Upsilon invariants of L-space cable knots

TL;DR

This paper provides a formula to compute the Upsilon invariant for L-space cable knots, linking it to known invariants of the base knot and torus knots, and explores its properties as a concordance invariant.

Contribution

It introduces a new explicit formula for the Upsilon invariant of L-space cable knots, expanding the computational tools for knot concordance invariants.

Findings

Derived a formula relating Upsilon of cable knots to base and torus knots

Computed integral Upsilon values for iterated cable knots

Demonstrated the Upsilon invariant as a rational-valued concordance invariant

Abstract

We give a formula of the Upsilon invariant of any L-space cable knot $K_{p,q}$ using $p,\Upsilon_K$ and $\Upsilon_{T_{p,q}}$. The integral value of the Upsilon invariant gives a ${\mathbb Q}$-valued knot concordance invariant. We compute the integral values for L-space iterated cable knots.

Equations

$$(1-\omega )S+(1-\bar{\omega })S^T.$$

$$\sigma (K)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\Delta }_{K_{p,q}}(t)={\Delta }_K(t^p){\Delta }_{T_{p,q}}(t).$$

$${\sigma }_{K_{p,q}}(\omega )={\sigma }_K({\omega }^p)+{\sigma }_{T_{p,q}}(\omega ).$$

$${{\rm Y}}_{K_{p,q}}(t)={{\rm Y}}_K(s)+{{\rm Y}}_{T_{p,q}}(t),$$

$${{\rm Y}}_K(t)=\underset{m\in \{0,\cdots ,2g\}}{\max }\{-2\#(S_K\cap [0,m))-t(g-m)\}.$$

$${\widetilde{{\rm Y}}}_K(t,m)=-2\#(S_K\cap [0,m))-t(g-m).$$

$${\mu }_K:=\underset{0<m<2g}{\min }\frac{2\#(S_K\cap [0,m))}{m},$$

$$\delta :=q-(2g-1)p.$$

$$z_{i_1}^{i_2}=i_{1}p+i_{2}q.$$

$${{\rm Y}}_{p,q}^{\delta ,1}(t)=\underset{z_0^i-\delta <m\le z_0^i}{\max }{\widetilde{{\rm Y}}}_{T_{p,q}}(t,m),{{\rm Y}}_{p,q}^{\delta ,2}(t)=\underset{z_{-1}^i<m\le z_0^i-\delta }{\max }{\widetilde{{\rm Y}}}_{T_{p,q}}(t,m).$$

$$\max \left\{{{\rm Y}}_{p,q}^{\delta ,1}(t),{{\rm Y}}_{p,q}^{\delta ,2}(t)\right\}={{\rm Y}}_{T_{p,q}}(t)$$

$${{\rm Y}}_K^{tr}(s)=\underset{\nu \in \{1,2,\cdots ,2g-1\}}{\max }{\widetilde{{\rm Y}}}_K(s,\nu ).$$

$$\{\begin{array}{ll}0\le s\le 2-{\mu }_K& i=0\\ {\mu }_K\le s\le 2-{\mu }_K& 1\le i\le p-2\\ {\mu }_K\le s\le 2& i=p-1.\end{array}$$

$${{\rm Y}}_{K_{p,q}}(t)={{\rm Y}}_K(s)+{{\rm Y}}_{T_{p,q}}(t)$$

$$\{\begin{array}{ll}0\le s<{\mu }_K& 1\le i\le p-1\\ 2-{\mu }_K<s\le 2& 0\le i\le p-2.\end{array}$$

$${{\rm Y}}_{K_{p,q}}(t)=\max \left\{{{\rm Y}}_K(s)+{{\rm Y}}_{p,q}^{\delta ,1}(t),{{\rm Y}}_K^{tr}(s)+{{\rm Y}}_{p,q}^{\delta ,2}(t)\right\}.$$

$${{\rm Y}}_{K_{p,q}}(t)=\max \left\{{{\rm Y}}_K(s)+{{\rm Y}}_{p,q}^{\delta ,1}(2-t),{{\rm Y}}_K^{tr}(s)+{{\rm Y}}_{p,q}^{\delta ,2}(2-t)\right\}.$$

$${{\rm Y}}_{T_{p,q}}(t)+{{\rm Y}}_K^{tr}(s)\le {{\rm Y}}_{K_{p,q}}(t)\le {{\rm Y}}_{T_{p,q}}(t)+{{\rm Y}}_K(s)$$

$$\{\begin{array}{ll}0\le s\le 4/3& i=0\\ 2/3\le s\le 4/3& i=1\\ 2/3\le s\le 2& i=2,\end{array}$$

$$I(K)={\int }_0^2{{\rm Y}}_K(t)dt,$$

$${\int }_{S^1}{\sigma }_{T_{p,q}}(\omega )=-\frac{1}{3}\left(pq-\frac{p}{q}-\frac{q}{p}+\frac{1}{pq}\right)=4(s(q,p)+s(p,q)-s(1,pq)),$$

$$q/p=a_1+\frac{1}{a_2+\frac{1}{\cdots +\frac{1}{a_n}}}=:[a_1,\cdots ,a_n].$$

$$I(T_{p,q})=-\frac{1}{3}(pq-\sum _{i=1}^n{a}_i).$$

$$I(K({\mathbf{p}}_1,\cdots ,{\mathbf{p}}_n))=I(K)+\sum _{i=1}^{n}I(T_{p_i,q_i}).$$

$$\frac{{\Delta }_K(t)}{1-t}=\sum _{s\in S_K}{t}^s,$$

$$S_{K_n}=\{0,3,5,7,\cdots ,2n-1,2n+1,2n+2\}\cup {\mathbb{Z}}_{\ge 2n+4}.$$

$$S_{K_{p,q}}=p{S}_K\cup (p{S}_K+q)\cup (p{S}_K+2q)\cup \cdots ,$$

$${\Delta }_K(t)=(1-t)(1+t+t^{s}f(t))=1-t^2+t^s(1-t)f(t),$$

$${\mu }_K\le \frac{2\#(S_K\cap [0,2))}{2}=1.$$

Full text

The author is supported by JSPS KAKENHI Grant Number 26800031

1. Introduction

1.1. Knot concordance invariants

Let $K$ be a knot in $S^{3}$. Let $S$ be a Seifert matrix of $K$ and $\omega$ a complex number with $|\omega|=1$. The Tristram-Levine signature (TL-signature) $\sigma_{K}(\omega)$ is defined as the signature of the matrix

[TABLE]

The usual definition of the knot signature $\sigma(K)$ implies $\sigma_{K}(-1)$, hence we have $\sigma(K)=\sigma_{K}(-1)$. Thus the TL-signature is a refinement of $\sigma(K)$. It is classically well-known that $\sigma(K)$ can give a lower bound of the 4-ball genus of $K$.

By using the knot filtration of the knot Floer chain complex $CFK^{\infty}(S^{3},K)$, Ozsváth and Szabó defined a knot concordance invariant $\tau:\mathcal{C}\to{\mathbb{Z}}$ (the $\tau$-invariant), where $\mathcal{C}$ is a knot concordance group. In fact, $\tau(K)$ also has a similar lower bound for 4-ball genus, as mentioned in [19]. The estimate by $\tau(K)$ is sharper than the one by $\sigma(K)$. In [17], Ozsváth, Stipsicz and Szabó defined a knot concordance invariant ($\Upsilon$-invariant) $\Upsilon:\mathcal{C}\to C([0,2])$ ($K\mapsto\Upsilon_{K}$), where $C([0,2])$ is the group consisting of continuous functions over the closed interval $[0,2]$. Livingston in [15] gave a simpler definition of $\Upsilon_{K}$. The $\Upsilon$-invariant is defined essentially by using the doubly graded filtration of $CFK^{\infty}(S^{3},K)$ and is regarded as a refinement of the $\tau$-invariant. In fact, $\tau(K)=-\Upsilon_{K}^{\prime}(0)$ holds. $\Upsilon$-invariant has been also applied to finding knot concordance classes linearly independent mutually, for example, as seen in [3] and [17].

We have seen that the invariant $\tau(K)$ is a Heegaard Floer analog of $\sigma$-invariant. The TL-signature $\sigma_{K}(\omega)$ and $\Upsilon_{K}^{\prime}(t)$ are locally constant away from zeros of $\Delta_{K}(t)$, and refinements of $\sigma(K)$ and $\tau(K)$ respectively. We can schematically show these relationships as below (1):

[TABLE]

The minus before $\Upsilon^{\prime}_{K}(t)$ is due to the restriction $-\Upsilon^{\prime}_{K}(0)=\tau(K)$.

1.2. Cabling formula of invariants

Consider cabling formulas for several invariants. Let $K$ be a knot in $S^{3}$. Let $V$ be a tubular neighborhood of $K$. For integers $p,q$, the $(p,q)$-cable $K_{p,q}$ of $K$ is defined to be the simple closed curves on $\partial V$ whose homology class is $p\cdot{\bf l}+q\cdot{\bf m}$ in $H_{1}(\partial V)$, where ${\bf l}$ and ${\bf m}$ are classes represented by a longitude curve and a meridian curve on $\partial V$. If $p,q$ are coprime integers, then $K_{p,q}$ is a knot and we call it a $(p,q)$-cable knot. The cabling formula for Alexander polynomial is as follows:

[TABLE]

Since the knot Floer homology is a categorification of the Alexander polynomial, it is natural to try to find the cabling formula for the knot Floer homology. For example, as in [6] and [7] many studies have been done. However, it has not been completely succeeded yet. In general, it is difficult to give the cabling formula for the knot Floer homology.

Due to [14], the cabling formula of the TL-signature is known as follows:

[TABLE]

These cabling formulas for $\sigma_{K}(\omega)$ and $\Delta_{K}(t)$ both consist of the invariants of the companion knot $K$ and the ones of torus knots.

Here we introduce Hom’s cabling formula for the $\tau$-invariant. This formula uses additional information $\epsilon(K)$ to compute $\tau$. The cabling formula for knot Floer chain complex, if any, would be much more complicated than classical invariants $\sigma(K)$ or $\Delta_{K}(t)$. We state it here.

Theorem 1** (Hom [11]).**

Let $K\subset S^{3}$. Then $\tau(K_{p,q})$ is completely determined by $p,q,\tau(K)$, and $\epsilon(K)$ in the following manner.

1. (1)

If $\epsilon(K)=1$, then $\tau(K_{p,q})=p\tau(K)+(p-1)(q-1)/2$. 2. (2)

If $\epsilon(K)=-1$, then $\tau(K_{p,q})=p\tau(K)+(p-1)(q+1)/2$. 3. (3)

If $\epsilon(K)=0$, then $\tau(K)=0$ and

$\tau(K_{p,q})=\tau(T_{p,q})=\begin{cases}(p-1)(q+1)/2&q<0\\ (p-1)(q-1)/2&q>0.\end{cases}$**

1.3. Motivation

Chen in [3] gives an inequality for the $\Upsilon$-invariant of any cable knot. However, it is also hard to obtain a closed formula of $\Upsilon_{K_{p,q}}$. To do so in small cases, we focus on any L-space knot, which is defined as a knot $K$ whose positive surgery of $K$ is an L-space. Here a rational homology sphere $Y$ is an L-space, if $Y$ has the same Heegaard Floer homology as that of $S^{3}$ for any spinc structure on $Y$. According to [18], the knot Floer homology of any L-space knot is simple.

Here we recall a necessary and sufficient condition for a cable knot $K_{p,q}$ to be an L-space knot by Hedden and Hom.

Theorem 2** (Hedden [7] and Hom [10]).**

Let $K$ be a knot with the Seifert genus $g$. $K_{p,q}$ is an L-space knot if and only if $K$ is an L-space knot with $(2g-1)p\leq q$.

Therefore, we reach the following natural problem.

Problem 3**.**

Find the cabling formula of the $\Upsilon$-invariant on L-space knots.

We shall consider the $\Upsilon$-invariant of any L-space cable knot (Theorem 4, 6 and 7). We, first, give a cabling formula of L-space cable knots $K$ with $2gp\leq q$ with the Seifert genus $g$. After that, we consider a cabling formula in the case of $(2g-1)p<q<2gp$.

1.4. Cabling formula for $\Upsilon$ of L-space knots with $2gp\leq q$.

We give a cabling formula of $\Upsilon$ for $2gp\leq q$.

Theorem 4** (The case of $2gp\leq q$).**

Let $K$ be an L-space knot with the Seifert genus $g$. Let $p,q$ be relatively prime positive integers with $2gp\leq q$. Then the $\Upsilon$-invariant of $K_{p,q}$ is computed as follows:

[TABLE]

where $s$ is the real number with $s\equiv pt(\bmod 2)$ and $0\leq s\leq 2$.

Here $\Upsilon_{K}(s)$ part in this formula can be regarded as a $p$-fold amalgamated function of $\Upsilon_{K}(t)$ in terms of a function in $t$. Here the amalgamated function means the deformation as in Figure 1.

In other words, by seeing $\Upsilon_{K}$ as a function over ${\mathbb{R}}/2{\mathbb{Z}}$, the $p$-fold amalgamated function means $\Upsilon_{K}(pt)$. This formula (4) is similar to the cabling formula (3).

To prove this formula, we use a simple expression of $\Upsilon_{K}(t)$ for any L-space knot $K$ by Borodzik and Livingston in [2]. They wrote down an $\Upsilon$-invariant formula for any L-space knot $K$ by using the formal semigroup $S_{K}$ as follows.

Proposition 5** ([2]).**

Let $K$ be an L-space knot with genus $g$. Then for any $t\in[0,2]$ we have

[TABLE]

We will explain the formal semigroup $S_{K}$ in Section 2.1. In this paper we put the following function:

[TABLE]

Hence the $\Upsilon$-invariant is written as $\Upsilon_{K}(t)=\underset{m\in\{0,\cdots,2g\}}{\max}\widetilde{\Upsilon}_{K}(t,m)$.

1.5. Cabling formula for $\Upsilon$ of L-space knots with $(2g-1)p<q<2gp$.

We set

[TABLE]

[TABLE]

and

[TABLE]

Here for any real number $t$ with $2i/p\leq t\leq 2(i+1)/p$ we define $\Upsilon^{\delta,1}_{p,q}(t)$ and $\Upsilon^{\delta,2}_{p,q}(t)$ to be

[TABLE]

Then,

[TABLE]

holds. We have $\Upsilon_{T_{p,q}}(t)=\underset{z^{i}_{-1}<m\leq z^{i}_{0}}{\max}\widetilde{\Upsilon}_{T_{p,q}}(t,m)$, due to Lemma 15. For any L-space knot $K$ we define the truncated $\Upsilon$-invariant as follows:

[TABLE]

Here we state the second main theorem in this article.

Theorem 6**.**

Let $K$ be an L-space knot with the Seifert genus $g$. Let $p,q$ be relatively prime integers with $(2g-1)p<q<2gp$. Let $t$ be a real number with $0\leq t\leq 2$. Suppose that $s$ is a real number and $i$ is an integer with the property that $pt\equiv s\bmod 2$ and $0\leq s\leq 2$ and $i=(pt-s)/2$.

Further, suppose that $s$ satisfies either of the following conditions:

[TABLE]

Then

[TABLE]

holds.

We clarify the invariant $\Upsilon_{K_{p,q}}(t)$ in the remaining case.

Theorem 7**.**

Let $K,g,p,q,t,s$, and $i$ be parameters satisfying the condition in the first paragraph in Theorem 6.

Further, suppose that $s$ satisfies either of

[TABLE]

In the former case, the following is satisfied:

[TABLE]

In the latter case, the following is satisfied:

[TABLE]

Actually, this formula holds in the case of Theorem 6. Then, $\Upsilon_{K}^{tr}(s)$ is equal to $\Upsilon_{K}(s)$ (Lemma 23), hence, as a result, the formula in Theorem 6 holds. Here we give an inequality of L-space cabling formula for $\Upsilon$-invariant.

Corollary 8**.**

Let $K$ be an L-space knot with the Seifert genus $g$. We assume that $(2g-1)p<q<2gp$. Let $t$ be a real number with $0\leq t\leq 2$, and $i$ and $s$ an integer and a real number with $2i/p\leq t\leq 2(i+1)/p$, $2i+s=pt$ and $0\leq s\leq 2$. Then

[TABLE]

holds.

In particular if $\mu_{K}\leq s\leq 2-\mu_{K}$, then the inequalities become the equalities.

1.6. Example $(T_{3,7})_{3,35}$.

Here we verify the formulas in Theorem 6 and 7 in the case of $(2g-1)p<q<2gp$. Consider the $(3,35)$-cable knot of $K=T_{3,7}$. Then $p=3$, $q=35$, and $g(K)=6$ hold. Therefore $K_{3,35}$ is an L-space knot from the Hedden-Hom criterion. We compare the functions $\Upsilon_{K_{3,35}}(t)-\Upsilon_{T_{3,35}}(t)$ and $\Upsilon_{K}(3t)$. Here we compute the two functions with the aid of Mathematica program by [13]. See Figure 2.

The value $\mu_{K}$ defined above is $2/3$. Let $t$ and $s$ be real numbers with $0\leq t,s\leq 2$ and $pt\equiv s\bmod 2$ and $i$ an integer $i=(pt-s)/2$.

If $0\leq i\leq 2$, $2i+s=3t$ and

[TABLE]

then $\Upsilon_{K_{3,35}}(t)=\Upsilon_{T_{3,35}}(t)+\Upsilon_{K}(s)$ holds, as described in Figure 2.

On the other hand, for the remaining regions, e.g., $i=1$ and $0<s<2/3$ or $4/3<s<2$, the $\Upsilon_{K_{3,35}}(t)$ violates the formula (4). In Section 5, we try to compute some of the actual functions of $\Upsilon_{K_{3,35}}(t)$ over the following regions: $i=1$ and $0<s<2/3$ and $i=0$ and $4/3<s<2$.

1.7. Integral value of $\Upsilon_{K}(t)$ on $[0,2]$.

Here we propose a knot concordance invariant that it is easy to compute from the cabling formula. We compute the integral value of $\Upsilon_{K}(t)$ on the interval $[0,2]$:

[TABLE]

which is also a knot concordance invariant. The motivation of this value is inspired by the $S^{1}$-integral value $\int_{S^{1}}\sigma_{K}(\omega)$. The $S^{1}$-integral value $\int_{S^{1}}\sigma_{T_{p,q}}(\omega)$ is computed as follows:

[TABLE]

where the function $s$ is the Dedekind sum. This computation has been done by many topologists for example [12], [16], [1] and [4].

On the other hand, $I(T_{p,q})$ is computed as follows:

Proposition 9**.**

Let $p,q$ be relatively prime positive integers. Let $a_{i}\geq 0$ be the $i$-th term of the non-negative continued fraction of $q/p$:

[TABLE]

Then we have

[TABLE]

Let ${\bf p}_{i}$ be a pair $(p_{i},q_{i})$ of coprime integers. We give a formula of $I$ for iterated cable L-space knots $K({\bf p}_{1},\cdots,{\bf p}_{n}):=(\cdots(K_{p_{1},q_{1}})_{p_{2},q_{2}}\cdots)_{p_{n},q_{n}}$.

Theorem 10**.**

Let $K:=L_{0}$ be an L-space knot and $L$ an iterated cable knot $K({\bf p}_{1},\cdots,{\bf p}_{n})$. We put $L_{i}:=K({\bf p}_{1},\cdots,{\bf p}_{i})$ for any integer $1\leq i\leq n$. If ${\bf p}_{i}=(p_{i},q_{i})$ satisfies $2g(L_{i-1})p_{i}\leq q_{i}$ for any integer $i$ with $1\leq i\leq n$, then the integral $I(K({\bf p}_{1},\cdots,{\bf p}_{n}))$ is computed as follows:

[TABLE]

For the $S^{1}$-integral value of $\sigma_{L}(\omega)$ of iterated torus knots $L$, a similar formula holds. See [1].

In Theorem 10 we deal with L-space iterated torus knots $L=T_{p,q}({\bf p}_{1},\cdots,{\bf p}_{n})$ satisfying $2g(L_{i-1})p_{i}\leq q_{i}$ for any integer $i$ with $1\leq i\leq n$.. Are these knots different from general L-space iterated torus knots? As an application of Theorem 10, we prove that there exists an L-space iterated torus knot that is not knot concordant to any L-space iterated torus knots satisfying $2g(L_{i-1})p_{i}\leq q_{i}$ (Proposition 26).

Remark 11**.**

On the other hand, in terms of the diagram (1), we might consider integral values $-\int_{0}^{1}\Upsilon_{K}^{\prime}(s)ds=-\Upsilon_{K}(1)$ or $\int_{S^{1}}\Sigma(s)ds$, where $\Sigma(s)$ is a piecewise linear continuous function with $\Sigma(0)=0$ and $\Sigma^{\prime}(\omega)=\sigma(\omega)$ for $\omega\in S^{1}$ away from finite points.

Acknowledgements

This work was started by computing the integral values of the $\Upsilon$-invariants of any torus knots. The author thanks for Min Hoon Kim. He told me the $\Upsilon$-invariant formula for the torus knots and the reference [5]. This became my motivation to compute the $\Upsilon$-invariants of the L-space cable knots. Furthermore, he gave me many useful comments for my earlier manuscript. The author would like to thank an anonymous referee for indicating several unclear points.

2. Preliminaries

In this section we introduce tools to prove our main theorem (Theorem 4).

2.1. Formal semigroup

Let $K$ be an L-space knot with the Seifert genus $g(K)=g$. Expanding the rational function $\Delta_{K}(t)/(1-t)$ as follows:

[TABLE]

we obtain a subset $S_{K}\subset{\mathbb{Z}}_{\geq 0}$. Hence, the coefficients of the right hand side are [math] or $1$. This subset $S_{K}$ is called the formal semigroup of $K$. The following properties hold:

Fundamental facts:

• •

Any algebraic knot is an L-space knot. If $K$ is an algebraic knot, then $S_{K}$ is a semigroup (by [21]).

• •

$S_{K}\cap{\mathbb{Z}}_{\geq 2g}={\mathbb{Z}}_{\geq 2g}$.

• •

$s\in S_{K}\Leftrightarrow 2g(K)-1-s\not\in S_{K}$.

• •

There is a cabling formula for formal semigroup (Proposition 12).

For example, if $K$ is a right-handed torus knot $T_{p,q}$, then $S_{T_{p,q}}$ is the semigroup generated by the positive integers $p,q$, namely, $S_{T_{p,q}}=\langle p,q\rangle=\{pa+qb\mid a,b\in{\mathbb{Z}}_{\geq 0}\}$ holds. There exists an L-space but not-algebraic knot. For example, for $n\geq 1$ the $(-2,3,2n+1)$ pretzel knot $K_{n}$ is an L-space knot, and the formal semigroup is as follows:

[TABLE]

$K_{1}=T_{3,4}$ and $K_{2}=T_{3,5}$ are only two algebraic knots in this sequence. For $n\geq 3$, we can easily see that $S_{K_{n}}$ is not a semigroup. The Alexander polynomials of $(-2,3,2n+1)$-pretzel knots can be found, for example, in [9].

Wang, in [22], proved the cabling formula for the formal semigroup of any L-space knot as follows:

Proposition 12** (A cabling formula for formal semigroup [22]).**

Let $K$ be a nontrivial L-space knot. Suppose $p\geq 2$ and $p(2g(K)-1)\leq q$. Then $S_{K_{p,q}}=pS_{K}+q{\mathbb{Z}}_{\geq 0}:=\{pa+qb\mid a\in S_{K},b\in{\mathbb{Z}}_{\geq 0}\}$.

Hence, $S_{K_{p,q}}=pS_{K}+q{\mathbb{Z}}_{\geq 0}$ can be decomposed as follows:

[TABLE]

where $pS+x:=\{pa+x|a\in S\}$ for a set $S\subset{\mathbb{R}}$.

Here we prove the following lemma.

Lemma 13**.**

Let $S_{K}$ be a formal semigroup of a non-trivial L-space knot $K$. Then $1\not\in S_{K}$ holds.

**Proof. ** If $1\in S_{K}$, then the Alexander polynomial of the L-space knot is computed as follows:

[TABLE]

where $s\geq 2$ and $f(t)$ is a series. Thus the coefficient of $t$ in $\Delta_{K}(t)$ vanishes. The coefficient of $t$ of the Alexander polynomial of a non-trivial L-space knot is $-1$ due to [8]. Thus $K$ must be the trivial knot. $\Box$

In the case of lens space knots, there would be some restrictions to $S_{K}$. The results in [20] can give some restrictions.

Here we claim that $\mu_{K}\leq 1$ if $K$ is a non-trivial knot, where $\mu_{K}$ is defined in Section 1.5. For, from the third in the fundamental facts above and Lemma 13, we have

[TABLE]

2.2. Proof of Theorem 4.

Let $K$ be an L-space knot with the Seifert genus $g$. Throughout this section we assume that the relatively prime positive integers $p,q$ satisfy $2gp\leq q$. In particular, $K_{p,q}$ is also an L-space knot.

For any L-space knot $K$ we put

[TABLE]

and

[TABLE]

Here we prove the following lemma.

Lemma 14**.**

Let $\nu$ be an integer with $0\leq\nu\leq 2g$ and $p$ a positive integer. Then we have

[TABLE]

[TABLE]

**Proof. ** Let $\bar{S}_{K}$ be the complement of $S_{K}$ in ${\mathbb{Z}}$. (11) and (12) are due to the following equalities:

[TABLE]

[TABLE]

∎

According to Proposition 5, the $\Upsilon$-invariant of an L-space knot $K$ is rewritten as follows:

[TABLE]

Extending the function $\varphi_{K}(m)$ as $\varphi_{K}(m)\equiv 0$ if $m<0$, we can define $\Phi_{K}(t,m)$ over $m\in{\mathbb{Z}}$. We note that the function $\Phi_{K}(t,m)$ satisfies the following:

[TABLE]

The last row is due to the third fundamental fact in Section 2.1. In other words, this fact means that since the exact half of ${\mathbb{Z}}\cap[0,2g)$ is included in $S_{K}\cap[0,2g)$, we have $\#(S_{K}\cap[0,2g))=\#({\mathbb{Z}}\cap[0,2g))-g=g$.

Thus, if a subset $S\subset{\mathbb{Z}}$ includes $\{0,1,\cdots,2g\}$ then we have

[TABLE]

The genus $g(K_{p,q})=:g_{p,q}$ is equal to the degree of $\Delta_{K_{p,q}}(t)$ since $K_{p,q}$ is an L-space knot. Thus from the cabling formula (2), we have

[TABLE]

We denote $\varphi_{K_{p,q}}(m)$ by $\varphi(m)$ and $\Phi_{K_{p,q}}(t,m)$ by $\Phi(t,m)$.

Here we give a first setting to prove cabling formulas.

Setting:

[TABLE]

Here we recall the definition of $z^{i_{2}}_{i_{1}}$ in (5).

Lemma 15**.**

Let $K$ be an L-space knot with $g=g(K)$. Let $p,q$ be relatively prime integers with $2gp\leq q$. Let $t,s,i$ be parameters satisfying (14). Then we have

[TABLE]

**Proof. ** Take parameters $t,s,i$ satisfying (14). The parameter $t$ is fixed here. We prove the following claim:

Claim 16**.**

If $m\leq z^{i}_{-1}(\Leftrightarrow m+p\leq z^{i}_{0})$ holds, then $\#(S_{K_{p,q}}\cap[m,m+p))\leq i$ holds.

**Proof. ** Using the decomposition (9) right after Proposition 12 we obtain Figure 3. It describes the local picture of $S_{K_{p,q}}$ around $m=z^{i}_{0}$. The top line in Figure 3 consists of the components of $pS_{K}+iq$ and the second top line consists of $pS_{K}+(i-1)q$. The other elements $\cup\{pS_{K}+jq|j=0,\cdots,i-2\}$ are omitted in Figure 3. The shaded circles in Figure 3 present $S_{K_{p,q}}$ as points projected to the $m$-axis. The empty circles correspond to $(p{\mathbb{Z}}_{\geq 0}+q{\mathbb{Z}}_{\geq 0})\setminus S_{K_{p,q}}$.

In the case of $m\leq z^{i}_{-1}$, since $(pS_{K}+jq)\cap[m,m+p)$ is at most one point, we have

[TABLE]

∎

Suppose that $m$ is an integer with $z^{i}_{2g-1}<m\leq 2g(K_{p,q})$. Similar to Claim 16, we prove the following claim:

Claim 17**.**

If $z^{i}_{2g-1}<m\leq 2g(K_{p,q})$, then $\#(S_{K_{p,q}}\cap[m,m+p))\geq i+1$ holds.

**Proof. ** If $z^{i}_{2g-1}<m$, then for a non-negative integer $j$ with $j\leq i$, $(pS_{K}+jq)\cap[m,m+p)$ is exact one point from the second of the fundamental facts in Section 2.1. Thus

[TABLE]

holds. $\Box$

We go back to the proof of Lemma 15. In the case of $m\leq z^{i}_{-1}$, using Claim 16 and (12) we have $\Phi(t,m+p)-\Phi(t,m)\leq i-tp/2\leq 0$. In the case of $z^{i}_{2g-1}<m\leq 2g(K_{p,q})$, using Claim 17 and (12), we have $\Phi(t,m+p)-\Phi(t,m)=\#(S_{K_{p,q}}\cap[m,m+p))-tp/2\geq i+1-tp/2\geq 0$.

Thus the minimum value of $\Phi(t,m)$ over $0\leq m\leq 2g(K_{p,q})$ is equal to the minimum value over $z^{i}_{-1}<m\leq z^{i}_{2g}$. $\Box$

As a corollary of this lemma, it follows that if $K$ is the unknot, then we have

[TABLE]

Next, we investigate the minimum value of $\Phi(t,m)$ in the region

[TABLE]

We prepare the following claim to prove Theorem 4:

Claim 18**.**

The minimum value of $\Phi(t,m)$ over $I_{i}$ coincides with

[TABLE]

**Proof. ** The behavior of $\Phi(t,m)$ among $I_{i}$ becomes Figure 4. Let $\omega_{\nu}$ be the minimum value of $\Phi(t,m)$ among $z^{i}_{\nu-1}<m\leq z^{i}_{\nu}$. If $\nu\not\in S_{K}$, then $\Phi(t,m+p)-\Phi(t,m)=\#(S_{K_{p,q}}\cap[m,m+p))-tp/2=i-tp/2\leq 0$, i.e., $\omega_{\nu+1}\leq\omega_{\nu}$ holds. If $\nu\in S_{K}$, then $\Phi(t,m)-\Phi(t,m-p)=\#(S_{K_{p,q}}\cap[m-p,m))-tp/2=i+1-tp/2\geq 0$, i.e., $\omega_{\nu}\leq\omega_{\nu+1}$ holds. Hence, we have only to consider the minimum values of $\omega_{\nu}$ in the case of $\nu\in S_{K}$ and $\nu-1\not\in S_{K}$.∎

This minimum value over $I_{i}$ is:

[TABLE]

where $\mu_{i}$ is the minimum value of $\Phi(t,m)$ over $(z^{i}_{-1},z^{i}_{0}]$. The function $\epsilon(\nu)$ is defined as follows:

[TABLE]

Here, the case of $m=-1$ for the summation in the minimum (16) means that the sum is [math]. Since $\sum_{l=0}^{m}(\epsilon(l)+1)=\#(S_{K}\cap[0,m+1))$ holds, the summation part in (16) is computed as follows:

[TABLE]

Then we have

[TABLE]

Hence, we obtain

[TABLE]

Here we claim the following. $\Phi_{p,q}(t,m)$ means $\Phi_{T_{p,q}}(t,m)$.

Claim 19**.**

We have

[TABLE]

**Proof. ** Let $S_{K}^{c}$ be the set $\{n\in{\mathbb{Z}}_{\geq 0}|n\not\in S_{K}\}$.

[TABLE]

is constructed by removing $pS_{K}^{c}\cup(pS_{K}^{c}+q)\cup(pS_{K}^{c}+2q)\cup\cdots$ from $S_{T_{p,q}}=p{\mathbb{Z}}_{\geq 0}\cup(p{\mathbb{Z}}_{\geq 0}+q)\cup(p{\mathbb{Z}}_{\geq 0}+2q)\cup\cdots$. Since $\#S_{K}^{c}=g$ holds, $\#(pS_{K}^{c}\cup(pS_{K}^{c}+q)\cup(pS_{K}^{c}+2q)\cup\cdots\cup(pS_{K}^{c}+(i-1)q))=ig$. Hence, $\mu_{i}=\underset{z^{i}_{-1}<m\leq z^{i}_{0}}{\min}\Phi(t,m)=\underset{z^{i}_{-1}<m\leq z^{i}_{0}}{\min}\Phi_{p,q}(t,m)-ig$.∎

Hence, we go back to the proof of Theorem 4. Using (15) we have,

[TABLE]

Therefore by using the formula (13), we obtain the following:

[TABLE]

∎

3. The case of $(2g-1)p<q<2gp$.

3.1. Minimum value of $\Phi(t,m)$.

Let $K$ be an L-space knot with the Seifert genus $g$. Throughout this section we assume that $p,q$ are relatively prime positive integers satisfying $(2g-1)p<q<2gp$. In particular, $K_{p,q}$ is an L-space knot. We consider $\Phi(t,m)=\#(S_{K_{p,q}}\cap[0,m))-tm/2$. Setting $\delta=q-(2g-1)p$, we have $0<\delta<p$.

We put

[TABLE]

This $I^{\delta}_{i}$ is slightly smaller than $I_{i}$ and is included in $I_{i}$. Here $z_{0}^{i}-\delta=z^{i-1}_{2g-1}$.

In the case of $(2g-1)p<q<2gp$, the behavior of $S_{K_{p,q}}$ around $z^{i}_{0}$ is different from the case of $2gp\leq q$ as in Figure 5.

Here we prove the following lemma analogous to Lemma 15.

Lemma 20**.**

Let $t,s$, and $i$ be parameters satisfying (14). Then we have

[TABLE]

**Proof. ** In the same way as Lemma 15, if $m$ satisfies $m\leq z^{i}_{0}-\delta$, then $\varphi(t,m+p)-\varphi(t,m)=\#(S_{K_{p,q}}\cap[m,m+p))\leq i$ holds. Hence we have, using (12),

[TABLE]

If $m$ satisfies $z^{i+1}_{-1}<m\leq 2g(K_{p,q})-p$, then we can compute as follows: $\varphi(m+p)-\varphi(m)=\#(S_{K_{p,q}}\cap[m,m+p))\geq i+1$. Hence we have

[TABLE]

Thus the minimum value of $\Phi(t,m)$ coincides with the minimum value over $I_{i}^{\delta}$. ∎

4. Proof of Theorem 6

Let $p,q$ be relatively prime positive integers with $(2g-1)p<q<2gp$. Let $t,s$, and $i$ be parameters satisfying (14). We decompose the case of $(2g-1)p<q<2gp$ into the following two parts:

[TABLE]

[TABLE]

Actually, this decomposition corresponds to whether the difference of $\Upsilon_{K_{p,q}}(t)$ and $\Upsilon_{K}(s)+\Upsilon_{T_{p,q}}(t)$ is equal or not. The case of (19) satisfies with $\Upsilon_{K_{p,q}}(t)=\Upsilon_{K}(s)+\Upsilon_{T_{p,q}}(t)$. In order to explain the decomposition, we also use the figure in Figure 2. The conditions (19) and (20) correspond regions in Figure 7 in this case of $(T_{3,7})_{3,35}$.

Theorem 6 or 7 correspond to the part (19) or (20) respectively. These will be proven in Section 4.2 and Section 4.3 respectively.

4.1. Modification of $\Phi(t,m)$

In $z^{i}_{-1}<m\leq z^{i}_{2g}$, the (local) behaviors of $\Phi(t,m)$ in the two cases of $2gp\leq q$ and $(2g-1)p\leq q<2gp$ are different. The different points are

[TABLE]

To compute the minimum of $\Phi(t,m)$ in the case of $(2g-1)p<q<2gp$, we modify $\Phi(t,m)$ into $\Phi^{i}(t,m)$ as defined in the next paragraph. For, $\Phi(t,m)$ does not satisfy this type of the cabling formula (4) any more because of the different behavior above. On the other hand, the modified $\Phi^{i}(t,m)$ has the same local behavior as the one in the case of $2gp\leq q$. Hence, $\Phi^{i}(t,m)$ becomes applicable to the formula (4).

We put

[TABLE]

and

[TABLE]

Two functions $\Phi^{i}(t,m)$ and $\Phi(t,m)$ coincide over $I_{i}^{\delta}$, while $\Phi^{i}(t,m)$ is the $-1$-shift of $\Phi(t,m)$ over the complement of $I_{i}^{\delta}$ when $1<i<p-1$. Figure 8 presents this.

Here we consider the following conditions.

Condition 1**.**

$\underset{z^{i}_{0}-\delta<m\leq z^{i+1}_{0}}{\min}\Phi(t,m)\leq\underset{z^{i}_{-1}<m\leq z^{i}_{0}-\delta}{\min}\Phi^{i}(t,m)$* holds.*

This condition is equivalent to

[TABLE]

Actually, since $\Phi(t,m)=\Phi^{i}(t,m)$ holds over $m\in I^{\delta}_{i}$, we have

[TABLE]

Condition 2**.**

$\underset{z^{i}_{0}-\delta<m\leq z^{i+1}_{0}}{\min}\Phi(t,m)\leq\underset{z^{i+1}_{0}<m\leq z^{i}_{2g}}{\min}\Phi^{i}(t,m)$* holds.*

In the same way as above this condition is equivalent to

[TABLE]

Now, we prove the following Claim 21.

Claim 21**.**

Let $t,s$, and $i$ be parameters satisfying (14).

Let $m$ be an integer with $z^{i}_{-1}<m\leq z^{i}_{0}$. Then, for $0\leq\nu\leq 2g$ we have

[TABLE]

Let $m$ be an integer with $z^{i}_{2g-1}=z^{i+1}_{0}-\delta<m\leq z^{i}_{2g}$. Then, for $0\leq\nu\leq 2g$ we have

[TABLE]

**Proof. ** Let $m$ be an integer with $z^{i}_{-1}<m\leq z^{i}_{0}$. Notice that (12) is satisfied even if we use $\Phi^{i}(t,m)$ instead of $\Phi(t,m)$. Then for any non-negative integer $\nu$ we have

[TABLE]

We explain the second equality in Claim 21. For any integer $j$ with $0\leq j<\nu$, $\#(S^{i}_{K_{p,q}}\cap[m+jp,m+(j+1)p))$ is

[TABLE]

Using the following relationship $z_{j}^{i}\in S_{K_{p,q}}\Leftrightarrow j\in S_{K}$, we have $\#(S^{i}_{K_{p,q}}\cap[m,m+\nu p))=i\nu+\#(S_{K}\cap[0,\nu))$.

Let $m$ be an integer with $z^{i}_{2g-1}<m\leq z^{i}_{2g}$. Let $\bar{S}_{K}$ be the complement of $S_{K}$ in ${\mathbb{Z}}$. In the same way as above, we have the following:

[TABLE]

∎

Next, we prove the following claim using Claim 21.

Claim 22**.**

The following holds:

• •

The inequality $s\geq\mu_{K}$ implies Condition 1.

• •

The inequality $s\leq 2-\mu_{K}$ implies Condition 2.

**Proof. ** If $s\geq\mu_{K}=\underset{1\leq\nu\leq 2g-1}{\min}\frac{2\varphi_{K}(\nu)}{\nu}$, then there exists an integer $\nu$ with $1\leq\nu\leq 2g-1$ satisfying $\Phi_{K}(s,\nu)\leq 0$. Then using the first formula in Claim 21, we have $\Phi^{i}(t,m+\nu p)-\Phi^{i}(t,m)=\Phi_{K}(s,\nu)\leq 0$. Thus applying this to $z^{i}_{-1}<m\leq z^{i}_{0}-\delta$, we have

[TABLE]

Therefore, Condition 1 is satisfied.

In the same way, we prove the second one. If $s\leq 2-\mu_{K}$, then there exists an integer $\nu$ with $1\leq\nu\leq 2g-1$ satisfying $\Phi_{K}(2-s,\nu)\leq 0$. In the same way as above we have $\Phi^{i}(t,m)-\Phi^{i}(t,m-\nu p)=-\Phi_{K}(2-s,\nu)\geq 0$.

[TABLE]

Therefore, Condition 2 is satisfied. ∎

4.2. Proof of Theorem 6.

In this section we prove Theorem 6.

4.2.1. Suppose that $\mu_{K}\leq s\leq 2-\mu_{K}$ and $1\leq i\leq p-2$ are satisfied.

Then, Claim 22 means Condition 1 and 2 are satisfied. Thus we have

[TABLE]

By using the same argument as the case of $2gp\leq q$,

[TABLE]

Here $\mu_{i}$ is the same thing as (16). Thus we have

[TABLE]

4.2.2. Suppose that $s\leq 2-\mu_{K}$ and $i=0$ are satisfied.

In this case, if $m\leq q$, then $\Phi^{0}(t,m)=\Phi(t,m)$ is satisfied from the definition of $\Phi^{0}(t,m)$. This implies for any $-p<m\leq-\delta$, we have $\Phi(t,m)=-tm/2\geq 0=\Phi(t,0)$. This means that Condition 1 is satisfied and

[TABLE]

Then $\underset{z^{0}_{-1}\leq m\leq z^{1}_{0}}{\min}\Phi(t,m)=\underset{-p<m\leq 2gp}{\min}\Phi^{0}(t,m)=\underset{m\in I_{0}}{\min}\Phi^{0}(t,m)$ holds, hence using Equation (17), and Claim 19 for $\underset{m\in I_{0}}{\min}\Phi^{0}(t,m)$ we have

[TABLE]

4.2.3. Suppose that $\mu_{K}\geq s$ are satisfied and $i=p-1$.

From the symmetry of $\Upsilon_{K_{p,q}}(t)$ and exchanging $s$ and $2-s$, we have

[TABLE]

∎

Considering Sections 4.2.1, 4.2.2 and 4.2.3, we can, therefore, finish the proof of Theorem 6.∎

4.3. Proof of Theorem 7.

First, we prove symmetry of $\Upsilon_{K}^{tr}(s)$.

Lemma 23**.**

*Let $K$ be an L-space knot. Then, we have $\Upsilon_{K}^{tr}(s)=\Upsilon_{K}^{tr}(2-s)$ *

**Proof. ** From the third of fundamental facts in Section 2.1,

[TABLE]

Using Lemma 11, we have

[TABLE]

∎ From Section 4.3.1, we give a proof of Theorem 7. Let $t,s$, and $i$ be parameters satisfying (14).

4.3.1. We suppose $0\leq s<\mu_{K}$ and $1\leq i\leq p-1$.

By using Equation (10), we have $s<\mu_{K}\leq 2-\mu_{K}$. Then for any integer $\nu,m$ with $1\leq\nu\leq 2g-1$ and $z^{i}_{-1}<m\leq z^{i}_{0}$, the following

[TABLE]

is satisfied.

Here, let $\mu_{i}^{1}(t)$ and $\mu_{i}^{2}(t)$ be the minimum values of $\Phi^{i}(t,m)$ over $z^{i}_{0}-\delta<m\leq z^{i}_{0}$ and $z^{i}_{-1}<m\leq z^{i}_{0}-\delta$ respectively. Since $\Phi_{K}(s,2g)-\Phi_{K}(s,0)=\#(S_{K}\cap[0,2g))-gs=g(1-s)\geq 0$, we obtain

[TABLE]

and

[TABLE]

Since $s<2-\mu_{K}$ holds as mentioned as above, there exists an integer $\nu$ with $1\leq\nu\leq 2g-1$ such that $\frac{2\varphi_{K}(\nu)}{\nu}<2-s$ holds. Hence, we have

[TABLE]

Hence

[TABLE]

Applying this, we obtain

[TABLE]

Here Claim 19 can be applied in our case:

[TABLE]

[TABLE]

As a result, we can compute $\Upsilon$-invariants as follows:

[TABLE]

4.3.2. We suppose $2-\mu_{k}<s\leq 2$ and $0\leq i\leq p-2$.

Reflecting $t$ as $t\mapsto 2-t$, other parameters are changed as $s\mapsto 2-s$ and $i\mapsto p-1-i$. Then using the symmetry of $\Upsilon$-invariant and the right previous formula and Lemma 23, we have

[TABLE]

At this point, we completed the proof of Theorem 7. ∎

Here we prove a corollary stated in Section 1.

Proof of Corollary 8. If $0<s<\mu_{K}$ or $2-\mu_{K}<s<2$ hold, then for all $\nu\in\{1,2,\cdots,2g-1\}$ $\Phi_{K}(s,\nu)>0=\Phi_{K}(s,0)=\Phi_{K}(s,2g)$ holds. Thus $\Upsilon_{K}^{tr}(s)<\Upsilon_{K}(s)$ holds. Therefore, we have

[TABLE]

Since for any $1<\nu<2g-1$, the inequality $\Phi^{i}(t,m)<\Phi^{i}(t,m+\nu p)$ for $z^{i}_{-1}<m\leq z^{i}_{0}-\delta$ holds. This means that $\Upsilon_{K_{p,q}}(t)\leq\Upsilon_{K}(s)+\Upsilon_{T_{p,q}}(t)$. $\Box$

5. The example $(T_{3,7})_{3,35}$.

5.1. Computation of $\Upsilon_{(T_{3,7})_{3,35}}$

In Section 1.6, we checked the failure of $\Upsilon$-invariant formula of Theorem 4 in the case of $(T_{3,7})_{3,35}$ by illustrating the difference of two graphs $\Upsilon_{T_{3,7}}(s)$ and $\Upsilon_{(T_{3,7})_{3,35}}(t)-\Upsilon_{T_{3,35}}(t)$ in Figure 2. We, here, investigate in terms of Theorem 6. Applying the cabling formula in Theorem 6 and 7 to this example, we compute precise function of $\Upsilon_{(T_{3,7})_{3,35}}(t)$. We put $K=T_{3,7}$. The genera are computed as $g=g_{3,7}=6$, $g_{3,35}=34$, and $g(K_{3,35})=52$. Then $S_{K}=\{0,3,6,7,9,10\}\cup{\mathbb{Z}}_{n\geq 12}$ holds, where $K=T_{3,7}$. When $\nu=0,1,2,\cdots,12$, the sequence $\varphi_{K}(\nu)$ is as follows:

[TABLE]

Hence, we have $\mu_{K}=2/3$ and $\delta=2$.

First, we consider $2/3<t<4/3$, namely this corresponds to the case of $i=1$ in Theorem 6. Furthermore we assume $2+s=3t$ and $0<s<2/3$. This means $2/3<t<8/9$. Then we have

[TABLE]

Since $\varphi_{T_{3,35}}(34)=\varphi_{T_{3,35}}(35)=12$, we have

[TABLE]

Since $\varphi_{T_{3,35}}(33)=11$, we have

[TABLE]

If $0<s<2/3$, then we have

[TABLE]

and while we have $\Upsilon_{K}(s)=-6s$.

Here we explain the second Equality (5.1). We consider several candidates of functions which give the maximum in $\{-2\varphi_{K}(\nu)-(6-\nu)s|\nu=1,2,\cdots,11\}$. During the set of $N_{i}:=\{s\in\{0,1,\cdots,11\}|\varphi_{K}(s)=i\}$ for $i\in{\mathbb{N}}$ the maximum function $-2\varphi_{K}(\nu)-(6-\nu)s$ is the one of the maximum $\nu$ in $N_{i}$. This coincides with $S_{K}\cap[1,11]=\{3,6,7,9,10\}$.

Suppose that $\varphi_{K}(\nu-1)<\varphi_{K}(\nu)$. Then since $-2\varphi_{K}(\nu-1)-(g-\nu+1)s>-2\varphi_{K}(\nu)-(g-\nu)s$ for any $0<s<2$. The function for such $\nu\in S_{K}$ is not a candidate of the maximum function. Thus we have only to consider $\{3,6,7,9,10\}-\{4,7,8,10,11\}=\{3,6,9\}$.

As a result, we have

[TABLE]

and

[TABLE]

Hence, when $2/3<t<8/9$, the $\Upsilon_{K_{p,q}}(t)$ is the following:

[TABLE]

Secondly, in $4/9<t<2/3$, applying (7) in Theorem 6, we compute $\Upsilon_{K_{3,35}}(t)$ as follows:

[TABLE]

6. Toward a further cabling formula

Let $K$ be an L-space knot. When positive relatively prime integers $p,q$ satisfy $q<(2g(K)-1)p$, the cable knot $K_{p,q}$ is not an L-space knot. In this case, to compute the $\Upsilon$-invariant $\Upsilon_{K_{p,q}}$, we would require the different formula. For example, consider the family $\Upsilon_{(T_{2,3})_{2,q}}$ for $q\in 2{\mathbb{Z}}+1$. Then the paper can give the following equalities

[TABLE]

where $2t\equiv s\bmod 2$ and $0\leq s\leq 2$. Furthermore, since we have $\Delta_{(T_{2,3})_{2,3}}(t)=\Delta_{T_{3,4}}(t)$, we obtain

[TABLE]

Furthermore, we obtain $\Upsilon_{(T_{2,3})_{2,1}}(t)$ as the graph in Figure 9. This is due to Hedden’s formula in [7]. This function coincides with

[TABLE]

because $(T_{2,3})_{2,1}$ is $\nu^{+}$-equivalent to $T_{3,4}\#(-T_{2,3})$. The $CFK^{\infty}((T_{2,3})_{2,1})$ is computed in [6].

These equalities can be generalized in other cases of cable knots of torus knots. For example, for $K=T_{2,5}$ and $g=2$ we have

[TABLE]

where $2t=s\bmod 2$ and $0\leq s\leq 2$. However, $\Upsilon_{K_{2,7}}(t)$ does not equal to the $\Upsilon$-invariant of any L-space cable knot $(T_{a,b})_{p,q}$ of any torus knot $T_{a,b}$ with $q\geq 2g_{a,b}p$. See Proposition 26.

Here we raise the following question.

Question 24**.**

Let $K$ be an L-space knot. Suppose that the integers $q,Q$ satisfy $q<(2g(K)-1)p<Q$. Does there exist any method to compute the $\Upsilon_{K_{p,q}}(t)$ by using $\Upsilon_{K_{p,Q}}(t)$ and so on?

7. Proofs of Proposition 9 and Theorem 10.

In [5], Feller and Krcatovich proved that the recurrence formula $\Upsilon_{T_{p,q}}(t)=\Upsilon_{T_{p,q-p}}(t)+\Upsilon_{T_{p,p+1}}(t)$. By using this formula, they proved the following closed formula of $\Upsilon$-invariant of torus knots.

Proposition 25** (Proposition 2.2 in [5]).**

Let $a_{i}$ be the same coefficient defined in (8) and $p_{i}$ the denominator of $[a_{i},a_{i+1},\cdots,a_{n}]$. Then we have

[TABLE]

Note that the formula depends on the way of taking the continued fraction in general, but it does not depend on the way to take the non-negative integral continued fraction expansions $q/p=[a_{i},\cdots,a_{n}]$, i.e., $a_{i}\geq 0$ for any $i$. Here we prove Proposition 9 by using the formula (22).

**Proof. ** From the torus knot formula, we immediately have

[TABLE]

Comparing the first derivative of (22) at $t=0$, we have

[TABLE]

The direct computation for $T_{p,p+1}$ implies the following:

[TABLE]

Thus, we have

[TABLE]

Since $p_{i-1}=a_{i}p_{i}+p_{i+1}$,

[TABLE]

Thus using (23) we get the following:

[TABLE]

$\Box$

Next, we prove Theorem 10 using Theorem 4.

**Proof. ** We put $L^{\prime}=L_{n-1}$. First we obtain the equality:

[TABLE]

This equality can be justified by regarding $\Upsilon_{K}(pt)$ as an extended function in such a way that we extend $\Upsilon_{K}(t)$ naturally to the periodic function over ${\mathbb{R}}$ with the period $2$. Using Theorem 4 and this computation we have

[TABLE]

By iterating this relationship we have

[TABLE]

$\Box$

Here we give an application.

Proposition 26**.**

Let $L$ be any iterated torus knot

[TABLE]

such that $(p_{i},q_{i})$ is any relatively prime integers satisfying $0<2g_{i}p_{i+1}\leq q_{i+1}$ for $1\leq i\leq n-1$, where $g_{i}$ the Seifert genus of the iterated torus knot

[TABLE]

Then $(T_{2,5})_{2,7}$ is not knot concordant to $L$.

**Proof. ** Let $K$ be $(T_{2,5})_{2,7}$. We give computation $I(K)=-17/3$ by using the formula in Theorem 7. If the assertion is not satisfied, then $I(K)$ is a linear combination of several $I(T_{p,p+1})$’s with positive integer coefficients. Since $I(T_{2,3})=-1$, $I(T_{3,4})=-8/3$, $I(T_{4,5})=-5$ and $I(T_{p,p+1})\leq-8$ for $p\geq 5$, the following is unique candidate:

[TABLE]

The denominator $p_{1}$ of $q_{1}/p_{1}$ is $2$ or $3$. Since the continued fractions of $q_{1}/p_{1}$ are $1+1/2$, $2+1/2$, $3+1/2$, $1+1/3$ or $1+1/(1+1/2)$. These are

[TABLE]

respectively. If the next rational $q_{2}/p_{2}$ satisfies $q_{2}/p_{2}\geq 2g_{2}=2$, $4$, $6$, $6$ and $8$ respectively. Since we cannot make either of the above five fractions with $q_{2}/p_{2}\geq 4$, we have $q_{1}/p_{1}=3/2$. The remaining value is

[TABLE]

The candidate of $q_{2}/p_{2}$ is $5/2$ only. The next condition is $q_{3}/p_{3}\geq 4$. We cannot make either of any five fractions with $q_{3}/p_{3}\geq 4$. Therefore $3I(T_{2,3})+I(T_{3,4})$ is not any integral values of such iterated torus knots. Since $I$ is a knot concordance invariant, $K$ is not knot concordant to any such iterated torus knot. $\Box$