2g(L_{00})=2g(L_{0})$, \hskip 31.2982pt if $m_{00}=m_{0}-1,\;L_{0}=L_{0-}\;{\rm and}\;\sigma(L_{0})>0$; $\\ 2g(L_{00})=2g(L_{0})-2$,;;; ; if $m_{00}=m_{0}+1\;L_{0}=L_{0+}\;{\rm and}\;\sigma(L_{0})<0$; $\\ 2g(L_{00})=2g(L_{0})-2$,;;; ; if $m_{00}=m_{0}+1\;L_{0}=L_{0-}\;{\rm and}\;\sigma(L_{0})>0$. \end{cases}

Therefore, only the following three cases are possible

1. (1)

$-\chi(L_{0+})=1-\chi(L_{00})>-\chi(L_{0-})$ if $L_{0}=L_{0+}$ and $\sigma(L_{0})<0$; 2. (2)

$-\chi(L_{0-})=1-\chi(L_{00})>-\chi(L_{0+})$ if $L_{0}=L_{0-}$ and $\sigma(L_{0})>0$; 3. (3)

$-\chi(L_{0+})=-\chi(L_{0-})=1-\chi(L_{00})$ if $L_{0}=L_{0+}$ and $\sigma(L_{0})<0$ or $L_{0}=L_{0-}$ and $\sigma(L_{0})>0$.

Hence, we have that:

(1) Assume that $-\chi(L_{0+})=1-\chi(L_{00})>-\chi(L_{0-})$. Then $L_{0}=L_{0+}$ and $\sigma(L_{0})<0$. Therefore, by Lemma 2.14, there are taut Seifert surfaces $S$, $S_{0}$ for $L_{0+}$ and $L_{00}$, respectively, which appear as in Figure 6, i.e $S$ is obtained from $S_{0}$ by plumbing on a positive Hopf band. Since $L_{0}$ has a unique minimal genus Seifert surface, by Theorem 2.19 $L_{00}$ has also a unique minimal genus Seifert surface.

(2) Assume that $-\chi(L_{0-})=1-\chi(L_{00})>-\chi(L_{0+})$. Then $L_{0}=L_{0-}$ and $\sigma(L_{0})>0$. Therefore by Lemma 2.14, there are taut Seifert surfaces $S$, $S_{0}$ for $L_{0}$ and $L_{00}$, respectively, which appear as in Figure 6, i.e $S$ is obtained from $S_{0}$ by plumbing on a negative Hopf band. Since $L_{0}$ has a unique minimal genus Seifert surface, by Theorem 2.19 $L_{00}$ has also a unique minimal genus Seifert surface.

(3) Assume that $-\chi(L_{0+})=-\chi(L_{0-})=1-\chi(L_{00})$. In this case $L_{0}$ can be equal to $L_{0+}$ or $L_{0-}$. Assume that $L_{0}=L_{0+}$. Then $\sigma(L_{0})<0$. By Lemma 2.15, if $S_{0}$ is any Seifert surface of $L_{00}$ which is taut in $\mathbb{S}^{3}\setminus K$ (as in Figure 7), then we have two subcases:

• •

$S_{0}$ remains taut in the manifold obtained by doing [math]-surgery to $K$. In this case, the surface $S_{-}$ obtained by altering $S_{0}$ locally as in Figure 7 is a taut Seifert surface for $L_{0-}$ in $\mathbb{S}^{3}$ and $\chi(S_{-})=\chi(S_{0})+1$. Since $-\chi(L_{0+})=-\chi(L_{0-})=1-\chi(L_{00})$ then $S_{0}$ cannot be taut in $\mathbb{S}^{3}$ because in that case the surface $S_{-}$ obtained by altering $S_{0}$ locally as in Figure 7 is a taut Seifert surface for $L_{0-}$ in $\mathbb{S}^{3}$ and $\chi(L_{0})=\chi(L_{0-})=\chi(S_{-})=\chi(S_{0})+1$. Let $L_{2}$ be the link obtained by plumbing a positive $T(2,4)$ torus link to the Seifert surface $S_{0}$. Then $|\sigma(L_{2})|=|\sigma(L_{00})-1|=|\sigma(L_{0})|=2g(L_{0})+m_{0}-1$ and by Theorem 2.12 either $\chi(L_{2})=\chi(L_{0})=\chi(L_{0-})$ or $\chi(L_{2})=\chi(L_{00})-1$ which implies that $\chi(L_{2})=\chi(L_{0})=\chi(L_{0-})=\chi(L_{00})-1$ and $g(L_{2})=g(L_{0})=g(L_{0-})$, continuing in this way we can plumb on a positive $T(2,2l)$ torus link and still get a link $L^{\prime}$ with same genus as $L_{0}$. Hence we can assume that $L_{0}$ is as in case (1) and then use the same argument.

• •

$S_{0}$ remains taut in $\mathbb{S}^{3}$. Since $-\chi(L_{0+})=1-\chi(L_{00})$, the Seifert surface $S$ for $L_{0}$ obtained from $S_{0}$ by plumbing on a positive Hopf band is taut. Since $L_{0}$ has a unique minimal genus Seifert surface, by Theorem 2.19 $L_{00}$ has also a unique minimal genus Seifert surface.

The case where $L_{0}=L_{0-}$ is done in the same way using (2).

Therefore, the link $L_{0}$ has a quasi-alternating crossing $c_{0}$ such that $L_{00}$ has a unique minimal genus Seifert surface. Thus $L_{0}$ is a strongly quasipositive link by the induction hypothesis. Since $L_{0}$ has a unique minimal genus Seifert surface then, using a similar argument as in the proof of Theorem 1.1, we show that $L$ is also strongly quasipositive. ∎

Remark 2.20**.**

Since every quasi-alternating link gives rise to an infinite family of quasi-alternating links obtained by replacing a quasi-alternating crossing with an alternating rational tangle [ChK], Theorem 1.1 and Theorem 1.2 can be extended to more classes of quasi-alternating links.**

Let L be a link and $D$ a link diagram of $L$. Checkerboard color the regions of the complement of the diagram in $\mathbb{R}^{2}$. Assume that the unbounded region $X_{0}$ is colored white. The other white regions will be called by $X_{1}$, $X_{2}$, $\cdots$, $X_{n}$. To any crossing $p$ of L we associate the number $\chi(p)$ which is +1 or -1 according to the convention in the Figure 1.

Let $B(D)$ be the following graph: a planar graph whose vertices are in one-to-one correspondence with the black regions in our checkerboard coloring, and whose edges correspond to crossings in the diagram. Call this graph the black graph of $D$.

Theorem 2.21**.**

(4.6 in [Cr])* Let $Tr(D)$ be the set of maximal trees in $B(D)$ (that is trees which contain all the vertices of $B(D)$) and $E$ be the set of all edges of $B(D)$. Then*

[TABLE]

Proposition 2.22**.**

Let $L$ be a non-alternating quasi-alternating link diagram with a quasi-alternating crossing c such that the links $L^{c}_{0}$ and $L^{c}_{\infty}$ are alternating. If the Conway notation of $L$ has only one negative sign, then $c$ is positive.

Proof.

Assume that the Conway notation of $L$ has only one negative sign. By contradiction let $c$ be the negative crossing and $A$ be the set of all maximal trees of $B(D)$ which contains $c$. Then by Theorem 2.21, we have also that,

[TABLE]

Therefore, the crossing $c$ is positive. ∎

Remark 2.23**.**

Proposition 2.22 states that for any non-alternating, quasi-alternating diagram $D$, the diagrams $D_{0}$ and $D_{\infty}$ are not alternating. They can be alternating only up to isotopy.

Proposition 2.24**.**

Let $L$ be non-alternating quasi-alternating link diagram with a quasi-alternating crossing c such that $L^{c}_{0}$ and $L^{c}_{\infty}$ are alternating (up to isotopy) and whose checkerboard color has only two negative crossings of the form [-2] or [-$\frac{1}{2}$]. If $L=L_{{\rm[-\frac{1}{2}]}}$ and $c$ is negative then $L_{{\rm[-2]}}$ is not quasi-alternating at the two negative crossings and vice versa, where $L_{{\rm[-2]}}$ is the link diagram obtained from $L_{{\rm[-\frac{1}{2}]}}$ by replacing the elementary tangle [-$\frac{1}{2}$] by [-2].

Proof.

Assume that $L=L_{{\rm[-\frac{1}{2}]}}$ and $c$ is negative. We show that $L^{\prime}=L_{{\rm[-2]}}$ is not quasi-alternating at the two negative crossings. Let $c_{1}$ and $c_{2}$ be the two negative crossings of $L_{{\rm[-2]}}$. Without loss of generality we can assume that the path $c_{1}c_{2}$ form a loop in $B(L^{\prime})$. Any maximal tree of $B(L^{\prime})$ contains at most one of them. Let $A$ be the set of all maximal trees of $B(L^{\prime})$ which contains $c_{1}$ and $A^{\prime}$ be the set of all maximal trees of $B(L^{\prime})$ which contains $c_{2}$. We have that

[TABLE]

Here the resolutions are applied in the two negative crossings in both $L^{\prime}=L_{{\rm[-2]}}$ and $L=L_{{\rm[-\frac{1}{2}]}}$.

In a similar way, we show that ${\rm det}\;L=|-2{\rm det}\;L_{0}+{\rm det}\;L_{\infty,\infty}|$, ${\rm det}\;L^{\prime}_{0}={\rm det}\;L_{\infty}=|-{\rm det}\;L_{\infty,\infty}+{\rm det}\;L_{0}|$, ${\rm det}\;L^{\prime}_{\infty}={\rm det}\;L_{\infty,\infty}$ and ${\rm det}\;L_{0}={\rm det}\;L^{\prime}_{0,0}$. Since $L=L_{{\rm[-\frac{1}{2}]}}$ is quasi-alternating and $c$ is negative then ${\rm det}\;L={\rm det}\;L_{\infty}+{\rm det}\;L_{0}=|-{\rm det}\;L_{\infty,\infty}+{\rm det}\;L_{0}|+{\rm det}\;L_{0}=|-2{\rm det}\;L_{0}+{\rm det}\;L_{\infty,\infty}|$, so ${\rm det}\;L=2{\rm det}\;L_{0}-{\rm det}\;L_{\infty,\infty}$, which implies that ${\rm det}\;L_{\infty}={\rm det}\;L_{0}-{\rm det}\;L_{\infty,\infty}$. Hence, if $L^{\prime}=L_{{\rm[-2]}}$ was quasi-alternating, then ${\rm det}\;L^{\prime}={\rm det}\;L^{\prime}_{\infty}+{\rm det}\;L^{\prime}_{0}={\rm det}\;L_{\infty,\infty}+{\rm det}\;L_{0}-{\rm det}\;L_{\infty,\infty}={\rm det}\;L_{0}={\rm det}\;L^{\prime}_{0,0}$, and this contradict the fact that ${\rm det}\;L^{\prime}=|-2{\rm det}\;L^{\prime}_{\infty}+{\rm det}\;L^{\prime}_{0,0}|$.

∎

3. strongly quasipositive montesinos links

A Montesinos link, denoted $M(e;t_{1},t_{2},\cdots,t_{r})$, is a link having a diagram of the form depicted in Figure 9, where $t_{i}=\frac{\beta_{i}}{\alpha_{i}}=[c_{1}^{i},\cdots,c_{m_{i}}^{i}]$, each $t_{i}$ represents a rational tangle (for the definition of rational tangle see Figure 10), and $e\in\mathbb{Z}$ represents $|e|$-half twists. We make the following assumptions in all this paper: $\alpha_{i}>1$ and $-\alpha_{i}<\beta_{i}<\alpha_{i}$, with gcd$(\alpha_{i},\beta_{i})=1$.

If $T(\frac{\beta}{\alpha})$ is a rational tangle of slope $\frac{\beta}{\alpha}$ with $\alpha>1$, $-\alpha<\beta<\alpha$ and gcd$(\alpha,\beta)=1$, then the upper left and the lower left corners of $T(\frac{\beta}{\alpha})$ are connected by a strand if and only if $\alpha$ is even. Therefore, for $K=M(e;t_{1},t_{2},\cdots,t_{r})$ to be a knot rather than a link, at most one of $\alpha_{1}$, $\alpha_{2}$,$\cdots$, $\alpha_{r}$ can be even. Since a cyclic permutation of indices does not change the knot type, we assume that $\alpha_{2}$,$\cdots$, $\alpha_{r}$ are odd if $K=M(e;t_{1},t_{2},\cdots,t_{r})$ is a knot.

Let $T(\frac{\beta}{\alpha})$ be a rational tangle of slope $\frac{\beta}{\alpha}$ with $\alpha>1$, $-\alpha<\beta<\alpha$ and gcd$(\alpha,\beta)=1$, such that

[TABLE]

The tangle $T(\frac{\beta}{\alpha})$ is represented in Figure 10.

Therefore, up to now on each of the $t_{i}$ box of the Montesinos link $M(e;t_{1},t_{2},\cdots,t_{r})$ in Figure 9 will be represented by a rational tangle as the one in Figure 10.

Remark 3.1**.**

If $r\leq 2$ then the Montesinos link $M(e;t_{1},t_{2},\cdots,t_{r})$ is a 2-bridge link. Therefore, throughout this paper we can assume that $r\geq 3$. **

Let $C=[c_{1},c_{2},\cdots,c_{m}]$ be a continued fraction. We say that $C$ is an even continued fraction if all the $c_{j}$ are even. We say that $C$ is a strict continued fraction if

1. (1)

$c_{j}$ is even for any odd $j$, and 2. (2)

$c_{j}c_{j+1}<0$ whenever $j$ is odd and $|c_{j}|=2$.

Lemma 3.2**.**

(Proposition 2.5 in [HM])* Let $\frac{\beta}{\alpha}$ be a rational number such that $\alpha$ is odd, $-\alpha<\beta<\alpha$, gcd$(\alpha,\beta)=1$, and $2|\beta|<\alpha$. Then $\frac{\beta}{\alpha}$ has a strict continued fraction.*

Proof of Proposition 1.5.

(1) In this case $L$ is a 2-component link. Since for any $i=1,\cdots,r$, $t_{i}=[2q_{1}^{i},2s_{1}^{i},2q_{2}^{i},2s_{2}^{i},\cdots,2q_{n_{i}}^{i},2s_{n_{i}}^{i}]$ where $q_{j}^{i}<0$, $s_{j}^{i}<0$, then each crossing of each tangle $T(t_{i}=\frac{\beta_{i}}{\alpha_{i}})$ has the form (a) as in Figure 11. Since $e\geq 0$ then each crossing of $e$ has the form (b) as in Figure 11. Since $e$ is even there exists an orientation of $L$ which makes it positive. Therefore, by [Ru2] $L$ is strongly quasipositive.

(2) In this case $L$ is a $r$-components link. Since for any $i=1,\cdots,r$, $t_{i}=[2q_{1}^{i},2s_{1}^{i},2q_{2}^{i},2s_{2}^{i},\cdots,2q_{n_{i}}^{i}]$ where $q_{j}^{i}<0$, $s_{j}^{i}<0$, then each crossing of each tangle $T(t_{i}=\frac{\beta_{i}}{\alpha_{i}})$ has the form (a) as in Figure 11. Since $e\geq 0$ then each crossing of $e$ has the form (b) as in Figure 11. Since $e$ is even, there exists an orientation of $L$ which makes it positive. Therefore, by [Ru2] $L$ is strongly quasipositive.

(3) In this case $L$ is a knot. Since all the continued fractions are even and $q_{j}^{i}<0$, $s_{j}^{i}<0$, then each crossing of each tangle $T(t_{i}=\frac{\beta_{i}}{\alpha_{i}})$ has the form (a) as in Figure 11. Since $e\geq 0$ then each crossing of $e$ has the form (b) as in Figure 11. Hence, there exists an orientation of $L$ which makes it positive. Therefore, by [Ru2] $L$ is strongly quasipositive. ∎

Proof of Proposition 1.6.

We prove this Proposition in the case where min$\{m_{i_{0}},m_{i_{0}+1}\}=2$ because the general case is done similarly. With some orientation of $K$ , the $i_{0}^{th}$ and $({i_{0}+1})^{th}$ strands are oriented as in Figure 12, which illustrates the case $c_{1}^{i_{0}}>0$ and $c_{1}^{i_{0}+1}<0$.

Performing a band move as shown in Figure 12, gives a punctured 2-sphere $A$ in $\mathbb{S}^{3}\times[0,1]$ such that $A\cap(\mathbb{S}^{3}\times\{1\})=K$ and $A\cap(\mathbb{S}^{3}\times\{0\})=K^{\prime}\#L(\frac{q}{p})$ where

[TABLE]

and $L(\frac{q}{p})$ is the two-bridge link with rational slope

[TABLE]

or

[TABLE]

Since $m_{i_{0}+1}$ and $m_{i_{0}}$ are even, $L(\frac{q}{p})$ bounds a Seifert surface $\Gamma$ of genus

[TABLE]

In $\mathbb{S}^{3}\times\{0\}$, $K^{\prime}\#L(\frac{q}{p})$ bounds the boundary connected sum $S$ of $\Gamma$ with a minimal genus Seifert surface for $K^{\prime}$. Therefore, taking the union of this surface $S$ with the punctured 2-sphere $A$ shows that

[TABLE]

To complete the proof we compare the genus $g_{4}(K)$ and $g(K)$, and to compute the genus of $K$ and $K^{\prime}$ we use [HM], we have three cases:

First case: If $e\neq 0$, then

[TABLE]

Second case: If $e=0$ and ($c_{1}^{1},c_{1}^{2},\cdots,c_{1}^{r})\neq\pm(1,-1,\cdots,1,-1)$ then

[TABLE]

Third case: If $e=0$ and ($c_{1}^{1},c_{1}^{2},\cdots,c_{1}^{r})=\pm(1,-1,\cdots,1,-1)$ then $g(K)=\frac{1}{2}(1+\sum_{i=1}^{r}m_{i})-(p+1)$, where $p=min\{p_{1},\cdots,p_{r}\}$, $p_{i}$ is the number of leading $2$’s in $t_{i}$ if $i$ is odd, or the number of leading $-2$’s, if $i$ is even. Therefore, we have that

[TABLE]

Since $g(K)>g_{4}(K)$ in all cases then $K$ is not strongly quasipositive. ∎