1.1. Primary decomposition for topologically slice knots

In what follows, we state primary decomposition conjectures and main results for ${\mathcal T}$ .

For a knot K in $S^3$ , denote the Alexander polynomial by $\Delta _K$ and regard it as an element in the Laurent polynomial ring $\mathbb {Q}[t^{\pm 1}]$ . Then $\Delta _K$ is well defined up to associates. Recall that $\lambda $ and $\mu \in \mathbb {Q}[t^{\pm 1}]$ are associates if $\lambda = at^k \mu $ for some $a\in \mathbb {Q}\smallsetminus \{0\}$ and $k\in \mathbb {Z}$ . The standard involution on $\mathbb {Q}[t^{\pm 1}]$ is defined by $\bigl (\sum a_i t^i\bigr )^* = \sum a_i t^{-i}$ . We say that $\lambda $ and $\mu \in \mathbb {Q}[t^{\pm 1}]$ are $*$ -associates if $\lambda $ is an associate of either $\mu $ or $\mu ^*$ .

Denote the smooth concordance class of a knot K by $[K]$ . Recall that $\Delta = \{[K] \in {\mathcal T} \mid \Delta _K$ is trivial $\}$ . For an irreducible $\lambda $ in $\mathbb {Q}[t^{\pm 1}]$ , let

$$ \begin{align*} {\mathcal T}_{\lambda} &=\{[K] \in {\mathcal T} \mid \Delta_K \text{ is an associate of } (\lambda\lambda^*)^k\text{ for some }k\ge 0\}, \\ {\mathcal T}^{\kern3pt\lambda} &= \{[K] \in {\mathcal T} \mid \Delta_K \text{ is relatively prime to }\lambda\}. \end{align*} $$

We remark that the product $\lambda \lambda ^*$ in the definition of ${\mathcal T}_{\lambda }$ reflects the Fox-Milnor condition that $\Delta _K$ is an associate of $ff^*$ for some $f\in \mathbb {Q}[t^{\pm 1}]$ when K is topologically slice. Note that ${\mathcal T}_{\lambda }$ and ${\mathcal T}^{\kern3pt\lambda }$ are subgroups containing $\Delta $ for every irreducible $\lambda $ . Also, ${\mathcal T}_{\lambda }={\mathcal T}_{\mu }$ and ${\mathcal T}^{\kern3pt\lambda }={\mathcal T}^{\kern3pt\mu }$ if $\lambda $ and $\mu $ are $*$ -associates.

Primary decomposition for topologically slice knots concerns natural homomorphisms

$$ \begin{align*} \Phi_L\colon \bigoplus_{[\lambda]} {\mathcal T}_{\lambda}/\Delta {\longrightarrow} {\mathcal T}/\Delta \quad\text{and}\quad \Phi_R\colon {\mathcal T}/\Delta {\longrightarrow} \bigoplus_{[\lambda]} {\mathcal T}/{\mathcal T}^{\kern3pt\lambda}. \end{align*} $$

Here the index $[\lambda ]$ of the direct sums varies over the $*$ -associate classes of irreducibles $\lambda $ that arise as the factor of an Alexander polynomial of a knot. The homomorphism $\Phi _L$ is defined to be the sum of the inclusions ${\mathcal T}_{\lambda }/\Delta \hookrightarrow {\mathcal T}/\Delta $ . Because $\Delta _K$ is a product of finitely many irreducibles, the quotient epimorphisms ${\mathcal T}/\Delta \twoheadrightarrow {\mathcal T}/{\mathcal T}^{\kern3pt\lambda }$ induce a homomorphism into the direct sum, which is our $\Phi _R$ above. This formulation is influenced by earlier work in the literature, particularly Levine [Reference Levine35, Reference Levine34] and Cochran, Harvey and Leidy [Reference Cochran, Harvey and Leidy14, Reference Cochran, Harvey and Leidy13].

We remark that Definition A.1 in the appendix generalises the notion of left and right primary decomposition to a broader context. Also, regarding the relationship of Question 1.1(1) and (2), see Lemma A.4 in the appendix.

The first main result of this article, which is given as Theorem A, says that there is a large subgroup of ${\mathcal T}$ for which the answers to the above questions are affirmative and many primary parts of ${\mathcal T}$ are highly nontrivial. To state the result, we use the following notation. For a subgroup ${\mathcal S}$ of ${\mathcal T}$ and an irreducible $\lambda \in \mathbb {Q}[t^{\pm 1}]$ , let ${\mathcal S}_{\lambda }$ be the subgroup of $[K]\in {\mathcal S}$ with $\Delta _K$ a power of $\lambda \lambda ^*$ and ${\mathcal S}^{\lambda }$ be the subgroup of $[K]\in {\mathcal S}$ with $\gcd (\Delta _K,\lambda )=1$ . That is, ${\mathcal S}_{\lambda } = {\mathcal S}\cap {\mathcal T}_{\lambda }$ and ${\mathcal S}^{\lambda } = {\mathcal S}\cap {\mathcal T}^{\kern3pt\lambda }$ . Then one can ask Question 1.1 for ${\mathcal S}$ in place of ${\mathcal T}$ . Let

(1.1) $$ \begin{align} \Lambda=\{(m+1)t-m \mid m \text{ is a positive integer}\}. \end{align} $$

Note that $\Lambda $ is an infinite collection of pairwise non- $*$ -associate irreducibles $\lambda $ such that $\lambda \lambda ^*$ is an Alexander polynomial of a knot.

Theorem A. There is a subgroup ${\mathcal S}$ in ${\mathcal T}$ containing $\Delta $ that satisfies the following:

1. (1) For every $\lambda \in \Lambda $ , ${\mathcal S}_{\lambda }/\Delta \cong \mathbb {Z}^{\infty }$ , ${\mathcal S}/{\mathcal S}^{\lambda } \cong \mathbb {Z}^{\infty }$ and the composition ${\mathcal S}_{\lambda }/\Delta \hookrightarrow {\mathcal S}/\Delta \twoheadrightarrow {\mathcal S}/{\mathcal S}^{\lambda }$ is an isomorphism.

2. (2) The inclusions ${\mathcal S}_{\lambda }/\Delta \to {\mathcal S}/\Delta $ induce an isomorphism $\bigoplus _{\lambda \in \Lambda } {\mathcal S}_{\lambda }/\Delta \to {\mathcal S}/\Delta $ .

3. (3) The surjections ${\mathcal S}/\Delta \to {\mathcal S}/{\mathcal S}^{\lambda }$ induce an isomorphism ${\mathcal S}/\Delta \to \bigoplus _{\lambda \in \Lambda } {\mathcal S}/{\mathcal S}^{\lambda }$ .

(1.2)

From Theorem A it follows that each of the primary parts ${\mathcal T}_{\lambda } /\Delta $ and ${\mathcal T}/{\mathcal T}^{\kern3pt\lambda }$ has a subgroup isomorphic to $\mathbb {Z}^{\infty }$ for all $\lambda \in \Lambda $ . An immediate consequence is the main result of [Reference Hedden, Livingston and Ruberman25] that ${\mathcal T}/\Delta $ has a subgroup isomorphic to $\mathbb {Z}^{\infty }$ .

Note that Theorem A(1) implies the following.

Indeed, Theorem B is equivalent to Theorem A, by an elementary formal argument.

Proof that Theorem B implies Theorem A. Suppose that Theorem B holds. Let ${\mathcal S}$ be the subgroup in ${\mathcal T}$ that is generated by $\Delta $ and the two-parameter family $\{K_{\lambda ,i}\}_{\lambda \in \Lambda , i\in {\mathbb N}}$ given by Theorem B.

We claim that, for each $\lambda \in \Lambda $ , ${\mathcal S}_{\lambda }$ is equal to the subgroup generated by $\Delta $ and the one-parameter family $\{K_{\lambda ,i}\}_{i\in {\mathbb N}}$ . To see this, first observe that for irreducibles $\lambda $ and $\mu $ that are not $*$ -associates, ${\mathcal S}_{\lambda }\subset {\mathcal S}^{\mu }$ , and so the composition

(1.3) $$ \begin{align} {\mathcal S}_{\lambda}/\Delta \hookrightarrow {\mathcal S}/\Delta \twoheadrightarrow {\mathcal S}/{\mathcal S}^{\mu} \end{align} $$

is zero. If a linear combination $[J] + \sum _{\mu } \sum _i r_{\mu ,i} [K_{\mu ,i}]$ ( $[J]\in \Delta $ , $r_{\mu ,i} \in \mathbb {Z}$ ) lies in ${\mathcal S}_{\lambda }$ , then for each $\mu \ne \lambda $ , the image of the linear combination in ${\mathcal S}/{\mathcal S}^{\mu }$ , which is represented by $\sum _{i} r_{\mu ,i} [K_{\mu ,i}]$ , should be zero by the observation. Therefore, $r_{\mu ,i}=0$ for all i and $\mu \ne \lambda $ , by the conclusion of Theorem B that the classes $[K_{\mu ,i}]$ are linearly independent in ${\mathcal S}/{\mathcal S}^{\mu }$ . This proves the claim.

Fix $\lambda \in \Lambda $ and temporarily denote by $\mathbb {Z}^{\infty }$ the free abelian group generated by the collection $\{K_{\lambda ,i}\}_i$ . The assignment $K_{\lambda ,i} \mapsto [K_{\lambda ,i}]$ gives rise to an epimorphism $\mathbb {Z}^{\infty } \twoheadrightarrow {\mathcal S}_{\lambda }/\Delta $ by the claim. Because ${\mathcal S}$ is generated by $\{K_{\lambda ,i}\}_{\lambda ,i}$ and $[K_{\mu ,i}]=0$ in ${\mathcal S}/{\mathcal S}^{\lambda }$ for $\mu \ne \lambda $ , the composition

(1.4) $$ \begin{align} \mathbb{Z}^{\infty} \twoheadrightarrow {\mathcal S}_{\lambda}/\Delta \hookrightarrow {\mathcal S}/\Delta \twoheadrightarrow {\mathcal S}/{\mathcal S}^{\lambda} \end{align} $$

is surjective. Moreover, by the linear independence of $[K_{\lambda ,i}]$ in ${\mathcal S}/{\mathcal S}^{\lambda }$ , (1.4) is an isomorphism. From this, it follows that both $\mathbb {Z}^{\infty } \to {\mathcal S}_{\lambda }/\Delta $ and ${\mathcal S}_{\lambda }/\Delta \to {\mathcal S}/{\mathcal S}^{\lambda }$ are isomorphisms. This shows that Theorem A(1) holds.

Because (1.3) is zero for $\mu \ne \lambda \in \Lambda $ , the composition of two horizontal arrows on the top row of (1.2) is the direct sum of the isomorphisms ${\mathcal S}_{\lambda }/\Delta \to {\mathcal S}/{\mathcal S}^{\lambda }$ . So the top row composition in (1.2) is an isomorphism. Also, the first horizontal arrow in the top row is surjective by the definition of ${\mathcal S}$ . From this, it follows that Theorem A(2) and (3) hold.

1.2. Primary decomposition for the bipolar filtration

The method of this article provides further information on the structure of ${\mathcal T}$ . To discuss this, we consider the bipolar filtration of ${\mathcal T}$ , which was defined by Cochran, Harvey and Horn [Reference Cochran, Harvey and Horn11]. It is a descending filtration

$$ \begin{align*} \{0\} \subset \cdots \subset {\mathcal T}_n \subset \cdots \subset {\mathcal T}_1 \subset {\mathcal T}_0 \subset {\mathcal T}, \end{align*} $$

where the subgroup ${\mathcal T}_n$ consists of concordance classes of certain knots called n-bipolar (see Definition 2.1). It is known that various modern smooth invariants vanish for knots in the subgroups ${\mathcal T}_0$ and ${\mathcal T}_1$ , but the associated graded groups $\mathrm {gr}_n({\mathcal T}\kern3pt):={\mathcal T}_n/{\mathcal T}_{n+1}$ are nontrivial for all n [Reference Cochran, Harvey and Horn11, Reference Cha and Kim6]. Indeed, the abelian group $\mathrm {gr}_n({\mathcal T}\kern3pt)$ is known to have infinite rank for $n=0$ [Reference Cochran and Horn15] and for all $n\ge 2$ [Reference Cha and Kim6].

As an attempt to understand the structure of the filtration, we formulate and study the primary decomposition of $\mathrm {gr}_n({\mathcal T}\kern3pt)$ . For an n-bipolar knot K, denote its class in $\mathrm {gr}_n({\mathcal T}\kern3pt)$ by $[K]$ . Similar to the case of ${\mathcal T}$ , for an irreducible element $\lambda $ in $\mathbb {Q}[t^{\pm 1}]$ , consider the following subgroups of $\mathrm {gr}_n({\mathcal T}\kern3pt)$ :

$$ \begin{align*} \mathrm{gr}_n({\mathcal T}\kern3pt)_{\lambda} &:=\{[K] \in \mathrm{gr}_n({\mathcal T}\kern3pt) \mid \Delta_K = (\lambda\lambda^*)^k \text{ for some } k \ge 0\}, \\ \mathrm{gr}_n({\mathcal T}\kern3pt)^{\lambda} &:= \{[K] \in \mathrm{gr}_n({\mathcal T}\kern3pt) \mid \Delta_K \text{ is relatively prime to }\lambda\}. \end{align*} $$

Also, let $\mathrm {gr}_n(\Delta ) =\{[K] \in \mathrm {gr}_n({\mathcal T}\kern3pt) \mid \Delta _K\ \text{is trivial}\}$ . The injections $\mathrm {gr}_n({\mathcal T}\kern3pt)_{\lambda }/\mathrm {gr}_n(\Delta ) \hookrightarrow \mathrm {gr}_n({\mathcal T}\kern3pt)/\mathrm {gr}_n(\Delta )$ and the surjections $\mathrm {gr}_n({\mathcal T}\kern3pt)/\mathrm {gr}_n(\Delta ) \twoheadrightarrow \mathrm {gr}_n({\mathcal T}\kern3pt)/\mathrm {gr}_n({\mathcal T}\kern3pt)^{\lambda }$ induce homomorphisms

$$ \begin{align*} \Phi^n_L\colon & \bigoplus\nolimits_{[\lambda]} \mathrm{gr}_n({\mathcal T}\kern3pt)_{\lambda}/\mathrm{gr}_n(\Delta) \longrightarrow \mathrm{gr}_n({\mathcal T}\kern3pt)/\mathrm{gr}_n(\Delta), \\ \Phi^n_R\colon & \mathrm{gr}_n({\mathcal T}\kern3pt)/\mathrm{gr}_n(\Delta) \longrightarrow \bigoplus\nolimits_{[\lambda]} \mathrm{gr}_n({\mathcal T}\kern3pt)/\mathrm{gr}_n({\mathcal T}\kern3pt)^{\lambda}. \end{align*} $$

The following is an analogue of Question 1.1.

The following result supports affirmative answers. Similar to the case of ${\mathcal T}$ , for a subgroup ${\mathcal S}$ in $\mathrm {gr}_n({\mathcal T}\kern3pt)$ , let ${\mathcal S}_{\lambda }={\mathcal S}\cap \mathrm {gr}_n({\mathcal T}\kern3pt)_{\lambda }$ and ${\mathcal S}^{\lambda }={\mathcal S}\cap \mathrm {gr}_n({\mathcal T}\kern3pt)^{\lambda }$ . Recall that the collection $\Lambda $ has been defined in (1.1) above.

Theorem C. Let $n\ge 2$ . Then there is a subgroup ${\mathcal S}$ in $\mathrm {gr}_n({\mathcal T}\kern3pt)$ that contains $\mathrm {gr}_n(\Delta )$ such that

$$ \begin{align*} \bigoplus_{\lambda\in\Lambda} {\mathcal S}_{\lambda}/\mathrm{gr}_n(\Delta) \xrightarrow{\cong} {\mathcal S}/\mathrm{gr}_n(\Delta) \xrightarrow{\cong} \bigoplus_{\lambda\in\Lambda} {\mathcal S} / {\mathcal S}^{\lambda} \end{align*} $$

and ${\mathcal S}_{\lambda }/\mathrm {gr}_n(\Delta ) \cong \mathbb {Z}^{\infty } \cong {\mathcal S} / {\mathcal S}^{\lambda }$ for every $\lambda \in \Lambda $ .

By the same argument as the proof that Theorem A is equivalent to Theorem B, it is seen that Theorem C is equivalent to the following statement.

Also, Theorem D implies Theorem B. Indeed, if a linear combination $\#_i\, a_i K_{\lambda ,i}$ of the knots $K_{\lambda ,i}$ in Theorem D is smoothly concordant to a knot L with $\Delta _L$ relatively prime to $\lambda $ , then the class $[L]$ automatically lies in the subgroup ${\mathcal T}_n$ of ${\mathcal T}$ because so are $K_{\lambda ,i}$ , and thus $[L]$ is zero in the quotient $\mathrm {gr}_n({\mathcal T}\kern3pt)/\mathrm {gr}_n({\mathcal T}\kern3pt)^{\lambda }$ . It follows that $a_i=0$ for all i, by Theorem D. Therefore, to obtain Theorems A, B and C, it suffices to prove Theorem D.

The remaining part of this article is organised as follows. Sections 2–4 are devoted to the proof of Theorem D. In the appendix, we discuss a general formulation of the notion of primary decomposition.

2.1. Construction of a family of knots $\{K_i\}$

We start the proof of Theorem D with a construction of knots that will be shown to generate the promised infinite rank free abelian subgroup.

Fix an integer $n\ge 2$ , as in Theorem D. Also, fix another integer $m\ge 1$ . Several objects we will use below depend on $(n,m)$ , but we omit it from notation because $(m,n)$ is fixed in our arguments. Let

$$ \begin{align*} \lambda(t) = \lambda_m(t) = (m+1)t-m \in \Lambda. \end{align*} $$

For each $(n,m)$ , we will construct an infinite family of knots $\{K_i\}$ indexed by integers $i>0$ , whose Alexander polynomial is $\lambda (t)\lambda (t^{-1})$ . The construction is similar to [Reference Cha and Kim6, Section 2.2]. (See also [Reference Cochran, Harvey and Horn11], which influenced the construction of [Reference Cha and Kim6] and ours.) Let $R(J,D)$ be the knot shown in Figure 1. Here J and D are knots that will be specified later. (For now, ignore the circles $\alpha _J$ and $\alpha _D$ .) The knot $R(J,D)$ bounds an obviously seen Seifert surface of genus one, which consists of a 0-handle and two 1-handles. In Figure 1, the two 1-handles are untwisted and cross each other $2m+1$ times. So, $S=\left [\begin {smallmatrix}0 & m+1\\m & 0\end {smallmatrix}\right ]$ is a Seifert matrix. We remark that [Reference Cha and Kim6] and [Reference Cochran, Harvey and Horn11] use the particular case of $m=1$ .

Figure 1 The knot $R(J,D)$ .

Because $tS-S^T$ presents the Alexander module of $R(J,D)$ , a routine computation shows that $R(J,D)$ has (integral) Alexander module

(2.1) $$ \begin{align} H_1(M(R(J,D));\mathbb{Z}[t^{\pm1}]) = \mathbb{Z}[t^{\pm1}]/\langle \lambda(t)\rangle \oplus \mathbb{Z}[t^{\pm1}]/\langle \lambda(t^{-1})\rangle, \end{align} $$

and the summands $\mathbb {Z}[t^{\pm 1}]/\langle \lambda (t)\rangle $ and $\mathbb {Z}[t^{\pm 1}]/\langle \lambda (t^{-1})\rangle $ are equal to the subgroups $\langle \alpha _J\rangle $ and $\langle \alpha _D\rangle $ generated by the loops $\alpha _J$ and $\alpha _D$ shown in Figure 1. It follows that $\Delta _{R(J,D)} = \lambda (t)\lambda (t^{-1})$ . Moreover, the Blanchfield pairing

$$ \begin{align*} B\ell\colon {\mathcal A} \times {\mathcal A} \longrightarrow \mathbb{Q}(t)/\mathbb{Q}[t^{\pm1}] \end{align*} $$

on the Alexander module ${\mathcal A} := H_1(M(R(J,D));\mathbb {Q}[t^{\pm 1}])$ has exactly two metabolisers, $\langle \alpha _J\rangle $ and $\langle \alpha _D\rangle $ . Here, a submodule $P\subset {\mathcal A}$ is called a metaboliser if P is equal to $P^{\perp }:=\{x\in {\mathcal A} \mid B\ell (P,x)=0\}$ . The above observations on $R(J,D)$ hold for any choice of J and D.

As another ingredient of the construction of the promised knots $K_i$ , we will use a family of knots $\{J^i_k\}$ , which was given in [Reference Cha and Kim6]. For $k=0$ , an explicit construction of such $\{J^i_0\}_i$ is given in [Reference Cha and Kim6, Section 4]. We need that $\{J^i_0\}_i$ satisfies the following conditions (J1), (J2) and (J3) for some sequence of increasing primes $\{p_i\}$ . For a knot J and an integer p, let $\sigma _J(\omega )\in \mathbb {Z}$ be the Levine-Tristram signature function of J at $\omega \in S^1\subset \mathbb {C}$ , and let

(2.2) $$ \begin{align} \rho(J,\mathbb{Z}_d) = \frac1d \sum_{k=0}^{d-1} \sigma_J(e^{2\pi k\sqrt{-1}/d}). \end{align} $$

• (J1) For each i, $J^i_0$ is $0$ -negative.

• (J2) For each i, $|\rho (J^i_0,\mathbb {Z}_{p_i})|> 69\,713\,280\cdot (6n+8m+86)$ .

• (J3) For $i<j$ , $\rho (J^j_0,\mathbb {Z}_{p_i})=0$ .

For $k=0,\ldots ,n-2$ , $J^i_{k+1}$ is defined inductively by $J^i_{k+1} = P_k(\eta _k,J^i_k)$ , where $P_k(\eta _k,J^i_k)$ is the satellite knot shown in the right of Figure 2. The left of Figure 2 shows the pattern $P_k$ in the exterior of an unknotted circle $\eta _k$ , which is a standard solid torus. The companion is the knot $J^i_k$ .

Figure 2 The stevedore’s pattern $(P_k,\eta _k)$ and the satellite knot $P_k(\eta _k,J^i_k)$ .

Now, let $K_i = R(J_{n-1}^i,D)$ , where $D=\mathrm {Wh}^+(T)$ is the positive Whitehead double of the right-handed trefoil T. Because D is topologically slice [Reference Freedman21], $K_i$ is topologically concordant to $R(J_{n-1}^i,U)$ , where U is the trivial knot. Because $R(J,U)$ is (smoothly) slice for any J, it follows that $K_i$ is topologically slice. By [Reference Cha and Kim6, Section 2.2, Lemma 2.3], property (J1) implies that each $K_i$ is n-negative and k-positive for all $k\ge 0$ . Because $\Delta _{K_i} = \lambda (t)\lambda (t^{-1})$ , it follows that the class $[K_i]$ lies in $\mathrm {gr}_n({\mathcal T}\kern3pt)_{\lambda } = ({\mathcal T}_n\cap {\mathcal T}_{\lambda }) / ({\mathcal T}_{n+1}\cap {\mathcal T}_{\lambda })$ . Therefore, to prove Theorem D, it suffices to show the following statement.

For the special case that L is a trivial knot and $m=1$ , the conclusion of Theorem 2.2 was shown in [Reference Cha and Kim6]. The general case of Theorem 2.2 requires substantially more sophisticated ideas and methods, which will be discussed in Sections 3 and 4.

2.2. Construction of a negaton

To prove Theorem 2.2 by contradiction, suppose that the knot K is $(n+1)$ -bipolar. Recall that $K_i$ is n-bipolar. The following observation will be useful. The knot L in the statement of Theorem 2.2 is automatically n-bipolar, because L is concordant to the sum of K, $-a_1K_1,\ldots$ $-a_{r-1}K_{r-1}$ , and $-a_rK_r$ and each of K and $K_i$ are n-bipolar.

We may assume $a_i\ne 0$ for all i, by removing $K_i$ when $a_i=0$ . In addition, by taking $-K$ instead of K, we may assume that $a_1>0$ . Under this assumption, we will prove that K is not $(n+1)$ -negative. To derive a contradiction, suppose that K is $(n+1)$ -negative. As done in [Reference Cha and Kim6, Section 2.3], we will construct a certain n-negaton for the first summand $K_1$ , which we will call $X^-$ below. Because the generality of Theorem 2.2 does not cause significant issues in this construction, we will closely follow [Reference Cha and Kim6], with minor additional changes related to L.

Use the n-bipolarity of L to choose an n-negaton, say, $Z^-_L$ , bounded by $-M(L)=M(-L)$ . Use the n-bipolarity of $K_i$ , choose an n-negaton $Z^-_i$ bounded by $-M(K_i)=M(-K_i)$ for each i for which $a_i>0$ and choose n-negaton $Z^-_i$ bounded by $M(K_i)$ for each i for which $a_i<0$ . Let $V^-$ be an $(n+1)$ -negaton bounded by $M(K)$ . There is a standard cobordism C bounded by the union of $\partial _-C := -M(K)$ and $\partial _+C := \bigl (\bigsqcup _i a_i M(K_i)\bigr ) \sqcup M(L)$ , which is associated with the connected sum expression $K=\bigl (\#_{i=1}^r \, a_i K_i\bigr ) \# L$ ; C is obtained by attaching to $\bigl (\bigsqcup _{i=1}^r a_i M(K_i) \times I\bigr ) \sqcup M(L)\times I$ , N 1-handles that connect the component and then attaching N 2-handles that make meridians of the involved $N+1$ knots parallel, where $N=\sum _i |a_i|$ . A detailed description of C can be found, for instance, from [Reference Cochran, Orr and Teichner17, p. 113]. Define

(2.3) $$ \begin{align} X^- := V^- \mathbin{\mathop{\cup}\limits_{\partial_-C}} C \mathbin{\mathop{\cup}\limits_{\partial_+C}} \bigg((a_1-1)Z^-_1 \sqcup \Bigl({\textstyle\bigsqcup\limits_{i>1} |a_i| Z^-_i} \Bigr) \sqcup Z^-_L \bigg). \end{align} $$

Figure 3 depicts the construction of $X^-$ . By (the argument of) [Reference Cha and Kim6, Lemma 2.4], $X^-$ is an n-negaton bounded by $M(K_1)$ .

Figure 3 The construction of $X^-$ . The sign of $M(K_i)$ equals that of $a_i$ .

Now, let

(2.4) $$ \begin{align} P := \operatorname{Ker} \{ H_1(M(K_1);\mathbb{Q}[t^{\pm1}]) \rightarrow H_1(X^-;\mathbb{Q}[t^{\pm1}]) \}. \end{align} $$

Because $X^-$ is an n-negaton with $n\ge 2$ , P is a metaboliser of the Blanchfield pairing on $H_1(M(K_1);\mathbb {Q}[t^{\pm 1}])$ , by [Reference Cochran, Harvey and Horn11, Theorem 5.8]. (See also the statement of [Reference Cha and Kim6, Lemma 2.5].) Because $K_1=R(J^1_{n-1},D)$ has exactly two metabolisers $\langle \alpha _J\rangle $ and $\langle \alpha _D\rangle $ , we have the following two cases: $P=\langle \alpha _D\rangle $ or $P=\langle \alpha _J\rangle $ . By deriving a contradiction for each case, the proof of Theorem 2.2 will be completed.

3.1. A localised mixed-type commutator series associated with X

Recall from Subsection 2.1 that $p_1$ denotes the first prime used in properties (J2) and (J3), and $\lambda (t)=(m+1)t-m$ . Let

$$ \begin{align*} \Sigma := \{f(t)\in \mathbb{Q}[t^{\pm1}] \mid f(1)\ne 0, \, \gcd(f(t),\lambda(t)\lambda(t^{-1}))=1 \}. \end{align*} $$

Obviously, $\Sigma $ is a multiplicative subset. Let $\mathbb {Q}[t^{\pm }]\Sigma ^{-1}$ be the localisation.

Definition 3.1. Let G be a group endowed with a homomorphism $G\to \pi _1(X)$ that induces an epimorphism $H_1(G) \twoheadrightarrow H_1(X)$ . For $i=0,1,\ldots ,n+1$ , define subgroups ${\mathcal P}^i G$ of G inductively as follows. Let ${\mathcal P}^0 G := G$ , and let ${\mathcal P}^1 G$ be the kernel of the composition

Let ${\mathcal P}^2G$ be the kernel of the composition

Here, $\operatorname {Im} H_1(Z_L^0;\mathbb {Q}[t^{\pm 1}]\Sigma ^{-1})$ is the image of $H_1(Z_L^0;\mathbb {Q}[t^{\pm 1}]\Sigma ^{-1}) \to H_1(X;\mathbb {Q}[t^{\pm 1}]\Sigma ^{-1})$ induced by the inclusion.

For $i=2,\ldots$ , $n-1$ , let

$$ \begin{align*} {\mathcal P}^{i+1}G := \operatorname{Ker} \left\{ {\mathcal P}^i G \twoheadrightarrow \frac{{\mathcal P}^i G}{[{\mathcal P}^i G,{\mathcal P}^i G]} \rightarrow \frac{{\mathcal P}^i G}{[{\mathcal P}^i G,{\mathcal P}^i G]} {\mathbin{\mathop{\otimes}_{{\mathbb{Z}}}}} \mathbb{Q} = H_1\bigl(G;\mathbb{Q}[G/{\mathcal P}^i G]\bigr) \right\}. \end{align*} $$

Finally, define

$$ \begin{align*} {\mathcal P}^{n+1}G := \operatorname{Ker} \left\{ {\mathcal P}^n G \twoheadrightarrow \frac{{\mathcal P}^n G}{[{\mathcal P}^n G,{\mathcal P}^n G]} \twoheadrightarrow \frac{{\mathcal P}^n G}{[{\mathcal P}^n G,{\mathcal P}^n G]} {\mathbin{\mathop{\otimes}_{{\mathbb{Z}}}}} \mathbb{Z}_{p_1} = H_1\bigl(G;\mathbb{Z}_{p_1}[G/{\mathcal P}^n G]\bigr) \right\}. \end{align*} $$

It is straightforward to verify inductively that ${\mathcal P}^k G$ is a normal subgroup of G and the standard kth derived subgroup $G^{(k)}$ lies in ${\mathcal P}^k G$ for all k.

We remark that the commutator series $\{{\mathcal P}^k G\}_k$ in Definition 3.1 is ‘localised at polynomials’ in the sense of [Reference Cochran, Harvey and Leidy14, Sections 3 and 4] and is of ‘mixed-coefficient’ type in the sense that we use both $\mathbb {Z}_{p_1}$ and $\mathbb {Q}$ (see [Reference Cha3, Section 4.1]).

In particular, define the subgroups ${\mathcal P}^i\pi _1(X_k)$ , by applying Definition 3.1 to the case $G=\pi _1(X_k) \to \pi _1(X)$ . The following properties, which we will state as Assertions A and B, are essential for our purpose. Let $\mu _k\subset M(J^1_k)$ be the meridian of $J^1_k$ . By the construction of $X_k$ , $M(J^1_k)$ is a component of $\partial X_k$ (see Figure 5), and thus $\mu _k$ represents an element in $\pi _1(X_k)$ for $k\le n-1$ . For brevity, let $J^1_n := K_1$ , so that the previous sentence holds for $k=n$ as well. Also, let $(P_{n-1},\eta _{n-1}):=(R(U,D),\alpha _D)$ , so that $J^1_k = P_{k-1}(\eta _{k-1},J^1_{k-1})$ holds for $k=n$ as well.

In the proof of Assertion A, the following fact will be useful. Let $\lambda _{k-1}$ be the zero-linking longitude of $J^1_{k-1}$ , which lies in

$$ \begin{align*} E(J^1_{k-1}) \subset E(J^1_{k-1})\cup E(P_{k-1}\sqcup \eta_{k-1}) = E(J^1_k) \subset M(J^1_k) \subset \partial X_k. \end{align*} $$

Assertion C. The inclusion $X_k \subset X_{k-1}$ induces an isomorphism ${\mathcal P}^i\pi _1(X_k)/\langle \lambda _{k-1} \rangle \cong {\mathcal P}^i\pi _1(X_{k-1})$ for all $i\le n-k+2$ . Consequently, we have

$$ \begin{align*} {\mathcal P}^{n-k+1}\pi_1(X_k)/{\mathcal P}^{n-k+2}\pi_1(X_k) \cong {\mathcal P}^{n-k+1}\pi_1(X_{k-1})/{\mathcal P}^{n-k+1}\pi_1(X_{k-1}). \end{align*} $$

Proof of Assertion C. Because $E_{k-1}$ is obtained by attaching a 2-handle to $M(J^1_k)\times [0,1]$ along $\lambda _{k-1}$ and then attaching a 3-handle (see Figure 4), $X_{k-1}=X_k\cup E_{k-1}$ is obtained from $X_k$ by the same handle attachments. It follows that $\pi _1(X_{k-1}) \cong \pi _1(X_k)/\langle \lambda _{k-1} \rangle $ . This shows the assertion for $i=0$ . Now, to proceed by induction, suppose that $i\ge 1$ and suppose that the assertion holds for $i-1$ . If $i=1$ , we have the following commutative diagram with exact rows:

Because the two rightmost horizontal arrows are to the same target $H_1(X)$ , there is an induced epimorphism ${\mathcal P}^i\pi _1(X_k) \twoheadrightarrow {\mathcal P}^i\pi _1(X_{k-1})$ and its kernel is equal to that of the epimorphism ${\mathcal P}^{i-1}\pi _1(X_k) \twoheadrightarrow {\mathcal P}^{i-1}\pi _1(X_{k-1})$ . So the assertion holds for $i=1$ . For $i=2$ , replace $H_1(X)$ in the above diagram by the quotient

$$ \begin{align*} H_1\bigl(X;\mathbb{Q}[t^{\pm1}]\Sigma^{-1}\bigr) / \operatorname{Im} H_1\bigl(Z_L^0;\mathbb{Q}[t^{\pm1}]\Sigma^{-1}\bigr) \end{align*} $$

and apply the same argument.

For $i\ge 3$ , we have

where $R=\mathbb {Q}$ or $\mathbb {Z}_{p_1}$ , depending on i. We claim that the rightmost vertical arrow is an isomorphism. From the claim, it follows that the assertion holds for i, once again by the argument used above. To show the claim, let $\gamma _{k-1}$ be the meridian of $J^1_{k-1}$ in the exterior $E(J^1_{k-1}) \subset M(J^1_k) \subset \partial X_k$ . Note that $\gamma _{k-1}$ is different from the meridian $\mu _{k-1}\subset M(J^1_{k-1})\subset \partial X_{k-1}$ used in the statement of Assertion A, but $\gamma _{k-1}$ and $\mu _{k-1}$ are isotopic in the cobordism $E_{k-1}$ . The meridian $\gamma _{k-1}$ is identified with the curve $\eta _{k-1}$ , which lies in the commutator subgroup $\pi _1(M(J^1_k))^{(1)}$ . Because $\pi _1(M(J^1_k))$ is normally generated by the meridian $\mu _k$ , $\gamma _{k-1}$ lies in $\langle \mu _k\rangle ^{(1)} = \langle \gamma _k\rangle ^{(1)}$ in $\pi _1(X_k)$ . By induction, it follows that $\gamma _{k-1}$ lies in $\langle \gamma _n\rangle ^{(n-k+1)}$ . Therefore, the image of $\pi _1(E(J^1_{k-1}))$ lies in $\pi _1(X_k)^{(n-k+1)}$ . Because the longitude $\lambda _{k-1}$ lies in $\pi _1(E(J^1_{k-1}))^{(1)}$ , it follows that

(3.1) $$ \begin{align} \lambda_{k-1} \in \pi_1(X_k)^{(n-k+2)}. \end{align} $$

Because $i\le n-k+2$ , (3.1) implies that $\lambda _{k-1} \in \pi _1(X_k)^{(i)} \subset ({\mathcal P}^{i-1}\pi _1(X_k))^{(1)}$ . Also, by the induction hypothesis, ${\mathcal P}^{i-1}\pi _1(X_k) \twoheadrightarrow {\mathcal P}^{i-1}\pi _1(X_{k-1})$ is an epimorphism with kernel $\langle \lambda _{k-1}\rangle $ . It follows that the rightmost vertical arrow in the above diagram is an isomorphism. This completes the proof of the assertion.

Proof of Assertion A. In the proof of Assertion C, we already showed that $\mu _k$ lies in $\pi _1(X_k)^{(n-k)}$ . This is the first part of Assertion A.

It remains to show that $\mu _k$ is nontrivial in ${\mathcal P}^{n-k}\pi _1(X_k)/{\mathcal P}^{n-k+1}\pi _1(X_k)$ . We will use reverse induction for $k=n$ , $n-1,\ldots$ , $0$ . For $k=n$ , $\pi _1(X_n)/{\mathcal P}^1\pi _1(X_n) = H_1(X)$ by definition, and the meridian $\mu _n$ of $J^1_n = K_1$ is a generator of $H_1(X)$ . So the assertion holds.

For the case $k=n-1$ , let $\Lambda =\mathbb {Q}[t^{\pm 1}]$ for brevity, and consider the following commutative diagram:

Here, $H_1(X_n;\Lambda \Sigma ^{-1}) \to H_1(X;\Lambda \Sigma ^{-1})$ is an isomorphism, because $\pi _1(X)=\pi _1(X_0)$ is isomorphic to $\pi _1(X_n)/\langle \lambda _0,\ldots ,\lambda _{n-1}\rangle $ by Assertion C, and the longitudes $\lambda _0,\ldots ,\lambda _{n-1}$ lie in $\pi _1(X_n)^{(2)}$ by (3.1). The same argument shows that the bottom horizontal arrow is an isomorphism. (Alternatively, one may use Mayer-Vietoris arguments to show that they are isomorphisms.) Also, consider the Mayer-Vietoris sequence for $X = \overline {X\smallsetminus Z_L^0} \cup _{M(L)} Z_L^0$ :

$$ \begin{align*} H_1(M(L);\Lambda\Sigma^{-1}) \longrightarrow H_1(X\smallsetminus Z_L^0;\Lambda\Sigma^{-1}) \oplus H_1(Z_L^0;\Lambda\Sigma^{-1}) \longrightarrow H_1(X;\Lambda\Sigma^{-1}) \longrightarrow 0. \end{align*} $$

We have $H_1(M(L);\Lambda \Sigma ^{-1}) = 0$ by the hypothesis that $\Delta _L(t)$ is relatively prime to the polynomial $\lambda (t)$ . It follows that the diagonal arrow on the right-hand side of the above diagram is an isomorphism.

By our hypothesis, the kernel P of

$$ \begin{align*} \langle \alpha_J\rangle \oplus \langle\alpha_D\rangle = H_1(M(K_1);\Lambda) \longrightarrow H_1(X_n;\Lambda) \end{align*} $$

is equal to the summand $\langle \alpha _D\rangle $ . So the other summand $\langle \alpha _J\rangle $ injects into $H_1(X_n;\Lambda )$ . Because $\langle \alpha _J\rangle \cong \Lambda /\langle \lambda \rangle $ is not annihilated by $\Sigma $ , this implies that $\langle \alpha _J\rangle $ injects into $H_1(X_n;\Lambda \Sigma ^{-1}) = H_1(X_n;\Lambda )\Sigma ^{-1}$ . By the above diagram, it follows that $\langle \alpha _J\rangle $ injects into $H_1(X\smallsetminus Z_L^0;\Lambda \Sigma ^{-1})$ . Thus, $\alpha _J$ is nontrivial in $H_1(X;\Lambda \Sigma ^{-1})/\operatorname {Im} H_1(Z_L^0;\Lambda \Sigma ^{-1})$ . Therefore, by the definition of ${\mathcal P}^2\pi _1(X_n)$ , $\alpha _J$ is nontrivial in the quotient

$$ \begin{align*} {\mathcal P}^1\pi_1(X_n)/{\mathcal P}^2\pi_1(X_n) \subset H_1(X;\Lambda\Sigma^{-1})/\operatorname{Im} H_1(Z_L^0;\Lambda\Sigma^{-1}). \end{align*} $$

By Assertion C, ${\mathcal P}^1\pi _1(X_n)/{\mathcal P}^2\pi _1(X_n) \cong {\mathcal P}^1\pi _1(X_{n-1})/{\mathcal P}^2\pi _1(X_{n-1})$ . Also, $\alpha _J$ is isotopic to the meridian $\mu _{n-1}$ in $X_{n-1}$ . It follows that $\mu _{n-1}$ is nontrivial in the quotient ${\mathcal P}^1\pi _1(X_{n-1})/{\mathcal P}^2\pi _1(X_{n-1})$ . This is exactly the promised conclusion for $k=n-1$ .

Now, suppose $0\le k \le n-2$ . The induction hypothesis is that $\mu _{k+1}$ is nontrivial in the quotient ${\mathcal P}^{n-k-1}\pi _1(X_{k+1})/{\mathcal P}^{n-k}\pi _1(X_{k+1})$ . To show that $\mu _k$ is nontrivial in ${\mathcal P}^{n-k}\pi _1(X_k)/{\mathcal P}^{n-k+1}\pi _1(X_k)$ , we use an argument that is essentially the same as [Reference Cha3, Proof of Theorem 4.14], which was influenced by [Reference Cochran, Harvey and Leidy12]. Let $R=\mathbb {Q}$ if $k\ge 1$ , and $R=\mathbb {Z}_{p_1}$ if $k=0$ . Let

$$ \begin{align*} B \colon H_1(M(J^1_{k+1});R[t^{\pm1}])\times H_1(M(J^1_{k+1});R[t^{\pm1}]) \longrightarrow R(t)/R[t^{\pm1}] \end{align*} $$

be the classical Blanchfield pairing of $J^1_{k+1}$ over R-coefficients. For brevity, let $G=\pi _1(X_{k+1})/{\mathcal P}^{n-k}\pi _1(X_{k+1})$ . Consider the noncommutative Alexander module ${\mathcal A}:= H_1(M(J^1_{k+1}); RG)$ . The noncommutative Blanchfield pairing ${\mathcal B} \colon {\mathcal A}\times {\mathcal A} \to {\mathcal K}/RG$ is defined following [Reference Cochran, Orr and Teichner16, Theorem 2.13], where ${\mathcal K}$ is the skew-quotient field of $RG$ . (For $R=\mathbb {Z}_{p_1}$ , see also [Reference Cha3, Section 5].) In our case, because $2\le n-k\le n$ , from Definition 3.1 it follows that $G=\pi _1(X_{k+1})/{\mathcal P}^{n-k}\pi _1(X_{k+1})$ is poly-torsion-free-abelian and, consequently, $RG$ is an Ore domain due to [Reference Cochran, Orr and Teichner16, Proposition 2.5] and [Reference Cha3, Lemma 5.2]. We will use the following known facts:

1. (1) The nontriviality of $\mu _{k+1}$ in ${\mathcal P}^{n-k-1}\pi _1(X_{k+1})/{\mathcal P}^{n-k}\pi _1(X_{k+1}) \subset G$ implies that ${\mathcal A} \cong RG {\mathbin {\mathop {\otimes }_{{R[t^{\pm 1}]}}}} H_1(M(J^1_{k+1});R[t^{\pm 1}])$ and that ${\mathcal B}(1\otimes x, 1\otimes y) = 0$ if and only if $B(x,y)=0$ . This is due to [Reference Leidy33, Theorem 4.7], [Reference Cha2, Theorem 5.16] and [Reference Cochran, Harvey and Leidy12, Lemma 6.5, Theorem 6.6]. See also [Reference Cha3, Theorem 5.4].

2. (2) The 4-manifold $X_{k+1}$ endowed with $\pi _1(X_{k+1}) \to G$ is a Blanchfield bordism in the sense of [Reference Cha4, Definition 4.11], due to an argument in [Reference Cha4, p. 3270] that uses [Reference Cha4, Theorem 4.13]. (The 4-manifold $W_{k+1}$ in [Reference Cha4] plays the role of our $X_{k+1}$ .) The only property of a Blanchfield bordism we need is the following: for all z in $\operatorname {Ker}\{{\mathcal A} \to H_1(X_{k+1};RG)\}$ , ${\mathcal B}(z,z)=0$ by [Reference Cha4, Theorem 4.12].

Recall that $E(P_k\sqcup \eta _k) \cup E(J^1_k) = E(J^1_{k+1}) \subset M(J^1_{k+1})\subset \partial X_{k+1}$ . Denote a zero-linking longitude of $\eta _k$ in $E(P_k\sqcup \eta _k)\subset M(J^1_{k+1})$ by $\eta _k$ , abusing notation. If $\eta _k$ is trivial in ${\mathcal P}^{n-k}\pi _1(X_{k+1})/{\mathcal P}^{n-k+1}\pi _1(X_{k+1})$ , then by the definition of ${\mathcal P}^{n-k+1}$ , $\eta _k=1\otimes \eta _k$ lies in the kernel of ${\mathcal A} \to H_1(X_{k+1};RG)$ . By (2), from this it follows that ${\mathcal B}(1\otimes \eta _k,1\otimes \eta _k) = 0$ and, consequently, by (1), $B(\eta _k,\eta _k)=0$ . Because $J^1_{k+1} = P_k(\eta _k,J^1_k)$ with $k\le n-2$ , the Alexander module $H_1(M(J^1_{k+1});R[t^{\pm 1}])$ is isomorphic to that of stevedore’s knot $P_k$ , which is a cyclic module generated by $\eta _k$ . It contradicts the nonsingularity of the classical Blanchfield pairing B. This shows that $\eta _k$ is nontrivial in ${\mathcal P}^{n-k}\pi _1(X_{k+1})/{\mathcal P}^{n-k+1}\pi _1(X_{k+1})$ , which is isomorphic to ${\mathcal P}^{n-k}\pi _1(X_k)/{\mathcal P}^{n-k+1}\pi _1(X_k)$ by Assertion C. Because $\eta _k$ is identified with $\mu _k$ , it follows that $\mu _k$ is nontrivial in the quotient ${\mathcal P}^{n-k}\pi _1(X_k)/{\mathcal P}^{n-k+1}\pi _1(X_k)$ . This completes the proof of Assertion A.

Proof of Assertion B. Recall that Assertion B says that $\pi _1(Z_0^L)^{(i)}$ maps to ${\mathcal P}^{i+1}\pi _1(X)$ for $1\le i \le n$ . To show this for $i=1$ , observe that the composition

$$ \begin{align*} \begin{aligned} \frac{\pi_1(Z_L^0)^{(1)}}{\pi_1(Z_L^0)^{(2)}} = H_1(Z_L^0;\mathbb{Z}[t^{\pm1}]) & \longrightarrow H_1(X;\mathbb{Z}[t^{\pm1}]) \longrightarrow H_1(X;\mathbb{Q}[t^{\pm1}]\Sigma^{-1}) \\ & \longrightarrow H_1(X;\mathbb{Q}[t^{\pm1}]\Sigma^{-1})/\operatorname{Im} H_1(Z_L^0;\mathbb{Q}[t^{\pm1}]\Sigma^{-1}) \end{aligned} \end{align*} $$

is obviously zero. By definition of ${\mathcal P}^2 \pi _1(X)$ , it follows that $\pi _1(Z_L^0)^{(1)}$ maps to ${\mathcal P}^2\pi _1(X)$ . Therefore, for all $1\le i\le n$ , $\pi _1(Z_L^0)^{(i)} = (\pi _1(Z_L^0)^{(1)})^{(i-1)}$ maps to ${\mathcal P}^2\pi _1(X)^{(i-1)}$ , which is a subgroup of ${\mathcal P}^{i+1}\pi _1(X)$ .

3.2. Obstruction from Cheeger-Gromov $\rho$ -invariants

Now, we will use the Cheeger-Gromov $\rho $ -invariant to derive a contradiction. We begin with some background. For a connected closed 3-manifold M and a homomorphism $\phi \colon \pi _1(M) \to \Gamma $ with $\Gamma $ arbitrary, the Cheeger-Gromov invariant $\rho ^{(2)}(M,\phi )\in \mathbb {R}$ is defined [Reference Cheeger and Gromov10]. The value of $\rho ^{(2)}(M,\phi )$ is preserved under composition with automorphisms of $\pi _1(M)$ , so one can view $\rho ^{(2)}(M,\phi )$ as an invariant of M equipped with (the homotopy class of) a map $\phi \colon M\to B\Gamma =K(\Gamma ,1)$ , instead of $\pi _1(M) \to \Gamma $ . Even when M is not connected, $\rho ^{(2)}(M,\phi )$ is defined for $\phi \colon M\to B\Gamma $ , with additivity under disjoint union: $\rho ^{(2)}(M,\phi )=\sum _i\rho ^{(2)}(M_i,\phi |_{M_i})$ where $M=\bigcup _i M_i$ with components $M_i$ .

In this article, we do not use the definition of $\rho ^{(2)}(M,\rho ^{(2)})$ given in [Reference Cheeger and Gromov10]. Instead, for our purpose, the following $L^2$ -signature defect interpretation is useful. If $M=\bigcup M_i$ bounds a 4-manifold W and $\phi \colon M\to B\Gamma $ factors through W, then $\rho ^{(2)}(M,\phi ) = \sum _i\rho ^{(2)}(M_i,\phi |_{M_i})$ is equal to the $L^2$ -signature defect $\smash {\bar \sigma ^{(2)}_{\Gamma }(W)}:=\smash {\operatorname {sign}^{(2)}_{\Gamma }(W)} -\operatorname {sign}(W)$ , where $\operatorname {sign}(W)$ is the ordinary signature and $\smash {\operatorname {sign}^{(2)}_{\Gamma }(W)}$ is the $L^2$ -signature of W over the group $\Gamma $ . We note that this approach can also be used to provide an alternative definition of $\rho ^{(2)}(M,\phi )$ for arbitrary $(M,\phi )$ . As references, see, for instance, [Reference Chang and Weinberger9], [Reference Cochran and Teichner18, Section 2], [Reference Harvey23, Section 3], [Reference Cha5, Section 2.1].

For our case, let $\Gamma := \pi _1(X) / {\mathcal P}^{n+1} \pi _1(X)$ . For a connected 3-dimensional submanifold M in X, denote the composition $\pi _1(M)\to \pi _1(X)\to \Gamma $ by $\phi $ , abusing notation.

Then, by the $L^2$ -signature defect interpretation for $\rho ^{(2)}(\partial X,\phi )$ , we have

(3.2) $$ \begin{align} \rho^{(2)}(M(J^1_0),\phi) &+ \sum_{k=0}^{n-2} \rho^{(2)}(M(P_k),\phi) + \rho^{(2)}(M(R(U,D)),\phi) = \bar\sigma^{(2)}_{\Gamma}(X) \end{align} $$

$$\begin{align*} &\quad = \bar\sigma^{(2)}_{\Gamma}(V^0) + \bar\sigma^{(2)}_{\Gamma}(C) + \bar\sigma^{(2)}_{\Gamma}(Z^0_L) + \sum_{i,j} \bar\sigma^{(2)}_{\Gamma}(Z^0_{i,j}) + \sum_{k=0}^{n-1} \bar\sigma^{(2)}_{\Gamma}(E_k),\notag \end{align*}$$

where the 4-manifolds $Z^0_{i,j}$ are copies of $Z^0_i$ used in the construction of X. (See Figure 5.) The second equality is obtained by Novikov additivity of $L^2$ -signatures (for instance, see [Reference Cochran, Orr and Teichner16 Lemma 5.9]).

Recall that $\mu _0$ is the meridian of $J^{1}_{0}$ in $M(J^{1}_{0})$ . By Assertion A, $\phi (\mu _{0})$ is nontrivial in $\Gamma $ . Because $\phi (\mu _{J^{1}_{0}})$ lies in the subgroup ${\mathcal P}^n\pi _{1}(X)/{\mathcal P}^{n+1} \pi _1(X)$ of $\Gamma $ , which is a vector space over $\mathbb {Z}_{p_1}$ by Definition 3.1, it follows that $\mu _{J^1_0}$ has order $p_1$ . So the image of $\pi _1(M(J^1_0))$ in $\Gamma $ under $\phi $ is isomorphic to $\mathbb {Z}_{p_1}$ . By [Reference Cha and Orr8, Lemma 8.7], this implies that

(3.3) $$ \begin{align} \rho^{(2)}(M(J^1_0),\phi) = \rho(J^1_0,\mathbb{Z}_{p_1}), \end{align} $$

where $\rho (J^1_0,\mathbb {Z}_{p_1})$ is defined by (2.2).

By the explicit universal bound for the Cheeger-Gromov invariants given in [Reference Cha5, Theorem 1.9], we have

(3.4) $$ \begin{align} \big|\rho^{(2)}\big(M(P_k),\phi\big)\big| \le 6\cdot 69\,713\,280 \end{align} $$

because stevedore’s knot $P_k$ has six crossings. Similarly, by [Reference Cha5, Theorem 1.9],

(3.5) $$ \begin{align} \big|\rho^{(2)}\big(M(R(U,D)),\phi\big)\big| \le (8m+92)\cdot 69\,713\,280 \end{align} $$

because $R(U,D)$ has a diagram with $8m+92$ crossings (see Figure 1).

By [Reference Cochran, Harvey and Leidy12, Lemma 2.4], the following holds for each k.

(3.6) $$ \begin{align} \bar\sigma^{(2)}_G(C)=0, \quad \bar\sigma^{(2)}_G(E_k)=0. \end{align} $$

To evaluate the terms $\bar \sigma ^{(2)}_{\Gamma }(V^0)$ and $\bar \sigma ^{(2)}_{\Gamma }(Z^0_L)$ in (3.2), we will use the following result.

In our case, $V^0$ is an integral $(n+1)$ -solution bounded by $M(K)$ . Also, the group $\Gamma $ lies in $D(\mathbb {Z}_{p_1})$ by [Reference Cha and Orr8, Lemma 6.8], and we have $\Gamma ^{(n+1)}=\{1\}$ because $\pi _1(X)^{(n+1)} \subset {\mathcal P}^{n+1}\pi _1(X)$ . The meridian of K has infinite order in $\Gamma $ because $\Gamma $ surjects onto $H_1(X)=\mathbb {Z}$ generated by the meridian. By Amenable Signature Theorem 3.3, it follows that

(3.7) $$ \begin{align} \bar\sigma^{(2)}_{\Gamma}(V^0) = \rho^{(2)}(M(K),\phi)=0. \end{align} $$

Now we will evaluate $\bar \sigma ^{(2)}_{\Gamma }(Z^0_L)$ , using the Amenable Signature Theorem again. Note that $\partial Z^0_L=M(L)$ . An important difference from the above paragraph is that the 4-manifold $Z^0_L$ is an integral n-solution, not $n+1$ . So, the Amenable Signature Theorem does not apply directly over $\Gamma $ , because $\Gamma ^{(n)}$ is not necessarily trivial. Instead, we proceed as follows.

Note that the map $\phi \colon \pi _1(M(L)) \to \pi _1(Z_L^0) \to \Gamma $ factors through $\pi _1(Z_L^0)/\pi _1(Z_L^0)^{(n)}$ , by Assertion B. Let G be the image of $\pi _1(Z_L^0)/\pi _1(Z_L^0)^{(n)}$ in $\Gamma $ , and let $\psi \colon \pi _1(M(L)) \to G$ be the map induced by $\phi $ . Because G injects into $\Gamma $ , we have $\rho ^{(2)}(M(L),\phi ) = \rho ^{(2)}(M(L),\psi )$ , by the $L^2$ -induction property (for instance, see [Reference Cheeger and Gromov10, Eq. 2.3]). Now, because G is a subgroup of $\Gamma $ that is in $D(\mathbb {Z}_{p_1})$ , G is in Strebel’s class $D({\mathbb {Z}}_{p_1})$ , too. Also, $G^{(n)}$ is trivial because it is the image of $\pi _1(Z_L^0)/\pi _1(Z_L^0)^{(n)}$ . The meridian $\mu _L$ of L has infinite order in G, because G surjects onto $H_1(X)=\mathbb {Z}$ , which $\mu _L$ generates. Therefore, Amenable Signature Theorem 3.3 (with n in place of $n+1$ ) applies to $(M(L),\psi )$ to conclude that

(3.8) $$ \begin{align} \bar\sigma^{(2)}_{\Gamma}(Z^0_L) = \rho^{(2)}(M(L),\phi) = \rho^{(2)}(M(L),\psi) = 0. \end{align} $$

By [Reference Cha and Kim6, Lemma 3.3], we may assume that each $Z^0_{i,j}$ has the property that $\bar \sigma ^{(2)}_{\Gamma }(Z^0_{i,j})$ is equal to either $0$ or $\rho (J^i_0,\mathbb {Z}_{p_1})$ . (Indeed, [Reference Cha and Kim6, Lemma 3.3] applies when every $J^i_0$ is $0$ -negative; it is the case by property (J1) in Subsection 2.1.) Moreover, by property (J3) in Subsection 2.1, we have

(3.9) $$ \begin{align} \bar\sigma^{(2)}_{\Gamma}(Z^0_{i,j}) = \begin{cases} 0 \text{ or } \rho(J^1_0,\mathbb{Z}_{p_1}) &\text{if }i=1, \\ 0 &\text{if }i>1. \end{cases} \end{align} $$

Now, combine (3.2)–(3.9) to obtain

$$ \begin{align*} N\cdot |\rho^{(2)}(J^1_0,\mathbb{Z}_{p_1})| \le (6n+8m+86)\cdot 69\;713\;280, \end{align*} $$

where N is one plus the number of the 4-manifolds $Z^0_{1,j}$ such that $\bar \sigma ^{(2)}(Z^0_{1,j})\ne 0$ . Because $N\ge 1$ , it contradicts property (J2) in Subsection 2.1. This completes the proof that P cannot be equal to $\langle \alpha _D \rangle $ . That is, P must be $\langle \alpha _J \rangle $ .

4.1. Finite cyclic covers and their d-invariants

We begin by applying a trick introduced in [Reference Cochran, Harvey and Horn11], which we describe below. Let $K_1=K_0(\alpha ,J)$ be a satellite knot, where the pattern satisfies $\mathrm {lk}(K_0,\alpha )=0$ . Then, the identity on $E(K\sqcup \alpha )$ extends to a map $E(K_1)=E(K\sqcup \alpha )\cup E(J) \to E(K\sqcup \alpha ) \cup E(U)=E(K_0)$ that induces an isomorphism $H_1(M(K_1);\mathbb {Q}[t^{\pm 1}]) \to H_1(M(K_0);\mathbb {Q}[t^{\pm 1}])$ , under which we will identify the Alexander modules. Essentially, [Reference Cochran, Harvey and Horn11, Lemma 8.2] says the following (see also [Reference Cha and Kim6, Lemma 5.1]): if $K_1=K_0(\alpha ,J)$ admits a $1$ -negaton $X_1$ bounded by $M(K_1)$ and if J is unknotted by changing some positive crossings to negative, then $K_0$ has a $1$ -negaton $X_0$ bounded by $M(K_0)$ such that the two maps

$$ \begin{align*} H_1(M(K_i);\mathbb{Q}[t^{\pm1}]) \longrightarrow H_1(X_i;\mathbb{Q}[t^{\pm1}]) \quad (i=0,1) \end{align*} $$

have the identical kernel. In particular, this applies to the satellite knot $K_1 = R(U,D)(\alpha _D,J^1_{n-1})$ defined in Subsection 2.1, because $J^1_{n-1}=P_{n-2}(\eta _n, J^1_{n-2})$ is unknotted by changing a single positive crossing (see Figure 2). Note that here we use that $n\ge 2$ . Therefore, in our case, the knot $K_0 := R(U,D)$ admits a $1$ -negaton bounded by $M(K_0)$ , say, W, such that

(4.1) $$ \begin{align} \langle\alpha_J\rangle = \operatorname{Ker}\{H_1(M(K_0);\mathbb{Q}[t^{\pm1}]) \rightarrow H_1(W;\mathbb{Q}[t^{\pm1}])\}. \end{align} $$

We will derive a contradiction from the existence of this 1-negaton W for $K_0$ .

The next step is to pass to finite-degree branched cyclic covers, to which Heegaard Floer homology machinery applies, following [Reference Cha and Kim6, Section 5.1]. Let $\Sigma _r$ be the r-fold branched cyclic cover of $(S^3,K_0)$ . The curves $\alpha _J$ and $\alpha _D$ in Figure 1 represent homology classes in $\Sigma _r$ , say, $x_1$ and $x_2\in H_1(\Sigma _r)$ , respectively. (The classes $x_1$ and $x_2$ are defined up to covering transformation, but it will cause no problem in our argument.) Due to [Reference Milnor37], we have $H_1(\Sigma _r) = H_1(M(K_0);\mathbb {Z}[t^{\pm 1}])/\langle t^r-1\rangle $ . Recall that $H_1(M(K_0);\mathbb {Z}[t^{\pm 1}])$ is given by (2.1). From this, by elementary computation, it follows that $H_1(\Sigma _r)=\mathbb {Z}_{(m+1)^r-m^r} \oplus \mathbb {Z}_{(m+1)^r-m^r}$ and the summands are generated by $x_1$ and $x_2$ . In particular, $\Sigma _r$ is a $\mathbb {Z}_2$ -homology sphere.

For a rational homology 3-sphere Y and a spin $^c$ structure $\mathfrak {t}$ on Y, Ozsváth and Szabó defined a correction term invariant $d(Y,\mathfrak {t})$ using the Heegaard Floer chain complex [Reference Ozsváth and Szabó40]. In case of a $\mathbb {Z}_2$ -homology sphere Y, the unique spin structure determines a canonical spin $^c$ structure on Y, which we denote by $\mathfrak {s}_Y$ , and all spin $^c$ structures of Y are given in the form $\mathfrak {s}_Y+c$ , where $c\in H^2(Y)$ and $+$ designates the action of $H^2(Y)$ on the set of spin $^c$ structures. For $x\in H_1(Y)$ , let $\widehat x\in H^2(Y)$ be the Poincaré dual of x. Techniques used in [Reference Cha and Kim6, Reference Cochran, Harvey and Horn11] give us the following d-invariant obstruction.

Proof. Attach a 2-handle to a given $1$ -negaton W, along the zero-framed meridian of $K_0$ , to obtain a 4-manifold, which we temporarily call V. Note that $\partial V = S^3$ and the cocore of the 2-handle is a slicing disk $\Delta \subset V$ bounded by $K_0$ . Take the r-fold branched cyclic cover of $(V,\Delta )$ and call it $V_r$ . Here the r-fold branched cyclic cover is defined because, using that W is a $1$ -negaton, we see that $H_1(W\smallsetminus \Delta )=\mathbb {Z}$ generated by a meridian of $\Delta $ . Indeed, if we denote by $W_r$ the r-fold cyclic cover of W, then $V_r$ is obtained by attaching a 2-handle to $W_r$ . The 4-manifold $V_r$ is bounded by $\Sigma _r$ .

The first key step is to relate the hypothesis $P=\langle \alpha _J\rangle $ , which is associated with the infinite cyclic cover, to the kernel

$$ \begin{align*} G:=\operatorname{Ker}\{H_1(\Sigma_r)\longrightarrow H_1(V_r)\} \end{align*} $$

associated with finite covers. The following is a modified version of [Reference Cha and Kim6, Lemma 5.2].

Assertion. Under the hypothesis that $P=\langle \alpha _J\rangle $ , $x_1\in G$ for all large primes r.

Although we do not use it, we remark that the assertion implies that $G=\langle x_1\rangle $ , because it is known that $|G|$ is equal to $|H_1(\Sigma _m)|^{1/2} = (m+1)^r-m^r\!.$

Proof of Assertion. Let $E_r$ be the r-fold cyclic cover of the exterior $E(K_0)$ for $r \le \infty $ . Consider the following commutative diagram:

The vertical arrows are induced by coverings and inclusions. Because $\alpha _J$ lies in the kernel of the composition of the top row by the hypothesis, $\alpha _J$ is $\mathbb {Z}$ -torsion in $H_1(W;\mathbb {Z}[t^{\pm 1}])$ . That is, $a\cdot \alpha _j=0$ in $H_1(W;\mathbb {Z}[t^{\pm 1}])$ for some nonzero $a\in \mathbb {Z}$ . By the above diagram, it follows that $a\cdot x_1 \in H_1(\Sigma _r)$ lies in the kernel G of $H_1(\Sigma _r)\to H_1(V_r)$ .

Suppose that r is a prime not smaller than any prime factor of a. Under this assumption, we claim that $\gcd (a, (m+1)^r - m^r)=1$ . From this it follows that $x_1$ lies in G, because $x_1$ has order $(m+1)^r - m^r$ in $H_1(\Sigma _r)$ . This proves the assertion, modulo the proof of the claim.

To show the claim, it suffices to show that every prime factor q of a is relatively prime to $(m+1)^r - m^r\!.$ It is obviously true, if $q\mid m$ or $q\mid m+1$ . So, suppose that q divides neither m nor $m+1$ but q divides $(m+1)^r - m^r\!.$ Let $u=m^*(m+1)$ , where $m^*$ is an arithmetic inverse of $m \bmod q$ . We have $u^r \equiv 1 \bmod q$ by the hypothesis and $u^{q-1} \equiv 1 \bmod q$ by Fermat’s little theorem. Because r is a prime and $u\not \equiv 1 \bmod q$ , it follows that $r\mid q-1$ . This contradicts the assumption that $r\ge q$ . This completes the proof of the claim.

The assertion enables us to invoke a result of Cochran, Harvey and Horn [Reference Cochran, Harvey and Horn11, Theorem 6.5], which says the following: if W is a $1$ -negaton bounded by $M(K_0)$ , then $d(\Sigma _r,\widehat x)\ge 0$ for all x lying in $G=\operatorname {Ker}\{H_1(\Sigma _r) \to H_1(V_r)\}$ . Applying this to $x=k\cdot x_1$ , the proof of Lemma 4.1 is completed.

On the other hand, Lemma 4.1 says that $d(\Sigma _r, \mathfrak {s}_{\Sigma _r} + k\widehat x_1)$ must be nonnegative. This contradiction implies that the kernel P defined in (2.4) cannot be equal to $\langle \alpha _J\rangle $ . This completes the proof of Theorem 2.2.

For $m=1$ , Theorem 4.2 was already shown in [Reference Cha and Kim6, Theorem 5.4]. (We remark that the symbol m in [Reference Cha and Kim6] denotes our r.) So, in the remaining part of this article, we will assume that $m>1$ .

Proof of Theorem 4.2. Let A and B be the 3-manifolds given by the surgery presentations in Figure 6. Let $Y_r$ be the connected sum of A, B and $r-3$ copies of the lens space $L_{2m+1,1}$ . (We use the orientation convention that $L_{p,1}$ is the p-framed surgery on the trivial knot U in $S^3$ .) Then, arguments in [Reference Cha and Kim6, Section 6.1] construct a negative definite 4-manifold W with $\partial W = Y_r \sqcup -\Sigma _r$ and $b_2(W)=(2m+1)r-4m+1$ and construct a spin $^c$ structure $\mathfrak {t}$ on W such that $c_1(\mathfrak {t})^2 = -r$ , $c_1(\mathfrak {t}|_{\Sigma _r}) = \widehat {x_1}$ and $c_1(\mathfrak {t}|_{Y_r}) = 0$ . Indeed, [Reference Cha and Kim6, Section 6.1] is the case of $m=1$ . We do not repeat the details here, because exactly the same method works under our assumption that $m\ge 1$ is odd. Perhaps the least obvious part is that the inverse matrix $P^{-1}$ in [Reference Cha and Kim6, Eq. (6.10)] should be replaced with a block matrix $P^{-1}=(R_{ij})_{1\le i,j\le r-1}$ with

$$ \begin{align*} R_{ij} = \frac{((m+1)^i-m^i)((m+1)^{r-j}-m^{r-j})}{(m+1)^r-m^r} \begin{bmatrix} 0 & (m+1)^{i-j} \\ m^{i-j} & 0 \end{bmatrix} \text{ for } i\ge j, \end{align*} $$

and $R_{ij}=R^T_{ji}$ for $i<j$ .

Figure 6 The 3-manifolds A and B. The box $2m+1$ represents $2m+1$ right-handed full twists between vertical strands.

By applying Ozsváth-Szabó’s d-invariant inequality [Reference Ozsváth and Szabó40, Theorem 9.6] to the negative definite 4-manifold W, and by using additivity of the d-invariant under connected sum, we have

$$ \begin{align*} \begin{aligned} d(\Sigma_r,\mathfrak{t}|_{\Sigma_r}) & \le d(Y_r,\mathfrak{t}|_{Y_r}) - \frac{c_1(\mathfrak{t})^2 + b_2(W)}{4} \\ & = d(A,\mathfrak{t}|_{A}) + d(B,\mathfrak{t}|_{B}) + (r-3)d(L_{2m+1,1},\mathfrak{t}|_{L_{2m+1,1}}) - \frac{2mr-4m+1}{4}. \end{aligned} \end{align*} $$

Because $c_1(\mathfrak {t}|_{Y_r})=0$ , we have $\mathfrak {t}|_{L_{2m+1,1}}= \mathfrak {s}_{L_{2m+1,1}}$ . By a d-invariant formula for lens spaces given in [Reference Ozsváth and Szabó40, Proposition 4.8], $d(L_{2m+1,1},\mathfrak {s}_{L_{2m+1,1}})=m/2$ . By Lemmas 4.3 and 4.4, which we will prove below, we have $d(A,\mathfrak {t}|_A) \le (m-7)/4$ and $d(B,\mathfrak {t}|_B) \le (m+2)/4$ . Combine them with the above inequality to obtain $d(\Sigma _r, \mathfrak {t}|_{\Sigma _r}) \le -\frac 32$ . Observe that $c_1(\mathfrak {t}|_{\Sigma _r})=\widehat {x_1} = 2k \widehat {x_1} = c_1(\mathfrak {s}_{\Sigma _r} + k \widehat {x_1})$ , because $2k\equiv 1\bmod (m+1)^r-m^r$ and $x_1$ has order $(m+1)^r-m^r\!.$ It follows that $\mathfrak {t}|_{\Sigma _r} = \mathfrak {s}_{\Sigma _r}$ , because $\Sigma _r$ is a $\mathbb {Z}_2$ -homology sphere.

Note that $|H_1(A)| = \bigl |\det \left [\begin {smallmatrix}m & -2m-1 \\ -2m-1 & m+1\end {smallmatrix}\right ]\bigr | = 3m^2+3m+1$ is odd, so there is a unique spin $^c$ structure $\mathfrak {s}$ of A such that $c_1(\mathfrak {s}) = 0$ .

For the case of B, because $|H_1(B)| = \bigl |\det \left [\begin {smallmatrix}m & -2m-1 \\ -2m-1 & m+2\end {smallmatrix}\right ] \bigr | = 3m^2+2m+1$ is even, there are exactly two spin $^c$ structures $\mathfrak {s}$ of B such that $c_1(\mathfrak {s}) = 0$ .

Proof of Lemma 4.3. Let $A'$ be the 3-manifold obtained by $(m+6)$ -surgery on the knot $T\#-D$ . See Figure 7. Let Y be the 3-manifold given by the last surgery diagram in Figure 7. The four diagrams in Figure 7 describe a cobordism from $A\# A'$ to Y. More precisely, start by taking ${(A\# A') \times I}$ . Attach a 2-handle to obtain a cobordism, say, V, from $A\#A'$ to the second surgery diagram in Figure 7. Apply handle slide, to change the second surgery diagram to the third. Observe that in the third surgery diagram, the two components with framing $m+6$ and $0$ give a $S^3$ summand, whereas the two components with framing m and $2m+7$ form a link concordant to the link in the last surgery diagram, which describes Y. It follows that there is a homology cobordism, say, $V'$ , from the third surgery diagram to Y. Now $V\cup V'$ is a cobordism from $A\#A'$ to Y.

Figure 7 A cobordism from $A\# A'$ to Y.

We claim that V is negative definite. To see this, view the surgery diagram of the 3-manifold $A\# A'$ as a Kirby diagram of a 4-manifold $V_0$ . That is, $V_0$ consists of one 0-handle and three 2-handles attached along the framed link diagram of $A\# A'$ in Figure 7. We have $\partial V_0=A\# A'$ . The second diagram in Figure 7 or, equivalently, the third diagram, is a Kirby diagram of the 4-manifold $V_0\cup _{A\# A'} V$ . The Kirby diagram of $V_0$ has linking matrix

$$ \begin{align*} L = \begin{bmatrix} m & -2m-1 & 0 \\ -2m-1 & m & 0 \\ 0 & 0 & m+6 \end{bmatrix}, \end{align*} $$

which has signature $1$ because $m>0$ and the top upper $2\times 2$ submatrix has negative determinant. It follows that $\operatorname {sign} V_0 = 1$ . The third diagram in Figure 7 has linking matrix

$$ \begin{align*} \begin{bmatrix} m & -2m-1 & 0 & 0\\ -2m-1 & 2m+7 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & m+6 \end{bmatrix}, \end{align*} $$

which has vanishing signature. So $\operatorname {sign} V_0\cup V = 0$ . By Novikov additivity, it follows that $\operatorname {sign} V=-1$ . Because $b_2(V)=1$ , this proves the claim that V is negative definite.

Because $V'$ is a homology cobordism, $H_*(V\cup V')=H_*(V)$ . Consequently, $V\cup V'$ is negative definite.

We will construct a generator of $H_2(V\cup V') = H_2(V)$ and use it to describe a certain spin $^c$ structure on $V\cup V'$ . Let $\sigma $ be the core of the 2-handle of $(V,A\#A')$ , and let $\alpha =\partial \sigma $ be its attaching circle, which lies in $A\#A'$ . See the second diagram in Figure 7, in which $\alpha $ is the zero-framed circle. Because the linking matrix L is a presentation for $H_1(A\#A')$ , it is seen that $H_1(A\#A')=\mathbb {Z}_{3m^2+3m+1}\oplus \mathbb {Z}_{m+6}$ , and $\alpha = (1,1)$ in $H_1(A\#A')$ . So, the order of $\alpha $ in $H_1(A\#A')$ is $(3m^2+3m+1)(m+6)/d$ , where $d:=\gcd (3m^2+3m+1, m+6)$ . (Indeed, it can be seen that d is either $91$ or $1$ .) From this, it follows that there is a 2-cycle z in $A\#A'$ such that

$$ \begin{align*} \partial z= \frac{(3m^2+3m+1)(m+6)}{d}\cdot \alpha. \end{align*} $$

Moreover, the 2-chain

$$ \begin{align*} E:= \frac{(3m^2+3m+1)(m+6)}{d} \cdot \sigma - z \end{align*} $$

is a generator of $H_2(V)=\mathbb {Z}$ , by a standard Mayer-Vietoris argument for the 2-handle attachment.

The self-intersection number $E\cdot E$ is equal to the intersection number (in $A\#A'$ ) of z and a pushoff of $\partial z$ , say, $\partial z'$ , taken along the 2-handle attachment framing (which is the zero framing in Figure 7). So $E\cdot E$ is equal to the linking number of $\partial z$ and its pushoff in the rational homology sphere $A\#A'$ . In addition, the linking number can be computed using the linking matrix L (for instance, see [Reference Cha and Ko7, Theorem 3.1]):

$$ \begin{align*} \begin{aligned} E\cdot E &= \mathrm{lk}_{A\#A'}(\partial z,\partial z') = \frac{(3m^2+3m+1)^2(m+6)^2}{d^2} \cdot \begin{bmatrix} 0 & -1 & 1 \end{bmatrix} L^{-1} \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} \\ & = \frac{(3m^2+3m+1)(m+6)(-2m^2+3m-1)}{d^2}. \end{aligned} \end{align*} $$

Because the factor $-2m^2+3m-1$ of the numerator is even and d is odd, $E\cdot E$ is even. From this, it follows that $0\in H^2(V\cup V')$ is characteristic. Therefore, there is a spin $^c$ structure $\mathfrak {t}$ on $V\cup V'$ such that $c_1(\mathfrak {t})=0$ . By the Ozsváth-Szabó inequality [Reference Ozsváth and Szabó40, Theorem 9.6], we have

$$ \begin{align*} d(Y,\mathfrak{t}|_Y) - d(A,\mathfrak{t}|_A) - d(A',\mathfrak{t}|_{A'}) \ge \frac 14. \end{align*} $$

Note that $A'$ is the $(m+6)$ -surgery on the knot $T\# -D$ , and $c_1(\mathfrak {t}|_{A'})=0$ because $c_1(\mathfrak {t})=0$ . By techniques of [Reference Cochran, Harvey and Horn11, p. 2150-2151] and [Reference Hedden, Kim and Livingston24, Appendix A], $d(A',\mathfrak {t}|_{A'}) = d(L_{m+6},\mathfrak {s}_{L_{m+6}})$ . By Ozsváth-Szabó’s formula for lens spaces [Reference Ozsváth and Szabó40, Proposition 4.8], we have $d(L_{m+6},\mathfrak {s}_{L_{m+6}}) = (m+5)/4$ . By Lemma 4.8, which we will prove below, we have $d(Y,\mathfrak {t}|_Y) \le (2m-1)/4$ . (Note that $c_1(\mathfrak {t}|_Y)=0$ because $c_1(\mathfrak {t})=0$ .) Combining these with the above inequality, it follows that $d(A,\mathfrak {s}) \le (m-7)/4$ .

4.2. A quick summary of Némethi’s method for Seifert 3-manifolds

For the reader’s convenience, we provide a summary of Némethi’s method to compute d-invariants [Reference Némethi38], which we will use in Subsection 4.3. We focus on the case of Seifert 3-manifolds, which is treated in [Reference Némethi38, Section 11], and [Reference Némethi38] provides techniques for a larger class of certain plumbed 3-manifolds.

Let Y be a Seifert 3-manifold. It is well known that Y admits a surgery presentation of a specific form, which is shown in the left side of Figure 8. The associated star-shaped plumbing graph, which is shown in the right side of Figure 8, is often used to describe the 3-manifold Y, as the boundary of a plumbed 4-manifold X: for each vertex, take a disk bundle over a 2-sphere whose Euler number is the integer decoration of the vertex. For each edge, perform $+1$ plumbing between the two disk bundles corresponding to the endpoints. The result is a 4-manifold X with $\partial X=Y$ .

Figure 8 A Seifert 3-manifold and its plumbing graph.

Let $\nu $ be the number of branches of the star-shaped graph in Figure 8. In this subsection, we assume that $\nu \ge 3$ and that X is negative definite. (We remark that not all Seifert 3-manifolds Y are described by a plumbing graph satisfying this assumption.)

We use the following notation. Let $e_0$ be the decoration of the root vertex. Let $s_{\ell }$ be the number of (non-root) vertices on the $\ell $ th branch, and let $e_{\ell ,1},\ldots$ , $e_{\ell ,s_{\ell }}$ be the decorations of those $s_{\ell }$ vertices. See Figure 8. We may assume that $e_{\ell ,j}\le -2$ for all $\ell $ , j (for instance, see [Reference Neumann39]). Let $b_0$ and $b_{\ell ,j} \in H_2(X)$ be the classes of 2-spheres corresponding to vertices with decoration $e_0$ and $e_{\ell ,j}$ respectively ( $1\le \ell \le \nu $ , $1\le j\le s_{\ell }$ ). They form a basis for the free abelian group $H_2(X)$ . Let $b^*_0$ and $b^*_{\ell ,j}\in H^2(X)$ be basis elements (hom) dual to $b_0$ and $b_{\ell ,j}$ . With respect to these bases, the intersection form $\lambda \colon H_2(X)\times H_2(X) \to X$ or its adjoint $\lambda \colon H_2(X) \to \mathrm {Hom}(H_2(X),\mathbb {Z}) = H^2(X)$ is given as follows: $\lambda (b_0,b_0)=e_0$ , $\lambda (b_{\ell ,j}, b_{\ell ,j}) = e_{\ell ,j}$ , and for $b\ne b'$ , $\lambda (b,b')=1$ if $\{b,b'\}=\{b_0,b_{\ell ,1}\}$ or $\{b_{\ell ,j},b_{\ell ,j+1}\}$ , and $\lambda (b,b')=0$ otherwise. For $1\le \ell \le \nu $ , define a continued fraction by

$$ \begin{align*} \frac{\alpha_{\ell}}{\omega_{\ell}} := [-e_{\ell,1},\ldots,-e_{\ell,s_{\ell}}] = -e_{\ell,1} - \cfrac{1}{-e_{\ell,2} - \cfrac{1}{\,\,\,\vdots\,\,\,}}. \end{align*} $$

The condition $e_{\ell ,j}\le -2$ implies that $\alpha _{\ell }/\omega _{\ell }>1$ . So we may assume $0\le \omega _{\ell } < \alpha _{\ell }$ . Then, a standard diagonalisation process applied to the intersection form $\lambda $ gives us a diagonal matrix whose all diagonals are easily seen to be negative, possibly except one, which is equal to

$$ \begin{align*} e := e_0 + \sum_{\ell=1}^{\nu} \omega_{\ell}/\alpha_{\ell}. \end{align*} $$

So, X is negative definite if and only if $e<0$ .

The set of characteristic elements in $H^2(X)$ is defined to be

$$ \begin{align*} \operatorname{Char}(X) := \{\xi \in H^2(X) \mid \xi(b_j)\equiv \lambda(b_j,b_j) \bmod 2 \text{ for all } j\}. \end{align*} $$

The Chern class $c_1\colon \operatorname {\mathrm {Spin}}^c(X) \to \operatorname {Char}(X)$ is bijective, because $H^2(X)$ does not have any 2-torsion. Under the identification via $c_1$ , the action of an element $c\in H^2(X)$ on $\operatorname {\mathrm {Spin}}^c(X)$ is given by $\xi \mapsto \xi +2c$ for $\xi \in \operatorname {Char}(X)=\operatorname {\mathrm {Spin}}^c(X)$ . In particular, the action of $x\in H_2(X)$ on $\operatorname {Char}(X)=\operatorname {\mathrm {Spin}}^c(X)$ via $\lambda \colon H_2(X)\to H^2(X)$ is given by $\xi \mapsto \xi +2\lambda (x)$ .

The Chern class $c_1\colon \operatorname {\mathrm {Spin}}^c(Y)\to H^2(Y)$ is not injective in general, so the standard identification of spin $^c$ structures of Y is given indirectly using X: we have a bijection

$$ \begin{align*} \operatorname{\mathrm{Spin}}^c(Y) \approx \operatorname{Char}(X)/2\lambda(H_2(X)). \end{align*} $$

Here, for a spin $^c$ structure $\xi \in \operatorname {\mathrm {Spin}}^c(X)=\operatorname {Char}(X)$ , the coset $[\xi ] = \xi +2\lambda (H_2(X))$ in $\operatorname {Char}(X)/2\lambda (H_2(X))$ corresponds to the restriction of $\xi $ on Y. Essentially, the bijectivity is a consequence of the fact that $H^2(Y)$ is the cokernel of $\lambda \colon H_2(X) \to H^2(X)$ .

In [Reference Némethi38], the notion of a distinguished representative is used to express a spin $^c$ structure of Y. Instead of the original definition (see Section 5, especially Definition 5.1 of [Reference Némethi38]), we will use a characterisation theorem as a definition. We need the following notation. For $1\le i\le j\le s_{\ell }$ , let $n^{\ell }_{i,j}/d^{\ell }_{i,j} := [-e^{\ell }_i, \ldots , -e^{\ell }_j]$ , where $n^{\ell }_{i,j}> 0$ and $\gcd (n^{\ell }_{i,j}, d^{\ell }_{i,j}) = 1$ . Note that $\alpha _{\ell }/\omega _{\ell } = n^{\ell }_{1,s_{\ell }}/d^{\ell }_{1,s_{\ell }}$ . Define an element $K\in \operatorname {\mathrm {Spin}}^c(X) = \operatorname {Char}(X)\subset H^2(X)$ by

(4.2) $$ \begin{align} K(b_0) = -e_0-2,\quad K(b_{\ell,j}) = -e_{\ell,j}-2 \text{ for } 1\le \ell\le \nu,\, 1\le j\le s_{\ell}. \end{align} $$

The element K is characteristic because $K(b_0) \equiv \lambda (b_0,b_0)$ and $K(b_{\ell ,j}) \equiv \lambda (b_{\ell ,j},b_{\ell ,j}) \bmod 2$ . In [Reference Némethi38], K is called the canonical spin $^c$ structure. We have $\operatorname {Char}(X) = K + 2H^2(X)$ . Consider a class $k_r\in \operatorname {Char}(X)$ of the form

(4.3) $$ \begin{align} k_r = K - 2 \bigg( a_0^{\vphantom{1}} b^*_0 + \sum_{\ell,j} a_{\ell,j}^{\vphantom{1}} b_{\ell,j}^* \bigg), \end{align} $$

where $a_0$ and $a_{\ell ,j}$ are integers. Let

(4.4) $$ \begin{align} a_{\ell} = \sum_{t=1}^{s_{\ell}-1} n^{\ell}_{t+1,s_{\ell}} a_{\ell,t}. \end{align} $$

Definition 4.5 [Reference Némethi38, Corollary 11.7]

The class $k_r$ in (4.3) called a distinguished representative if $0\le a_{\ell } < \alpha _{\ell }$ for all $\ell $ and if

(4.5) $$ \begin{align} 0\le a_0 \le -1 -i e_0 - \sum_{\ell=1}^{\nu} \Bigl[ \frac{i\omega_{\ell} + a_{\ell}}{\alpha_{\ell}} \Bigr] \end{align} $$

for all $i>0$ . Here $[x]$ is the largest integer not greater than x.

To state Némethi’s formula for the d-invariant, we need one more notation. Let

(4.6) $$ \begin{align} \tau(i) = \sum_{t=0}^{i-1} \bigg(a_0+1-te_0 + \sum_{\ell=1}^{\nu} \Bigl[ \frac{-t\omega_{\ell} + a_{\ell}}{\alpha_{\ell}} \Bigr] \bigg). \end{align} $$

In particular, $\tau (0)=0$ .

Theorem 4.5 Némethi [Reference Némethi38, p. 1038]

Suppose that X is negative definite and $\nu \ge 3$ . Suppose that the class $k_r$ given in (4.3) is a distinguished representative. Then, for the spin $^c$ structure $[k_r]$ of Y, the d-invariant is given by

$$ \begin{align*} d(Y,[k_r]) = \frac{k_r^2+b_2(X)}{4} - 2\cdot \min \{ \tau(i) \mid i\ge 0\}. \end{align*} $$

4.3. d-invariants of the 3-manifolds Y and B

Let Y be the 3-manifold given by the last surgery diagram in Figure 7 or, equivalently, by the first surgery diagram in Figure 9. The following lemma gives an estimate of the d-invariant of Y, which is used to complete the proof of Lemma 4.3. Note that there are two spin $^c$ structures on Y satisfying $c_1(\mathfrak {s})=0$ , because $|H_1(Y)| = 2m^2-3m+1$ is even.

Figure 9 Surgery diagram calculus that gives a plumbing tree for Y. The symbol $(*)$ means handle slides and elimination of components with 0-framed meridians.

Proof. The surgery diagram calculus in Figure 9 shows that Y is a Seifert 3-manifold. The last plumbing graph in Figure 9 describes a plumbed 4-manifold X with $\partial X=Y$ .

To compute the d-invariant, we will apply the method discussed in Subsection 4.2. Using the notation in Subsection 4.2, denote the basis of $H_2(X)$ by $\{b_0$ , $b_{1,1},\ldots$ , $b_{1,2m}$ , $b_{2,1}$ , $b_{3,1}$ , $b_{4,1}\}$ and the dual basis of $H^2(X)$ by $\{b^*_0$ , $b^*_{1,1},\ldots$ , $b^*_{1,2m}$ , $b^*_{2,1}$ , $b^*_{3,1}$ , $b^*_{4,1}\}$ . The intersection form $\lambda \colon H_2(X)\times H_2(X)\to \mathbb {Z}$ is computed straightforwardly from the plumbing graph:

$$ \begin{align*} \lambda = \left[ \begin{array}{c|ccccc|c|c|c} -2 & 1 & & & & & 1 & 1 & 1 \\ \hline 1 & -2 & 1 & & & & & & \\ & 1 & -2 & 1 & & & & & \\ & & \ddots & \ddots & \ddots & & & \\ & & & 1 & -2 & 1 & & & \\ & & & & 1 & -2 & & & \\ \hline 1 & & & & & & -2 & & \\ \hline 1 & & & & & & & -3 & \\ \hline 1 & & & & & & & & -m-1 \end{array} \right]_{(2m+4)\times(2m+4)}. \end{align*} $$

Also, using the definition in Subsection 4.2, it is routine to compute the following:

$$ \begin{align*} \Bigl( \frac{\alpha_1}{\omega_1},\, \frac{\alpha_2}{\omega_2},\, \frac{\alpha_3}{\omega_3},\, \frac{\alpha_4}{\omega_4} \Bigr) = \Bigl( \frac{2m+1}{2m},\,\frac21,\,\frac31,\,\frac{m+1}{1} \Bigr). \end{align*} $$

So, the orbifold Euler number is given by

$$ \begin{align*} e = -2 + \frac{2m}{2m+1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{m+1} = -\frac{(m-1)(2m-1)}{6(m+1)(2m+1)}. \end{align*} $$

For $m>1$ , we have $e<0$ , so X is negative definite.

We will describe two spin $^c$ structures $[k_1]$ and $[k_2]\in \operatorname {Char}(X)/2\lambda (H_2(X))$ . Let

$$ \begin{align*} k_1 &:= -2 b^*_{1,2m} - b^*_{3,1} + 2b^*_{4,1}, \\ k_2 &:= -2 b^*_{1,2m-1} + b^*_{3,1} - 2b^*_{4,1}. \end{align*} $$

It is straightforward to show that $k_1$ and $k_2$ are distinguished representatives in the sense of Definition 4.5. Indeed, in our case, the canonical spin $^c$ structure described in (4.2) is given by $K= b^*_{3,1} -(m+1) b^*_{4,1}$ , and $k_1$ is of the form (4.3), where

$$ \begin{align*} a_0 = 0,\quad (a_{1,1},\ldots,a_{1,2m}) = (0,\ldots,0,1), \quad a_{2,1}=0,\quad a_{3,1}=1,\quad a_{4,1}=1. \end{align*} $$

By (4.4), we have $(a_0,a_1,a_2,a_3,a_4) = (0,1,0,1,\tfrac {m-3}{2})$ . This satisfies the conditions in Definition 4.5, so $k_1$ is a distinguished representative. The class $k_2$ is shown to be a distinguished representative, too, by similar computation. In this case, we have

$$ \begin{align*} a_0 = 0,\quad (a_{1,1},\ldots,a_{1,2m}) = (0,\ldots,0,1,0), \quad a_{2,1}=0,\quad a_{3,1}=0,\quad a_{4,1}=m-2 \end{align*} $$

and $(a_0,a_1,a_2,a_3,a_4) = (0,2,0,0,m-2)$ .

Under the adjoint $\lambda \colon H_2(X) \to H^2(X)=\mathrm {Hom}(H_2(X),\mathbb {Z})$ of the intersection form of X, $k_1$ and $k_2$ are respectively the images of

$$ \begin{align*} x_1 &= 2b_0 + 2b_{1,1}+\cdots+2b_{1,2m} + b_{2,1} + b_{3,1}, \\ x_2 &= 4b_1 + 4b_{1,1}+\cdots+4b_{1,2m-1}+2b_{1,2m} + 2b_{2,1} + 2b_{3,1} + 2b_{4,1}. \end{align*} $$

Recall that $\operatorname {Char}(X)\subset H^2(X)$ is identified with $\operatorname {\mathrm {Spin}}^c(X)$ via $c_1$ , and thus for a spin $^c$ structure $[k] \in \operatorname {Char}(X)/2\lambda (H_2(X))$ of Y, we have $c_1([k]) = k|_Y$ . So, $c_1([k]) = 0$ if and only if k lies in the kernel of $H^2(X) \to H^2(Y)$ or, equivalently, k lies in the image of $\lambda \colon H_2(X) \to H^2(X)$ . From this observation, it follows that $c_1([k_i]) = 0$ for $i=1,2$ , because $k_i = \lambda (x_i)$ . Also, $[k_1] \ne [k_2]$ because $x_1-x_2\notin 2H_2(X)$ . Therefore, to complete the proof, it suffices to show $d(Y,[k_i]) \le (2m-1)/4$ for $i=1,2$ .

We have

$$ \begin{align*} k_1^2 = \lambda(x_1,x_1) = -3, \quad k_2^2 = \lambda(x_2,x_2) = -m-4. \end{align*} $$

The last thing we need is the minimum of the values of $\tau (i)$ defined in (4.6). Recall the notation $\Delta _i = \tau (i+1)-\tau (i)$ from Remark 4.7.

Assertion. For both $k_1$ and $k_2$ and for all $i\ge 0$ , $\Delta _i \ge 0$ .

We will provide a proof of the assertion for $m\ge 23$ . For $m<23$ , the assertion is verified by direct inspection using Remark 4.7 (indeed, the author used a computer program), so we omit details for $m<23$ . Note that it suffices to use $m\ge 23$ to prove the main results of this article, Theorems A, B, C and D.

For the case of $k_1$ , by Remark 4.7, we have $\Delta _i\ge 0$ for $i\ge R$ , where

$$ \begin{align*} R=8+(48m-6)/(2m-1)(m-1). \end{align*} $$

Because $m\ge 23$ , $R\le 10$ . So, $\Delta _i\ge 0$ for $i\ge 10$ . For $i\le 10$ , using (4.6), we have

$$ \begin{align*} \Delta_i &= 1 + 2i + \bigg[ \frac{-2mi+1}{2m+1} \bigg] + \bigg[ \frac{-i}{2} \bigg] + \bigg[ \frac{-i+1}{3} \bigg] + \bigg[ \frac{-i+(m-3)/2}{m+1} \bigg] \\ & \ge 1+2i-i+\frac{-i-1}{2} + \frac{-i-1}{3} = \frac{i+1}{6} \ge 0. \end{align*} $$

This shows the claim for $k_1$ . For the case of $k_2$ , we proceed in the same way. We have

$$ \begin{align*} R=7+(48m-6)/(2m-1)(m-1). \end{align*} $$

Because $m\ge 23$ , $R\ge 10$ , and thus $\Delta _i\ge 0$ for $i\ge 10$ by Remark 4.7. For $1\le i\le 10$ , using (4.6), we have

$$ \begin{align*} \Delta_i &= 1 + 2i + \bigg[ \frac{-2mi+1}{2m+1} \bigg] + \bigg[ \frac{-i}{2} \bigg] + \bigg[ \frac{-i}{3} \bigg] + \bigg[ \frac{-i+m-2}{m+1} \bigg] \\ & \ge 1+2i-i+\frac{-i-1}{2} + \frac{-i-2}{3} = \frac{i-1}{6} \ge 0. \end{align*} $$

For $i=0$ , a direct computation using (4.6) gives $\Delta _0=1$ . This completes the proof of the assertion.

From the assertion, it follows that $\min \{\tau (i)\mid i\ge 0\}=0$ because $\tau (0)=0$ . So, by Némethi’s Theorem 4.6,

$$ \begin{align*} d(Y,[k_i]) = \frac{k_i^2 + 2m+4}{4} = \begin{cases} (2m-1)/4 &\text{for } i=1, \\ m/4 &\text{for } i=2. \end{cases} \end{align*} $$

Therefore $d(Y,[k_i]) \le (2m-1)/4$ holds for $i=1,2$ .

Now, to complete the proof of Theorem 2.2, it only remains to prove of Lemma 4.4, which estimates d-invariants of the 3-manifold B in Figure 6. In the proof below, we will use that B is a Seifert 3-manifold as well.

Proof of Lemma 4.4. Recall that B is the 3-manifold described in Figure 6. Let $\mathfrak {s}$ be a spin $^c$ structure of B satisfying $c_1(\mathfrak {s})=0$ . The goal is to show that $d(B,\mathfrak {s})\le (m+2)/4$ .

As we will see in what follows, it will be useful to consider $-B$ instead of B. That is, we will show $d(-B,\mathfrak {s})\ge -(m+2)/4$ . Figure 10 shows that $-B$ is a Seifert 3-manifold. We use the notation of Subsection 4.2. The associated plumbed 4-manifold X has $H_2(X)=\mathbb {Z}^{2m}$ , with basis elements $b_0$ , $b_{1,1},\ldots$ , $b_{1,m}$ , $b_{2,1},\ldots$ , $b_{2,m-2}$ , $b_{3,1}$ . The associated decorations are $e_0=-2$ , $e_{1,j} = -2$ for all j, $e_{2,j}=-2$ for all j and $e_{3,1}=-2m-1$ . Let $b^*_0$ , $b^*_{\ell ,j}$ be the dual basis elements in $H^2(X)$ . Because

$$ \begin{align*} e = -2 + \frac{m}{m+1} + \frac{m-2}{m-1} + \frac{1}{2m+1} = \frac{-3m^2 -2m -1}{(m^2-1)(2m+1)} < 0, \end{align*} $$

the 4-manifold X is negative definite. This is why we use $-B$ instead of B.

Figure 10 Surgery diagram calculus showing that $-B$ is a Seifert 3-manifold. The symbol $(*)$ means handle slides (or Rolfsen twist [Reference Rolfsen41, p. 264]) followed by elimination of a component together with a 0-framed meridian.

The canonical spin $^c$ structure K of X is given by $K=(2m-1)b^*_{3,1}$ . Let

$$ \begin{align*} x_1 &= b_{1,1} + b_{1,3} + \cdots + b_{1,m} + b_{3,1}, \\ x_2 &= b_{2,1} + b_{2,3} + \cdots + b_{2,m-2} + b_{3,1}, \end{align*} $$

and let $k_i = \lambda (x_i) \in H^2(X)$ for $i=1$ , $2$ . Then $k_i \in \operatorname {Char}(X)=K+2H^2(X)$ , so that $[k_i] \in \operatorname {Char}(X)/2\lambda (H_2(X))=\operatorname {\mathrm {Spin}}^c(-B)$ is a spin $^c$ structure of $-B$ . Similar to the proof of Lemma 4.8, $c_1([k_i]) = k_i|_{-B} = 0$ . Also, $[k_1] \ne [k_2]$ in $\operatorname {\mathrm {Spin}}^c(-B)$ because $x_1-x_2 \notin 2H_2(X)$ . It follows that $[k_1]$ and $[k_2]$ are the two spin $^c$ structures of $-B$ with $c_1=0$ . So, it suffices to show that $d(-B,[k_i])\ge -(m+2)/4$ for $i=1$ , $2$ . Instead of determining the values exactly, we will present a simpler argument that gives the promised estimate. We have $k_1^2 = \lambda (x_1,x_1) = -3m-2$ . So, by Ozsváth-Szabó’s inequality [Reference Ozsváth and Szabó40, Theorem 9.6],

(4.7) $$ \begin{align} d(-B,[k_1]) \ge \frac{k_1^2 + b_2(X)}{4} = \frac{-m-2}{4}. \end{align} $$

Similarly, because $k_2^2 = \lambda (x_2,x_2) = -3m$ , we have

(4.8) $$ \begin{align} d(-B,[k_2]) \ge \frac{k_2^2 + b_2(X)}{4} = \frac{-m}{4}. \end{align} $$

So, we have $d(-B,[k_i]) \ge -(m+2)/4$ for $i=1$ , $2$ . (Indeed, it can be shown that the equality holds in (4.7) and (4.8), by using the technique described in Subsection 4.2, as we did in the proof of Lemma 4.8.)