Arc index of pretzel knots of type $(-p, q, r)$

Abstract

We computed the arc index for some of the pretzel knots $K=P(-p,q,r)$ with $p,q,r\ge 2$, $r\geq q$ and at most one of $p,q,r$ is even. If $q=2$, then the arc index $\alpha (K)$ equals the minimal crossing number $c(K)$. If $p\ge 3$ and $q=3$, then $\alpha (K)=c(K)-1$. If $p\ge 5$ and $q=4$, then $\alpha (K)=c(K)-2$.

1. Arc presentation

Let $D$ be a diagram of a knot or a link $L$. Suppose that there is a simple closed curve $C$ meeting $D$ in $k$ distinct points which divide $D$ into $k$ arcs $\alpha _1,\alpha _2,\ldots ,\alpha _k$ with the following properties:

(1)

Each $\alpha _i$ has no self-crossing.

(2)

If $\alpha _i$ crosses over $\alpha _j$ at a crossing, then $i>j$ and it crosses over $\alpha _j$ at any other crossings with $\alpha _j$.

(3)

For each $i$, there exists an embedded disk $d_i$ such that $\partial d_i=C$ and $\alpha _i\subset d_i$.

(4)

$d_i\cap d_j=C$, for distinct $i$ and $j$.

Then the pair $(D,C)$ is called an arc presentation of $L$ with $k$ arcs, and $C$ is called the axis of the arc presentation. Figure 1 shows an arc presentation of the trefoil knot. The thick round curve is the axis. It is known that every knot or link has an arc presentation 34. For a given knot or link $L$, the minimal number of arcs in all arc presentations of $L$ is called the arc index of $L$, denoted by $\alpha (L)$.

By removing a point $P$ from $C$ away from $L$, we may identify $C\setminus P$ with the $z$-axis and each $d_i\setminus P$ with a vertical half plane along the $z$-axis. This shows that an arc presentation is equivalent to an open-book presentation.

Figure 2.

An open-book presentation of the right-handed trefoil knot

Given a link $L$, let $c(L)$ denote the minimal crossing number of $L$.

Theorem 1.1 (Jin-Park).

A prime link $L$ is nonalternating if and only if

$$\alpha (L) \leq c(L).$$

2. Kauffman polynomial

The Kauffman polynomial $F_L(a,z)$ of an oriented knot or link $L$ is defined by

$$F_L(a,z)=a^{-w(D)}\Lambda _D(a,z)$$

where $D$ is a diagram of $L$, $w(D)$ the writhe of $D$ and $\Lambda _D(a,z)$ the polynomial determined by the rules K1, K2 and K3.

(K1)

$\Lambda _O(a,z)=1$ where $O$ is the trivial knot diagram.

(K2)

For any four diagrams $D_+$, $D_-$, $D_0$ and $D_\infty$ which are identical outside a small disk in which they differ as shown below,

we have the relation$$\Lambda _{D_+}(a,z)+\Lambda _{D_-}(a,z)=z(\Lambda _{D_0}(a,z)+\Lambda _{D_\infty }(a,z)).$$

(K3)

For any three diagrams $D_+$, $D$ and $D_-$ which are identical outside a small disk in which they differ as shown below,

we have the relation$$a\,\Lambda _{D_+}(a,z)=\Lambda _D(a,z)=a^{-1}\Lambda _{D_-}(a,z).$$

For a connected sum and a split union of two diagrams, $\Lambda$ satisfies the following properties:

(K4)

If $D$ is a connected sum of $D_1$ and $D_2$, then$$\Lambda _D(a,z) = \Lambda _{D_1}(a,z)\, \Lambda _{D_2}(a,z).$$

(K5)

If $D$ is the split union of $D_1$ and $D_2$, then$$\Lambda _D(a,z) = (z^{-1}a-1+z^{-1}a^{-1})\, \Lambda _{D_1}(a,z)\, \Lambda _{D_2}(a,z).$$

The Laurent degree in the variable $a$ of the Kauffman polynomial $F_L(a,z)$ is denoted by $\operatorname {spread}_a(F_L)$ and defined by the formula

$$\operatorname {spread}_a(F_L)= \operatorname {max-deg}_a (F_L) - \operatorname {min-deg}_a (F_L).$$

Notice that $\operatorname {spread}_a (F_L)=\operatorname {spread}_a (\Lambda _D)$ for any diagram $D$ of $L$. The following theorem gives an important lower bound for the arc index.

Theorem 2.1 (Morton-Beltrami).

Let $L$ be a link. Then

$$\alpha (L) \geq \operatorname {spread}_a (F_L)+2.$$

If $L$ is nonsplit and alternating, then the equality holds so that $\alpha (L)=c(L)+2$. This is shown by Bae and Park 1 using arc presentations in the form of wheel diagrams.

3. Pretzel knots

Given a sequence of integers $p_1,p_2,\ldots ,p_n$, we connect two disjoint disks by $n$ bands with $p_i$ half twists, $i=1,2,\ldots ,n$, so that the boundary of the resulting surface is a link as shown in Figure 3. This link is called the pretzel link of type $(p_1,p_2,\ldots ,p_n)$ and denoted by $P(p_1,p_2,\ldots ,p_n)$.

In the case $n=3$, the pretzel links satisfy the following properties:

Proposition 3.1.

Let $p$, $q$, and $r$ be nonzero integers.

(1)

The link type of $P(p,q,r)$ is independent of the order of $p,q,r$.

(2)

$P(p,q,r)$ is a knot if and only if at most one of $p,q,r$ is an even number.

In this work, we compute the arc index for the pretzel knots $K=P(-p, q, r)$ with $p, q,r\ge 2$. By Proposition 3.1(1), we may assume that $r\geq q$. By Theorem 3.2, we know that $P(-p,q,r)$ is a minimal crossing diagram of $K$, i.e., $c(K)=p+q+r$.

Theorem 3.2 (Lickorish-Thistlethwaite).

If a link $L$ admits a reduced Montesinos diagram having $n$ crossings, then $L$ cannot be projected with fewer than $n$ crossings.

This work was motivated by Theorem 3.3 which is a special case of Theorem 1.1.

Theorem 3.3 (Beltrami-Cromwell).

If $K=P(-p,q,r)$ is a knot with $p, q, r\ge 2$, then

$$\alpha (K)\leq c(K)=p+q+r.$$

By computing $\operatorname {spread}_a (F_K)$ and finding arc presentations of $K=P(-p,q,r)$ with the minimum number of arcs for various values of $p$, $q$ and $r$, we obtained sharper results.

4. Main results

Theorem 4.1.

If $K=P(-2, q, r)$ is a knot with $3\le q\le r$, then

$$\alpha (K)\le c(K)-1.$$

Theorem 4.2.

If $K=P(-p, 2, r)$ is a knot with $p\ge 3$, $r\ge 3$, then

$$\alpha (K)=c(K).$$

Theorem 4.3.

If $K=P(-p, 3, r)$ is a knot with $p\ge 3$, $r\ge 3$, then

$$\alpha (K)=c(K)-1.$$

Theorem 4.4.

If $K=P(-p, 4, r)$ is a knot with $p\ge 5$, $r\ge 5$, then

$$\alpha (K)=c(K)-2.$$

Theorem 4.5.

If $K=P(-3, 4, r)$ is a knot with $r\ge 7$, then

$$c(K)-4\le \alpha (K)\le c(K)-2.$$

5. Arc presentations of $P(-p,q,r)$

Proposition 5.1.

If $K=P(-2, q, r)$ is a knot with $3\le q\le r$, then $K$ has an arc presentation with $q+r+1$ arcs.

Proof.

Figure 4 shows a pretzel diagram of $P(-2,q,r)$ and its arc presentation with $q+r+1$ arcs. The thick curve is the axis of the arc presentation which cuts the knot at 1 place in the leftmost box, $q-1$ places in the second, 2 places in the third, and $r-1$ places in the fourth. The $q+r+1$ arcs of the knot satisfy the four properties of an arc presentation.

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Proposition 5.2.

If $K=P(-p,q,r)$ is a knot with $p\ge 3$ and $2\le q\le 3\le r$, then $K$ has an arc presentation with $p+r+2$ arcs.

Proof.

For each of $q=2,3$, Figure 5 shows a pretzel diagram of $P(-p,q,r)$ and its arc presentation with $p+r+2$ arcs. The thick curve is the axis of the arc presentation which cuts the knot at $p-1$ places in the leftmost box, $4$ places in the second, and $r-1$ places in the third. The $p+r+2$ arcs of the knot satisfy the four properties of an arc presentation.

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Proposition 5.3.

If $K=P(-p, q, r)$ is a knot with $p\geq 3$ and $4\le q\le r$, then $K$ has an arc presentation with $p+q+r-2$ arcs.

Proof.

In Figure 6, the diagram (a) shows a pretzel diagram of $P(-p,q,r)$ with $p\geq 3$ and $4\le q\le r$. The diagram (b) is obtained from (a) by two applications of the Reidemeister move of type 3. The diagram (c) shows an arc presentation with $p+q+r-1$ arcs. The diagram (d) is obtained from (c) by isotoping the arc labeled $x$ over the axis so that there are only $p+q+r-2$ arcs. Each of the seven boxes, from left to right, contains $1$, $p-3$, $2$, $3$, $q-3$, $1$, and $r-3$ arcs, respectively.

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6. The Kauffman polynomial of the pretzel knots $P(-p,q,r)$

For any link diagram $D$, the polynomial $\Lambda _D$ is of the form

$$\Lambda _D(a,z)=\sum _{i=m}^n f_i(z)a^i$$

where $m,n$ are integers with $m\le n$, and $f_i(z)$’s are polynomials in $z$ with integer coefficients such that $f_m(z)\ne 0$ and $f_n(z)\ne 0$. To simplify our computation of $\operatorname {spread}_a(\Lambda _D)$ we use the notation

$$f_m(z)=\langle k_m z^{h_m}\rangle ,$$

$$f_n(z)=\langle k_n z^{h_n}\rangle ,$$

$$\sum _{i=m}^n f_i(z)a^i=\left[\langle k_n z^{h_n}\rangle a^{n},\langle k_m z^{h_m}\rangle a^{m}\right],\quad (m<n)$$

where $k_mz^{h_m}$ and $k_nz^{h_n}$ are the highest degree terms in $f_m(z)$ and $f_n(z)$, respectively. For example, we write

$$z(z^2-1)a^{-1}+z^2a^{-2}-2za^{-3}=\left[\langle z^3\rangle a^{-1},\langle -2z\rangle a^{-3}\right].$$

We also use the notation $\Lambda _{(p_1,p_2,\ldots ,p_n)}$ for $\Lambda _D$ when $D=P(p_1,p_2,\ldots ,p_n)$.

Figure 7.

Links $P(-m,0)$, $P(0,n)$, and $P(-m,n)$

Let the polynomial $\Phi _i(z)$ be defined by $\Phi _0(z)=1$, $\Phi _1(z)=z$ and⁠¹

¹

$\Phi _i(2z)$’s are the Chebyshev’s polynomials of the second type.

✖

$$\Phi _{i+1}(z)=z\Phi _i(z)-\Phi _{i-1}(z).$$

Lemma 6.1.

Let $m,n$ be nonnegative integers. Then

$$\Lambda _{(-m,n)}= \begin{cases} \ \left[\langle z\rangle a^{k-1},\langle z^{k-1}\rangle a^{-1}\right] &\text{if } k=m-n>1, \\ \ a^{-1} &\text{if } k=m-n=1, \\ \ z^{-1}a-1+z^{-1}a^{-1} &\text{if } k=n-m=0, \\ \ a &\text{if } k=n-m=1, \\ \ \left[\langle z^{k-1}\rangle a,\langle z\rangle a^{-(k-1)}\right] &\text{if } k=n-m>1. \\ \end{cases}$$

More precisely,

$$\Lambda _{(-m,n)}= \begin{cases} \ \left[za^{m-1},z^{-1}\Phi _m(z)a^{-1}\right] &\text{if } m\geq 3,\ \ n=0, \\ \ \left[z^{-1}\Phi _n(z)a,za^{-(n-1)}\right] &\text{if } m=0,\ \ n\geq 3. \\ \end{cases}$$

Proof.

Using the skein relations K1, K2, and K3, we have the following

$$\Lambda _{(-1,0)}=a^{-1},\ \ \Lambda _{(0,1)}=a$$

$$\Lambda _{(0,0)}=z^{-1}a-1+z^{-1}a^{-1}.$$

Let $D_v$ be a vertical integer tangle which has $|v|$ times half-twists in the positive or negative direction according to the sign of $v$ for an integer $v$. $D_{+1}$, $D_{-1}$, $D_{0}$, and $D_{\infty }$ are exemplified in ${\mathrm{(K2)}}$. In the sublemma below, we use the following simplified notation:

$$\Lambda _v=\Lambda _{D_v},\ \ \Lambda _{\infty }=\Lambda _{D_{\infty }}.$$

Sublemma.

For $m\ge 2$ and $n\ge 2$, we have

$$\begin{eqnarray} \Lambda _{-m} &=& -\Phi _{m-2}(z)\,\Lambda _0+\Phi _{m-1}(z)\,\Lambda _{-1}+\left(\sum _{i=0}^{m-2}z\Phi _i(z)a^{m-1-i}\right)\,\Lambda _\infty , {\tag{6.1}}\\ \Lambda _{n} &=& -\Phi _{n-2}(z)\,\Lambda _0+\Phi _{n-1}(z)\,\Lambda _{+1}+\left(\sum _{j=0}^{n-2}z\Phi _j(z)a^{-(n-1-j)}\right)\,\Lambda _\infty . {\tag{6.2}} \end{eqnarray}$$

Proof of Sublemma.

As shown below, equation 6.1 holds when $m=2, 3$.

$$\begin{aligned} \Lambda _{-2} &= -\Lambda _0+z\Lambda _{-1}+za\Lambda _\infty \\ &= -\Phi _{0}(z)\,\Lambda _0+\Phi _{1}(z)\,\Lambda _{-1}+z\Phi _0(z)a\,\Lambda _\infty \\ \Lambda _{-3} &= -\Lambda _{-1}+z\Lambda _{-2}+za^2\Lambda _\infty \\ &= -\Phi _0(z)\,\Lambda _{-1}+z\left(-\Phi _{0}(z)\,\Lambda _0+\Phi _{1}(z)\,\Lambda _{-1}+z\Phi _0(z)a\,\Lambda _\infty \right)+za^2\Lambda _\infty \\ &= -\Phi _{1}(z)\,\Lambda _0+\Phi _{2}(z)\,\Lambda _{-1}+\left(\sum _{i=0}^1z\Phi _i(z)a^{2-i}\right)\,\Lambda _\infty .\\ \end{aligned}$$

Suppose that equation 6.1 holds for $2\le m\le k-1$ for some $k\ge 4$. Then

$$\begin{aligned} \Lambda _{-k} &= -\Lambda _{-(k-2)}+z\Lambda _{-(k-1)}+za^{k-1}\Lambda _\infty \\ &= -\left(-\Phi _{k-4}(z)\,\Lambda _0+\Phi _{k-3}(z)\,\Lambda _{-1}+\left(\sum _{i=0}^{k-4}z\Phi _i(z)a^{k-3-i}\right)\,\Lambda _\infty \right)\\ & \phantom {={}}+z\left(-\Phi _{k-3}(z)\,\Lambda _0+\Phi _{k-2}(z)\,\Lambda _{-1}+\left(\sum _{i=0}^{k-3}z\Phi _i(z)a^{k-2-i}\right)\,\Lambda _\infty \right)\! +\!za^{k-1}\Lambda _\infty \\ &= -\Phi _{k-2}(z)\,\Lambda _0+\Phi _{k-1}(z)\,\Lambda _{-1}+\left(\sum _{i=0}^{k-2}z\Phi _i(z)a^{k-1-i}\right)\,\Lambda _\infty .\\ \end{aligned}$$

This proves that equation 6.1 holds for $m\ge 2$. In a similar manner, we can prove that equation 6.2 holds for $n\ge 2$.■

For $m\geq 2$ and $n\geq 2$, using equations 6.1 and 6.2 on $m$ and $n$ respectively, we obtain

$$\begin{aligned} \Lambda _{(-m,0)} &= -\Phi _{m-2}(z)\Lambda _{(0,0)}+\Phi _{m-1}(z)\Lambda _{(-1,0)}+\sum _{i=0}^{m-2}z\Phi _{i}(z)a^{m-1-i} \\ &=\begin{cases} \left[(z-z^{-1})a,(z-z^{-1})a^{-1}\right] &\text{if } m=2,\\ \left[za^{m-1},\{\Phi _{m-1}(z)-z^{-1}\Phi _{m-2}(z)\}a^{-1}\right] &\text{if } m>2 \end{cases}\\ &=\begin{cases} \left[z^{-1}\Phi _2(z)a,z^{-1}\Phi _2(z)a^{-1}\right] &\text{if } m=2,\\ \left[za^{m-1},z^{-1}\Phi _m(z)a^{-1}\right] &\text{if } m>2. \end{cases} \end{aligned}$$

$$\begin{aligned} \Lambda _{(0,n)} &= -\Phi _{n-2}(z)\Lambda _{(0,0)}+\Phi _{n-1}(z)\Lambda _{(0,1)}+\sum _{j=0}^{n-2}z\Phi _{j}(z)a^{-(n-1-j)} \\ &=\begin{cases} \left[(z-z^{-1})a,(z-z^{-1})a^{-1}\right] &\text{if } n=2,\\ \left[\{\Phi _{n-1}(z)-z^{-1}\Phi _{n-2}(z)\}a,za^{-(n-1)}\right] &\text{if } n>2 \end{cases}\\ &=\begin{cases} \left[z^{-1}\Phi _2(z)a,z^{-1}\Phi _2(z)a^{-1}\right] &\text{if } n=2,\\ \left[z^{-1}\Phi _n(z)a,za^{-(n-1)}\right] &\text{if } n>2. \end{cases} \end{aligned}$$ Since $\Lambda _D(a,z)$ is an invariant under regular isotopy of diagrams, we have

$$\Lambda _{(-m,n)}= \begin{cases} \ \Lambda _{(-k,0)} &\text{if } k=m-n\ge 1, \\ \ \Lambda _{(0,0)} &\text{if } k=m-n=0, \\ \ \Lambda _{(0,k)} &\text{if } k=n-m\ge 1. \\ \end{cases}$$

This completes the proof.

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Lemma 6.2.

$$\begin{aligned} \Lambda _{(-p,0,r)}&= \begin{cases} \ 1 &\text{if } p=r=1, \\ \ \left[\langle z\rangle a^p,\langle z^{p-1}\rangle a^0\right] &\text{if } p>1,\ r=1, \\ \ \left[\langle z^r\rangle a^p,\langle z^{p}\rangle a^{-r}\right] &\text{if } p>1,\ r>1. \\ \end{cases}\\ \Lambda _{(-p,1,r)}&= \begin{cases} \ a^2 &\text{if } p=0,\ r=1, \\ \ \left[\langle z^{r-1}\rangle a^2,\langle z\rangle a^{-(r-2)}\right] &\text{if } p=0,\ r>1, \\ \ a^{-r} &\text{if } p=1,\ r\ge 1, \\ \ \left[\langle z^{r+1}\rangle a^p,\langle z^{p-1}\rangle a^{-r}\right] &\text{if } p>1,\ r\ge 1. \\ \end{cases} \end{aligned}$$

More precisely, for $p\geq 3$ and $r\geq 3$

$$\Lambda _{(-p,0,r)}=\left[\Phi _r(z)a^{p},\ \Phi _p(z)a^{-r}\right]$$

and for $r\geq 2$

$$\begin{aligned} \Lambda _{(-p,1,r)}&=\begin{cases} \left[z^{-1}\Phi _{r+2}(z)a^2,\ \Phi _{1}(z)a^{-r}\right] &\text{if } p=2,\\ \left[\Phi _{r+1}(z)a^{p},\ \Phi _{p-1}(z)a^{-r}\right] &\text{if } p>2. \end{cases} \end{aligned}$$

Proof.

Since $P(-p,0,r)=P(-p,0)\sharp P(0,r)$, we have $\Lambda _{(-p,0,r)}=\Lambda _{(-p,0)}\,\Lambda _{(0,r)}$. Therefore the formula about $\Lambda _{(-p,0,r)}$ follows from Lemma 6.1.

Now we consider the formula about $\Lambda _{(-p,1,r)}$. Three cases with $p=0$ or $p=1$ follow from K1, K2, K3, and Lemma 6.1. The other case is derived by equation (6.1). For $p\geq 2$, we have

$$\begin{aligned} \Lambda _{(-p,1,r)} &= -\Phi _{p-2}(z) \Lambda _{(0,1,r)}+\Phi _{p-1}(z)\Lambda _{(-1,1,r)}+\sum _{i=0}^{p-2}z\Phi _{i}(z)a^{p-1-i}\Lambda _{(1,r)}\\ &= -\Phi _{p-2}(z)a \Lambda _{(0,r)}+\Phi _{p-1}(z)a^{-r}+\sum _{i=0}^{p-2}z\Phi _{i}(z)a^{p-1-i}\Lambda _{(0,r+1)}\\ &=\begin{cases} \left[\{-z^{-1}\Phi _{r}(z)+\Phi _{r+1}(z)\}a^2,\ \Phi _{1}(z)a^{-r}\right] &\text{if } p=2,\\ \left[\Phi _{r+1}(z)a^{p},\ \Phi _{p-1}(z)a^{-r}\right] &\text{if } p>2 \end{cases}\\ &=\begin{cases} \left[z^{-1}\Phi _{r+2}(z)a^2,\ \Phi _{p-1}(z)a^{-r}\right] &\text{if } p=2,\\ \left[\Phi _{r+1}(z)a^{p},\ \Phi _{p-1}(z)a^{-r}\right] &\text{if } p>2. \end{cases} \end{aligned}$$

This completes the proof.

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Proposition 6.3.

$\operatorname {spread}_a (\Lambda _{(-p, 2, r)}(a,z)) =p+r$ for $p\geq 3$, and $r\geq 3$.

Proof.

For $p\ge 3$, we show that

$$\begin{equation} \Lambda _{(-p,2,r)}= \begin{cases} \left[\langle z^{3}\rangle a^p,\langle z^{p-1}\rangle a^{-2}\right] & (\text{if } r=1),\\ \left[\langle z^{4}\rangle a^p,\langle 3z^{p-2}\rangle a^{-2}\right] & (\text{if } r=2),\\ \left[\langle z^{r+2}\rangle a^p,\langle z^{p-2}\rangle a^{-r}\right] & (\text{if } r\ge 3). \end{cases} {\tag{6.3}} \end{equation}$$

Using K1, K2, K3 and Lemmas 6.1 and 6.2, we obtain

$$\begin{align*} \Lambda _{(-p,2,1)}&=-\Lambda _{(-p,0, 1)}+z\Lambda _{(-p,1, 1)}+za^{-1}\Lambda _{(-p,1)}\\ &=-\left[\langle z\rangle a^{p},\langle z^{p-1}\rangle a^{0}\right] + z\left[\langle z^2\rangle a^p,\langle z^{p-1}\rangle a^{-1}\right] +za^{-1}\left[\langle z\rangle a^{p-2},\langle z^{p-2}\rangle a^{-1}\right]\\ &=\left[\langle z^3\rangle a^p,\langle z^{p-1}\rangle a^{-2}\right],\\ \Lambda _{(-p,2,2)}&=-\Lambda _{(-p,0, 2)}+z\Lambda _{(-p,1, 2)}+za^{-1}\Lambda _{(-p,2)}\\ &=-\left[za^{p-1}, z^{-1}\Phi _p(z)a^{-1}\right]\left[z^{-1}\Phi _2(z)a,z^{-1}\Phi _2(z)a^{-1}\right]\\ & \qquad \qquad + z\left[\Phi _3(z)a^p,\Phi _{p-1}(z)a^{-2}\right]\\ & \qquad \qquad + za^{-1} \begin{cases} a^{-1} & (\text{if } p=3),\\ \left[z^{-1}\Phi _2(z)a,z^{-1}\Phi _2(z)a^{-1}\right] & (\text{if } p=4),\\ \left[za^{p-3}, z^{-1}\Phi _{p-2}(z)a^{-1}\right] & (\text{if } p\ge 5)\\ \end{cases}\\ &=-\left[\Phi _2(z)a^p,z^{-2}\Phi _2(z)\Phi _p(z)a^{-2}\right]+z\left[\Phi _3(z)a^p,\Phi _{p-1}(z)a^{-2}\right]\\ & \qquad \qquad + \begin{cases} za^{-2} & (\text{if } p=3),\\ za^{-1}\left[z^{-1}\Phi _2(z)a,z^{-1}\Phi _2(z)a^{-1}\right] & (\text{if } p=4),\\ za^{-1}\left[za^{p-3}, z^{-1}\Phi _{p-2}(z)a^{-1}\right] & (\text{if } p\ge 5)\\ \end{cases}\\ &=\left[\Phi _4(z)a^p,\langle 3z^{p-2}\rangle a^{-2}\right], \end{align*}$$

which prove the first two cases of (6.3). Now we show the third case of (6.3) by an induction on $r$. For $r=3$, we have

$$\begin{align*} \Lambda _{(-p,2,3)}&=-\Lambda _{(-p,2, 1)}+z\Lambda _{(-p,2, 2)}+za^{-2}\Lambda _{(-p,2)}\\ &=-\left[\langle z^3\rangle a^p,\langle z^{p-1}\rangle a^{-2}\right]+z\left[\langle z^4\rangle a^p,\langle 3z^{p-2}\rangle a^{-2}\right]\\ & \qquad \qquad + \begin{cases} za^{-3} & (\text{if } p=3),\\ \left[\langle z^2\rangle a^{p-5},\langle z^{p-2}\rangle a^{-3}\right] & (\text{if } p\ge 4)\\ \end{cases}\\ &=\left[\langle z^5\rangle a^p,\langle z^{p-2}\rangle a^{-3}\right], \end{align*}$$

and for $r\ge 4$, inductively, we have

$$\begin{align*} \Lambda _{(-p,2,r)}&=-\Lambda _{(-p,2,r-2)}+z\Lambda _{(-p,2,r-1)}+za^{-(r-1)}\Lambda _{(-p,2)}\\ &=-\left[\langle z^r\rangle a^p,\langle *\rangle a^{-(r-2)}\right]+z\left[\langle z^{r+1}\rangle a^p,\langle z^{p-2}\rangle a^{-(r-1)}\right] \\ & \qquad \qquad + \begin{cases} za^{-r} & (\text{if } p=3),\\ \left[\langle z^2\rangle a^{p-r-2},\langle z^{p-2}\rangle a^{-r}\right] & (\text{if } p\ge 4)\\ \end{cases}\\ &=\left[\langle z^{r+2}\rangle a^p,\langle z^{p-2}\rangle a^{-r}\right] \end{align*}$$

where $\langle *\rangle a^{-(r-2)}$ indicates that the lowest $a$-degree of $\Lambda _{(-p,2,r-2)}$ is not smaller than $-(r-2)$.

This completes the proof.

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Proposition 6.4.

$\operatorname {spread}_a (\Lambda _{(-p, 3, r)}(a,z)) =p+r$ for $p\geq 3$ and $r\geq 3$.

Proof.

We show that

$$\begin{equation*} \Lambda _{(-p,3,r)}= \begin{cases} \left[\langle z^6\rangle a^p,\langle 2z^{p-3}\rangle a^{-3}\right] &(\text{if } r=3),\\ \left[\langle z^{r+3}\rangle a^p,\langle z^{p-3}\rangle a^{-r}\right] &(\text{if } r\ge 4).\\ \end{cases} \end{equation*}$$

Using equation (6.2) and Lemmas 6.1 and 6.2, we obtain

$$\begin{aligned} \Lambda _{(-p,3,3)}&=-\Phi _1(z)\Lambda _{(-p,0,3)}+\Phi _2(z)\Lambda _{(-p,1,3)}+(za^{-2}+z^2a^{-1})\Lambda _{(-p,3)}\\ &=-\Phi _1(z)\left[za^{p-1}, z^{-1}\Phi _p(z)a^{-1}\right]\left[z^{-1}\Phi _3(z)a,za^{-2}\right]\\ &\qquad \qquad +\Phi _2(z)\left[\Phi _4a^{p},\Phi _{p-1}(z)a^{-3}\right]\\ &\qquad \qquad +(za^{-2}+z^2a^{-1}) \begin{cases} (z^{-1}a-1+z^{-1}a^{-1}) &(\text{if } p=3),\\ a^{-1} &(\text{if } p=4),\\ \left[\langle z\rangle a^{p-4},\langle z^{p-4}\rangle a^{-1}\right] &(\text{if } p\ge 5)\\ \end{cases}\\ &=\left[\langle z^6\rangle a^p,\Phi _{p-3}(z)a^{-3}\right]+ \begin{cases} \left[z,a^{-3}\right] &(\text{if } p=3),\\ \left[z^2a^{-2},za^{-3}\right] &(\text{if } p=4),\\ \left[\langle z^3\rangle a^{p-5},\langle z^{p-3}\rangle a^{-3}\right] &(\text{if } p\ge 5)\\ \end{cases}\\ &=\left[\langle z^{6}\rangle a^p,\langle 2z^{p-3}\rangle a^{-3}\right].\\ \end{aligned}$$

Using K1, K2, K3 and the results above, we obtain

$$\begin{aligned} \Lambda _{(-p,3,4)}&=-\Lambda _{(-p,2, 3)}+z\Lambda _{(-p,3, 3)}+za^{-3}\Lambda _{(-p,3)} \qquad (\because \Lambda _{(-p,3,2)}=\Lambda _{(-p,2, 3)})\\ &=-\left[\langle z^5\rangle a^p,\langle z^{p-2}\rangle a^{-2}\right]+z\left[\langle z^6\rangle a^p,\langle 2z^{p-3}\rangle a^{-3}\right]\\ &\qquad \qquad +za^{-3} \begin{cases} (z^{-1}a-1+z^{-1}a^{-1}) &(\text{if } p=3),\\ a^{-1} &(\text{if } p=4),\\ \left[\langle z\rangle a^{p-4},\langle z^{p-4}\rangle a^{-1}\right] &(\text{if } p\ge 5)\\ \end{cases}\\ &=\left[\langle z^7\rangle a^{p},\langle z^{p-3}\rangle a^{-4}\right], \end{aligned}$$

and, for $r\ge 5$, inductively, we have

$$\begin{aligned} \Lambda _{(-p,3,r)}&=-\Lambda _{(-p,3, r-2)}+z\Lambda _{(-p,3, r-1)}+za^{-(r-1)}\Lambda _{(-p,3)}\\ &=-\left[\langle z^{r+1}\rangle a^p,\langle *\rangle a^{-(r-2)}\right]+z\left[\langle z^{r+2}\rangle a^p,\langle z^{p-3}\rangle a^{-(r-1)}\right]\\ &\qquad \qquad +za^{-(r-1)} \begin{cases} (z^{-1}a-1+z^{-1}a^{-1}) &(\text{if } p=3),\\ a^{-1} &(\text{if } p=4),\\ \left[\langle z\rangle a^{p-4},\langle z^{p-4}\rangle a^{-1}\right] &(\text{if } p\ge 5)\\ \end{cases}\\ &=\left[\langle z^{r+3}\rangle a^{p},\langle z^{p-3}\rangle a^{-r}\right]. \end{aligned}$$

This completes the proof.

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Proposition 6.5.

$\operatorname {spread}_a (\Lambda _{(-3, 4, r)}(a,z)) =r+1$ for $r\ge 7$.

Proof.

$$\begin{aligned} \Lambda _{(-3,4,0)}=\Lambda _{(-3,0)}\Lambda _{(0,4)} =\left[\Phi _{4}(z)a^3,\Phi _{3}(z)a^{-4}\right].\\ \end{aligned}$$ Using equation (6.1) and K1, K3 and Lemma 6.1, we obtain

$$\begin{aligned} \Lambda _{(-3,4,1)}&=-\Phi _{1}(z)\Lambda _{(0,4,1)}+\Phi _{2}(z)\Lambda _{(-1,4,1)}+\left(\sum _{i=0}^{1}z\Phi _i(z)a^{2-i}\right)\Lambda _{(4,1)}\\ &=-za\Lambda _{(0,4)}+\Phi _{2}(z)a^{-4}+(za^2+z^2a)\Lambda _{(0,5)}\\ &=-za\left[z^{-1}\Phi _{4}(z)a,za^{-3}\right]+\Phi _{2}(z)a^{-4}+(za^2+z^2a)\left[z^{-1}\Phi _{5}(z)a,za^{-4}\right]\\ &=\left[\Phi _{5}(z)a^3,\Phi _{2}(z)a^{-4}\right].\\ \end{aligned}$$

Using equation (6.2) and the above two formulas, for $r\ge 7$ we obtain

$$\begin{aligned} \Lambda _{(-3,4,r)}&=-\Phi _{r-2}(z)\Lambda _{(-3,4,0)}+\Phi _{r-1}(z)\Lambda _{(-3,4,1)}+\sum _{i=0}^{r-2}z\Phi _{i}(z)a^{-(r-1-i)}\Lambda _{(-3,4)}\\ &=-\Phi _{r-2}(z)\left[\Phi _{4}(z)a^3,\Phi _{3}(z)a^{-4}\right]\\ &\phantom {={}} +\Phi _{r-1}(z)\left[\Phi _{5}(z)a^3,\Phi _{2}(z)a^{-4}\right]+\sum _{i=0}^{r-2}z\Phi _{i}(z)a^{-(r-2-i)}\\ &=\left[\langle z^{r+4}\rangle a^3,za^{-(r-2)}\right]. \end{aligned}$$

This completes the proof.

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Proposition 6.6.

$\operatorname {spread}_a (\Lambda _{(-p, 4, r)}(a,z)) =p+r$ for $p\geq 5$ and $r\ge 5$.

Proof.

Using equation (6.2) and Lemmas 6.1 and 6.2, we obtain

$$\begin{aligned} &\Lambda _{(-p,4,r)}=-\Phi _{2}(z)\Lambda _{(-p,0,r)}+\Phi _{3}(z)\Lambda _{(-p,1,r)}+\sum _{i=0}^{2}z\Phi _{i}(z)a^{-(3-i)}\Lambda _{(-p,r)}\\ &\quad =-(z^2-1)\left[\Phi _{r}(z)a^{p},\ \Phi _{p}(z)a^{-r}\right]\\ &\quad \phantom {={}} +(z^3-2z)\left[\Phi _{r+1}(z)a^{p},\ \Phi _{p-1}(z)a^{-r}\right]\\ &\quad \phantom {={}} +\{za^{-3}+z^2a^{-2}+(z^3-z)a^{-1}\}\begin{cases} \ \left[\langle z\rangle a^{k-1},\langle z^{k-1}\rangle a^{-1}\right] &\text{if } k=p-r>1, \\ \ a^{-1} &\text{if } k=p-r=1, \\ \ z^{-1}a-1+z^{-1}a^{-1} &\text{if } k=p-r=0, \\ \ a &\text{if } k=r-p=1, \\ \ \left[\langle z^{k-1}\rangle a,\langle z\rangle a^{-(k-1)}\right] &\text{if } k=r-p>1 \\ \end{cases}\\ &\quad =\left[\langle z^{r+4}\rangle a^{p},\{-(z^2-1)\Phi _{p}(z)+(z^3-2z)\Phi _{p-1}(z)\}a^{-r}\right]\\ &\quad =\left[\langle z^{r+4}\rangle a^{p},\Phi _{p-4}(z)a^{-r}\right]. \end{aligned}$$

This completes the proof.

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7. Proofs of main results and comments

Theorem 4.1 is proved by Proposition 5.1. Table 1 shows that the upper bound ‘$c(K)-1$’ for the arc index in Theorem 4.1 is best possible. It also shows that the lower bound ‘$\operatorname {spread}_a (F_K)+2$’ in Theorem 2.1 is best possible.

The proof of Theorem 4.2 is a combination of Propositions 5.2 and 6.3. The proof of Theorem 4.3 is a combination of Propositions 5.2 and 6.4. The proof of Theorem 4.4 is a combination of Propositions 5.3 and 6.6. The proof of Theorem 4.5 is a combination of Propositions 5.3 and 6.5. Table 2 shows that the upper bound ‘$c(K)-2$’ is best possible but the lower bound ‘$c(K)-4$’ may not be best possible.

Figure 1.

An arc presentation of the right-handed trefoil knot

Figure 3.

Pretzel links $P(p_1,p_2,\ldots ,p_n)$ and $P(-p,q,r)$

Figure 4.

An arc presentation of $P(-2,q,r)$

Figure 5.

Arc presentations of $P(-p,q,r)$ for $q=2,3$

   

Figure 6.

Arc presentations of $P(-p,q,r)$ with $p\geq 3$ and $4\le q\le r$

Table 1.

Examples of Theorem 4.1

Pretzel knot $K$	DT Name⁠2 2 The Dowker-Thistlethwaite name. See 10. ✖	$\operatorname {spread}_a (F_K)+2$	arc index	$c(K)-1$
$P(-2,3,3)$	$8n3$	$6$	$7$	7
$P(-2,3,5)$	$10n21$	$6$	$8$	9
$P(-2,3,7)$	$12n242$	$9$	$9$	11
$P(-2,5,5)$	$12n725$	$10$	$10$	11

Table 2.

Examples of Theorem 4.5

Pretzel knot $K$	DT Name	$c(K)-4$	arc index	$c(K)-2$
$P(-3,4,5)$	$12n475$	$8$	$10$	10
$P(-3,4,7)$	$14n12205$	$10$	$11$	12