2-torsion in the n-solvable filtration of the knot concordance group

TL;DR

This paper demonstrates that for the n-solvable filtration of the knot concordance group, the associated graded groups contain infinitely many elements of order 2, with specific properties and distinctions based on classical invariants.

Contribution

It proves the existence of infinitely many 2-torsion elements in the graded groups of the n-solvable filtration, expanding prior knowledge from lower levels to higher levels.

Findings

Each G_n contains infinite 2-torsion elements for n>1.

These elements are represented by negative amphichiral knots with trivial classical invariants.

The knots are distinguished by their Alexander polynomials and higher-order Alexander modules.

Abstract

In 1997 Cochran-Orr-Teichner introduced a natural filtration, called the n-solvable filtration, of the smooth knot concordance group, C. Its terms {F_n} are indexed by half integers. We show that each associated graded abelian group G_n=F_n/F_{n.5}, n>1, contains infinite linearly independent sets of elements of order 2 (this was known previously for n=0,1). Each of the representative knots is negative amphichiral, with vanishing s-invariant, tau-invariant, delta-invariants and Casson-Gordon invariants. Moreover each is smoothly slice in a rational homology 4-ball. In fact we show that there are many distinct such classes in G_n, distinguished by their classical Alexander polynomials and by the orders of elements in their higher-order Alexander modules.

Figures

Equations

$$\cdots \subset {\mathcal{F}}_{n+1}\subset {\mathcal{F}}_{n.5}\subset {\mathcal{F}}_n\subset \cdots \subset {\mathcal{F}}_1\subset {\mathcal{F}}_{0.5}\subset {\mathcal{F}}_0\subset \mathcal{C},$$

$$\cdots \subset {\mathcal{F}}_{n+1}\subset {\mathcal{F}}_{n.5}\subset {\mathcal{F}}_n\subset \cdots \subset {\mathcal{F}}_1\subset {\mathcal{F}}_{0.5}\subset {\mathcal{F}}_0\subset \mathcal{C}.$$

$$FM:\mathcal{C}\twoheadrightarrow {\mathbb{Z}}_2^{\infty }.$$

$$\mathcal{C}\twoheadrightarrow \mathcal{A}\mathcal{C}\cong {\mathbb{Z}}^{\infty }\oplus {\mathbb{Z}}_2^{\infty }\oplus {\mathbb{Z}}_4^{\infty },$$

$${\mathbb{G}}_0\cong {\mathbb{Z}}^{\infty }\oplus {\mathbb{Z}}_2^{\infty }\oplus {\mathbb{Z}}_4^{\infty }.$$

$${\mathbb{G}}_0\cong \underset{\begin{array}{c}p(t)\end{array}}{⨁}\left({\mathbb{Z}}^{r_p}\oplus {\mathbb{Z}}_2^{m_p}\oplus {\mathbb{Z}}_4^{n_p}\right)$$

$${\mathbb{Z}}^{\infty }\oplus {\mathbb{Z}}_2^{\infty }\subset {\mathbb{G}}_1.$$

$$\underset{\begin{array}{c}p(t)\end{array}}{⨁}{\mathbb{Z}}^{\infty }\subset {\mathbb{G}}_1.$$

$${\mathbb{Z}}^{\infty }\subset {\mathbb{G}}_n.$$

$$\underset{\begin{array}{c}{\mathbb{P}}_n\end{array}}{⨁}{\mathbb{Z}}^{\infty }\cong \underset{\begin{array}{c}\mathcal{P}\in {\mathbb{P}}_n\end{array}}{⨁}\mathcal{I}(\mathcal{P})\subset {\mathbb{G}}_n.$$

$${\mathbb{Z}}_2^{\infty }\subset {\mathbb{G}}_n,$$

$$\underset{\begin{array}{c}{\mathbb{P}}_{n-1}\end{array}}{⨁}{\mathbb{Z}}_2^{\infty }\subset {\mathbb{G}}_n,$$

$${\mathbb{Z}}_2^{\infty }\subset {\mathbb{G}}_1.$$

$$\left(\begin{array}{cc}m& 0\\ -1& -m\end{array}\right).$$

$${\mathfrak{R}}^m\equiv {\mathbb{E}}^m\#{\mathbb{E}}^m\cong {\mathbb{E}}^m\#-{\mathbb{E}}^m,$$

$$K^{n-1}\equiv R^{n-1}\circ \cdots \circ R^1(K^0).$$

$$R({\mathcal{F}}_i,\dots ,{\mathcal{F}}_i)\subset {\mathcal{F}}_{i+1}.$$

$${\mathcal{K}}^n\equiv {\mathcal{K}}^n(m_1,\dots ,m_n,K^0)\equiv {\mathfrak{R}}_{{\eta }_1,{\eta }_2}^{m_n}({\mathcal{K}}^{n-1},\overline{{\mathcal{K}}^{n-1}}),$$

$${\mathcal{R}}^{m_{n-1}}\circ \cdots \circ {\mathcal{R}}^{m_1}(K^0),$$

$${\mathcal{K}}^1={\mathcal{R}}_{\alpha }^{m_1}(K^0);$$

$${\mathcal{K}}^2={\mathcal{R}}_{\alpha }^{m_2}\circ {\mathcal{R}}_{\alpha }^{m_1}(K^0);$$

$$⋮$$

$${\mathcal{K}}^{n-1}={\mathcal{R}}_{\alpha }^{m_{n-1}}\circ \cdots \circ {\mathcal{R}}_{\alpha }^{m_1}(K^0);$$

$${\mathcal{K}}^n={\mathfrak{R}}_{{\eta }_1,{\eta }_2}^{m_n}({\mathcal{K}}^{n-1},\overline{{\mathcal{K}}^{n-1}}).$$

$$G_r^{(n+1)}=\ker \left(G_r^{(n)}\to \frac{G_r^{(n)}}{[G_r^{(n)},G_r^{(n)}]}\to \frac{G_r^{(n)}}{[G_r^{(n)},G_r^{(n)}]}{\otimes }_{\mathbb{Z}}\mathbb{Q}\right).$$

$$\cdots ⊲G_{\ast }^{(n+1)}⊲G_{\ast }^{(n)}⊲\cdots ⊲G_{\ast }^{(0)}\equiv G,$$

$$\cdots \subset {\mathcal{F}}_{n+1}^{\ast }\subset {\mathcal{F}}_{n.5}^{\ast }\subset {\mathcal{F}}_n^{\ast }\subset \cdots \subset {\mathcal{F}}_1^{\ast }\subset {\mathcal{F}}_{0.5}^{\ast }\subset {\mathcal{F}}_0^{\ast }\subset \mathcal{C}.$$

$${\mathcal{F}}_n\subset {\mathcal{F}}_n^{\ast }.$$

$$\begin{array}[]{lcc}G^{(0)}_{\mathcal{P}}\equiv G;\\ G^{(1)}_{\mathcal{P}}\equiv G^{(1)}_{r};\end{array}$$

$$G_{\mathcal{P}}^{(n+1)}\equiv \ker \left(G_{\mathcal{P}}^{(n)}\to \frac{G_{\mathcal{P}}^{(n)}}{[G_{\mathcal{P}}^{(n)},G_{\mathcal{P}}^{(n)}]}{\otimes }_{\mathbb{Z}[G/G_{\mathcal{P}}^{(n)}]}\mathbb{Q}[G/G_{\mathcal{P}}^{(n)}]S_{p_n}^{-1}\right).$$

$$S_{p_1}=S_{p_1}(G)=\{q_1(\mu )...q_r(\mu )|(p_1(t),q_j(t))=1;G/G_r^{(1)}\cong ⟨\mu ⟩\}\subset \mathbb{Q}[G/G_r^{(1)}];$$

$$S_{p_n}=S_{p_n}(G)=\{q_1(a_1)...q_r(a_r)|\widetilde{(p_n,q_j)}=1;q_j(1)\ne 0;a_j\in G_{\mathcal{P}}^{(n-1)}/G_{\mathcal{P}}^{(n)}\},$$

Full text

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called the ($n$)-solvable filtration. We show that each associated graded abelian group $\{\mathbb{G}_{n}=\mathcal{F}_{n}/\mathcal{F}_{n.5}~{}|~{}n\in\mathbb{N}\}$, $n\geq 2$ contains infinite linearly independent sets of elements of order $2$ (this was known previously for $n=0,1$). Each of the representative knots is negative amphichiral, with vanishing $s$-invariant, $\tau$-invariant, $\delta$-invariants and Casson-Gordon invariants. Moreover each is slice in a rational homology $4$-ball. In fact we show that there are many distinct such classes in $\mathbb{G}_{n}$, distinguished by their Alexander polynomials and, more generally, by the torsion in their higher-order Alexander modules.

*†*Partially supported by NSF DMS-0706929

*††*Partially supported by NSF CAREER DMS-0748458

*†††*Partially supported by NSF DMS-0805867

1. Introduction

A (classical) knot $K$ is the image of a smooth embedding of an oriented circle in $S^{3}$. Two knots, $K_{0}\hookrightarrow S^{3}\times\{0\}$ and $K_{1}\hookrightarrow S^{3}\times\{1\}$, are concordant if there exists a proper smooth embedding of an annulus into $S^{3}\times[0,1]$ that restricts to the knots on $S^{3}\times\{0,1\}$. Let $\mathcal{C}$ denote the set of (smooth) concordance classes of knots. The equivalence relation of concordance first arose in the early $1960$’s in work of Fox, Kervaire and Milnor in their study of isolated singularities of $2$-spheres in $4$-manifolds and, indeed, certain concordance problems are known to be equivalent to whether higher-dimensional surgery techniques “work” for topological $4$-manifolds [15][28][3]. It is well-known that $\mathcal{C}$ can be endowed with the structure of an abelian group (under the operation of connected-sum), called the smooth knot concordance group. The identity element is the class of the trivial knot. Any knot in this class is concordant to a trivial knot and is called a slice knot. Equivalently, a slice knot is one that is the boundary of a smooth embedding of a $2$-disk in $B^{4}$. In general, the abelian group structure of $\mathcal{C}$ is still poorly understood. But much work has been done on the subject of knot concordance (for excellent surveys see  [19] and  [37]). In particular,  [11] introduced a natural filtration of $\mathcal{C}$ by subgroups

[TABLE]

called the ($n$)-solvable filtration of $\mathcal{C}$ and denoted $\{\mathcal{F}_{n}\}$ (defined in Section 3). The non-triviality of $\mathcal{C}$ can be measured in terms of the associated graded abelian groups $\{\mathbb{G}_{n}=\mathcal{F}_{n}/\mathcal{F}_{n.5}~{}|~{}n\in\mathbb{N}\}$ (here we ignore the other “half” of the filtration, $\mathcal{F}_{n.5}/\mathcal{F}_{n+1}$, where almost nothing is known). This paper is concerned with elements of order two in $\mathcal{C}$ and, more generally, with elements of order two in $\mathbb{G}_{n}$.

We will review some of the history of $2$-torsion phenomena in $\mathcal{C}$ in the context of the $n$-solvable filtration. One of the earliest results concerning $\mathcal{C}$ was an epimorphism constructed by Fox and Milnor  [15]

[TABLE]

Soon thereafter, Levine constructed an epimorphism

[TABLE]

to a group, $\mathcal{AC}$, that became known as the algebraic knot concordance group. Any knot in the kernel of (1.1) is called an algebraically slice knot. In terms of the $n$-solvable filtration, Levine’s result is [11, Remark 1.3.2, Thm. 1.1]:

[TABLE]

It is known that there exist elements of order two in $\mathcal{C}$ that realize some of the above $2$-torsion invariants. Let $\overline{K}$ denote the mirror image of the oriented knot $K$, obtained as the image of $K$ under an orientation reversing homeomorphism of $S^{3}$; and let $r(K)$ denote the reverse of $K$, which is obtained by merely changing the orientation of the circle. Then it is known that $K\#r(\overline{K})$ is a slice knot, so the inverse of $[K]$ in $\mathcal{C}$, denoted $-[K]$, is represented by $r(\overline{K})$, denoted $-K$. A knot $K$ is called negative amphichiral if $K$ is isotopic to $r(\overline{K})$. It follows that, for any negative amphichiral knot $K$, $K\#K$ is a slice knot, since it is isotopic to $K\#-K$. Hence negative amphichiral knots represent elements of order either $1$ or $2$ in $\mathcal{C}$. It is a conjecture of Gordon that every class of order two in $\mathcal{C}$ can be represented by a negative amphichiral knot [19].

In fact the work of Milnor and Levine in the $1960$’s resulted in a more precise statement:

[TABLE]

where the sum is over all primes $p(t)\in\mathbb{Z}[t]$ where $p(t)\doteq p(t^{-1})$ and $p(1)=\pm 1$ [34, Sections 10,11,24][48][24, p.131]. That is, the algebraic concordance group (and $\mathbb{G}_{0}$) admits a certain $p(t)$-primary decomposition, wherein a knot has a nontrivial $p(t)$-primary part only if $p(t)$ is a factor of its Alexander polynomial. (Indeed, Levine and Stoltzfus classified $\mathbb{G}_{0}$ by first splitting the Witt class of the Alexander module (with its Blanchfield form) into its $p(t)$-primary parts).

In the $1970$’s the introduction of Casson-Gordon invariants in  [1][2] led to the discovery that the subgroup of algebraically slice knots was of infinite rank and contained infinite linearly independent sets of elements of order two [27][36]. In terms of the $n$-solvable filtration this implies the existence of

[TABLE]

Different $\mathbb{Z}^{\infty}$-summands were exhibited in [31][16]. More recent work of Se-Goo Kim  [29] on the “polynomial splitting” properties of Casson-Gordon invariants led to a generalization analogous to the result of Milnor-Levine:

[TABLE]

Thus there is evidence that $\mathbb{G}_{1}$ also exhibits a $p(t)$-primary decomposition. Further strong evidence is given in [30]. Although a similar statement for the $2$-torsion in $\mathbb{G}_{1}$ has not appeared, it is expected from combining the work of  [29] and Livingston [36]. Several authors have shown that certain knots that projected to classes of order $2$ and $4$ in $\mathcal{AC}$ are in fact of infinite order in $\mathcal{C}$ [38][39][26][20][35]. A number of papers have addressed the non-triviality of $\{\mathbb{G}_{n}\}$, [18][17][31][16][11][12][13], culminating in  [10] where it was shown that, for any integer $n$, there exists

[TABLE]

Moreover the recent work [8] of the authors resulted in a generalization of the latter fact, along the lines of the Levine-Milnor primary decomposition and  [30]: for each “distinct” $n$-tuple $\mathcal{P}=(p_{1}(t),...,p_{n}(t))$ of prime polynomials with $p_{i}(1)=\pm 1$, there is a distinct subgroup $\mathbb{Z}^{\infty}\cong\mathcal{I}(\mathcal{P})\subset\mathbb{G}_{n}$, yielding a subgroup

[TABLE]

Given a knot $K$, such an $n$-tuple encodes the orders of certain submodules of the sequence of higher-order Alexander modules of $K$. Thus one can distinguish concordance classes of knots not only by their classical Alexander polynomials, but also, loosely speaking, by their higher-order Alexander polynomials. This result indicates that $\mathbb{G}_{n}$ decomposes not just according to the prime factors of the classical Alexander polynomial, but also according to types of torsion in the higher-order Alexander polynomials.

Here we show corresponding results for $2$-torsion. That is, for any $n\geq 2$, not only will we exhibit

[TABLE]

but we also will exhibit many distinct such subgroups

[TABLE]

parametrized by their Alexander polynomials and the types of torsion in the higher-order Alexander polynomials. The representative knots are distinguished by families of von Neumann signature defects associated to their classical Alexander polynomials and “higher-order Alexander polynomials”. The precise statement is given in Theorem 5.8. Each of these concordance classes has a negative amphichiral representative that is smoothly slice in a rational homology $4$-ball. Thus the classical signatures and the Casson-Gordon signature-defect obstructions [1] (indeed all metabelian obstructions) vanish for these knots [11, Theorem 9.11]. In addition, the $s$-invariant of Rasmussen [45], the $\tau$-invariant of Ozsváth-Szabó [43], and the $\delta_{p^{n}}$ invariants of Manolescu-Owens and Jabuka [41][25][42] vanish on these concordance classes, since each of these invariants induces a homomorphism $\mathcal{C}\to\mathbb{Z}$ and so must have value zero on classes representing torsion in $\mathcal{C}$. Our examples are inspired by those of Livingston, who provided examples that can be used to establish (1.3) in the case $n=1$  [36]. His examples are distinguished by their Casson-Gordon signature defects. Our examples are distinguished by higher-order $L^{(2)}$-signature defects. It is striking that elements of finite order can sometimes be detected by signatures. The key observation is that, unlike invariants such as the classical knot signatures, the $s$ invariant, the $\tau$-invariant, or the $\delta$-invariants, the invariants arising from higher-order signature defects (including Casson-Gordon invariants) are not additive under connected sum. Therefore there is no reason to expect that they would vanish on elements of finite order.

Our work is further evidence that $\mathbb{G}_{n}$ exhibits some sort of primary decomposition, but wherein not only the classical Alexander polynomial, but also some higher-order Alexander polynomials are involved.

We remark that [11] also defined a filtration, $\{\mathcal{F}_{n}^{top}\}$, of the topological concordance group, $\mathcal{C}^{top}$. Since it is known, by work of Freedman and Quinn, that a knot lies in $\mathcal{F}_{n}^{top}$ if and only if it lies in $\mathcal{F}_{n}$, all of the results of this paper apply equally well, without change, to the filtration $\{\mathcal{F}_{n}^{top}\}$. Therefore, for simplicity, in this paper we will always work in the smooth category.

2. The examples

Our examples are inspired by those of Livingston  [36], who exhibited an infinite “linearly independent” set of negative amphichiral algebraically slice knots. His examples can be used to establish the existence of the aforementioned

[TABLE]

2.1. The Building Blocks

Consider the knot shown on the left-hand side of Figure 2.1.

Here $J$ is an arbitrary pure two component string link [32][21]. The disk containing the letter $J$ symbolizes replacing the trivial $2$-string link by the $2$-string link $J$. Viewing the knot diagram as being in the $xy$-plane ($y$ being vertical), the mirror image can be defined as the image under the reflection $(x,y,z)\mapsto(x,y,-z)$, which alters a knot diagram by replacing all positive crossings by negative crossings and vice-versa. Recall that the image of $J$ under this reflection is denoted $\overline{J}$. We also consider a “flip” homeomorphism of $S^{3}$ which flips over a diagram, given by rotation of $180$ degrees about the $y$-axis or $f(x,y,z)=(-x,y,-z)$. Note that these homeomorphisms commute. Special cases of the following elementary observation appeared in  [36, Lemma 2.1]  [37, p.326] and  [4, p. 60].

Lemma 2.1**.**

Suppose $J$ is an arbitrary pure two component string link. Then the knot $K$ on the left-hand side of Figure 2.1 is negative amphichiral.

Proof.

The knot on the right-hand side of Figure 2.1 is a diagram for $r(\overline{K})$, since it is obtained by a reflection, in the plane of the paper, of the diagram for $K$, followed by a reversal of the string orientation. Here we use that $f$ commutes with the reflection. We claim that the result is isotopic to $K$. Flipping the diagram (rotating by $180$ degrees about the vertical axis in the plane of the paper), we arrive at the diagram shown on the left-hand side of Figure 2.2. This is identical to the original diagram of $K$ except that the left-hand band passes under the right-hand band instead of over. But the left-hand band can be “swung” around by an isotopy as suggested in the right-hand side of Figure 2.2, bringing it on top of the other band, at which point one arrives at the original diagram of $K$. ∎

The following result was shown for the figure-eight knot (the case that the string link $J$ is a single twist) by the first author (inspired by  [14]). It was extended, by Cha, to the case that $J$ is an arbitrary number of twists in  [4, p.63]. Our contribution here is just to note that Cha’s proof suffices to prove this more general result.

Lemma 2.2**.**

Each knot $K$ in the family shown in Figure 2.1 is slice in a rational homology $4$-ball.

Proof.

We follow the argument of [4], only indicating where our more general argument deviates. It suffices to show that the zero-framed surgery, $M_{K}$, as shown on the left-hand side of Figure 2.3, is rational homology cobordant to $S^{1}\times S^{2}$. After adding, to $M_{K}\times[0,1]$, a four-dimensional $1$-handle and $2$-handle (going algebraically twice over the $1$-handle) and performing certain handle slides (see [4, p.62-64]), one arrives at a $3$-manifold $M^{\prime}$ given by surgery on the $3$-component link drawn as the solid lines on the right-hand side of Figure 2.3. Therefore $M_{K}$ is rationally homology cobordant to $M^{\prime}$.

Next one shows, as follows, that this underlying $3$-component link, $L_{1}$, is concordant to the simple $3$-component link, $L_{4}$ shown on the right-hand side of Figure 2.4.

Ignoring the framings on $L_{1}$, add a band as shown by the dashed lines on the right-hand side of Figure 2.3, resulting in the $4$-component link, $L_{2}$, shown on the left-hand side of Figure 2.4. Here $-J$ is the image of the (unoriented) string link under the map $(x,y,z)\mapsto(x,-y,z)$. One must be careful here since $\overline{J}$, which is reflection in the plane of the paper, is not the correct notion of mirror image for a string link. Since our $y$-axis is the true axis of the string link (the $[0,1]$ factor in $D^{2}\times[0,1]$), $-J$ is the concordance inverse of $J$ in the string link concordance group [32][22], so $J+(-J)$ is concordant to the trivial $2$-string link. Hence the link $L_{2}$ is concordant to the $4$-component link, $L_{3}$, that would result from taking $J$ to be trivial. Capping off the right-most unknotted component of $L_{3}$, we arrive at the $3$-component link, $L_{4}$, shown on the right-hand side of Figure 2.4. This describes the desired concordance from $L_{1}$ to $L_{4}$. Consequently, $M^{\prime}$ is homology cobordant to the $3$-manifold described by the framed link on the right-hand side of Figure 2.4, which is known to homeomorphic to $S^{1}\times S^{2}$. ∎

In this paper we will only need the special case of these lemmas wherein the string link $J$ consists of two twisted parallels of a single knotted arc as indicated by the examples in Figures 2.5 and  2.6. Here an $m$ inside the rectangle indicates $m$ full positive twists between the two strands, and the $J$ inside the rectangle indicates that the trivial two component string link has been replaced by two parallel zero-twisted copies of a single knotted arc $J$. This is explained more fully in Subsection 2.2.

Proposition 2.3**.**

If $m$ and $n$ are distinct positive integers then the Alexander polynomials $\Delta_{m}(t)$ of $\mathbb{E}^{m}$ and $\Delta_{n}(t)$ of $\mathbb{E}^{n}$ are distinct and irreducible, hence coprime.

Proof.

A Seifert matrix for $\mathbb{E}^{m}$ with respect to the obvious basis is

[TABLE]

Thus the Alexander polynomial of $\mathbb{E}^{m}$ is $\Delta_{m}(t)=m^{2}t^{2}-(2m^{2}+1)t+m^{2}$. The discriminant $4m^{2}+1$ is easily seen, for $m\neq 0$, to never be the square of an integer, so the roots of $\Delta_{m}(t)$ are real and irrational. Hence $\Delta_{m}(t)$ is irreducible over $\mathbb{Q}[t,t^{-1}]$. It follows that if $\Delta_{m}(t)$ and $\Delta_{n}(t)$ had a common factor then they would be identical up to a unit. But the equations $m^{2}=qn^{2}$ and $2m^{2}+1=q(2n^{2}+1)$ imply $q=1$ so $m=\pm n$. ∎

2.2. Doubling Operators

To construct knots that lie deep in the $n$-solvable filtration, we use iterated generalized satellite operations.

Suppose $R$ is a knot in $S^{3}$ and $\vec{\alpha}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{m})$ be an ordered, oriented, trivial link in $S^{3}$, that misses $R$, bounding a collection of oriented disks that meet $R$ transversely as shown on the left-hand side of Figure 2.7. Suppose $(K_{1},K_{2},\ldots,K_{m})$ is an $m$-tuple of auxiliary knots. Let $R_{\vec{\alpha}}(K_{1},\ldots,K_{m})$ denote the result of the operation pictured in Figure 2.7, that is, for each $\alpha_{j}$, take the embedded disk in $S^{3}$ bounded by $\alpha_{j}$; cut off $R$ along the disk; grab the cut strands, tie them into the knot $K_{j}$ (such that the strands have linking number zero pairwise) and reglue as shown schematically on the right-hand side of Figure 2.7.

We will call this the result of infection performed on the knot $\boldsymbol{R}$ using the infection knots $\boldsymbol{K_{j}}$ along the curves $\boldsymbol{\alpha_{j}}$  [12]. In the case that $m=1$ this is the same as the classical satellite construction. This construction has an alternative description. For each $\alpha_{j}$, remove a tubular neighborhood of $\alpha_{j}$ in $S^{3}$ and glue in the exterior of a tubular neighborhood of $K_{j}$ along their common boundary, which is a torus, in such a way that (the longitude of) $\alpha_{j}$ is identified with the meridian, $\mu_{j}$, of $K_{j}$ and the meridian of $\alpha_{j}$ is identified with the reverse of the longitude, $\ell_{j}$, of $K_{j}$ as suggested by Figure 2.8. The resulting space can be seen to be homeomorphic to $S^{3}$ and the image of $R$ is the new knot.

It is well known that if the input knots $K_{1}$ and $K_{2}$ are concordant, then the output knots $R_{\alpha}(K_{1})$ and $R_{\alpha}(K_{2})$ are concordant. Thus the functions $R_{\vec{\alpha}}$ descend to $\mathcal{C}$.

Definition 2.4** ([9, 8]).**

A doubling operator, $R_{\vec{\alpha}}:\mathcal{C}\times\dots\times\mathcal{C}\to\mathcal{C}$ is a function, as in Figure 2.7, that is given by infection on a ribbon knot $R$ wherein, for each $i$, $lk(R,\alpha_{i})=0$. Often we suppress $\alpha_{i}$ from the notation.

These are called doubling operators because they generalize untwisted Whitehead doubling.

In particular we will consider the family of doubling operators $\mathfrak{R}^{m}_{\eta_{1},\eta_{2}}(-,-)$ shown in Figure 2.9.

Note that, since $\mathbb{E}^{m}$ is negative amphichiral by Lemma 2.1,

[TABLE]

which is well known to be a ribbon knot [46, Exercise 8E.30]. Thus $\mathfrak{R}^{m}$ is a negative amphichiral ribbon knot. For the case $m=1$, this was already noted in  [36].

We will also consider the family of doubling operators, $\mathcal{R}^{m}_{\alpha}$, shown in Figure 2.10 (where here the $-m$ inside a box symbolizes $m$ full negative twists between the bands but where the individual bands remain untwisted), equipped with a specified circle $\alpha$ that can be shown to generate its Alexander module.

2.3. Elements of order 2 in $\boldsymbol{\mathcal{F}_{n}}$

Now we describe large families of examples of negative amphichiral knots that lie in $\mathcal{F}_{n}$. Let $K^{0}$ be any knot with Arf($K^{0}$)$=0$. Let $K^{n-1}$ be the image of $K^{0}$ under the composition of any $n-1$ doubling operators (each requiring a single input), that is,

[TABLE]

Then, for any integer $m$ we define $K^{n}$ as in Figure 2.11, that is, $K^{n}\equiv\mathfrak{R}^{m}_{\eta_{1},\eta_{2}}(K^{n-1},\overline{K^{n-1}})$.

Proposition 2.5**.**

For any $n\geq 1$, any $m$, any composition of $n-1$ doubling operators and any Arf invariant zero input knot $K^{0}$, the knot $K^{n}$ of Figure 2.11 satisfies

• •

$K^{n}$* is negative amphichiral;*

• •

$K^{n}\in\mathcal{F}_{n}$;

• •

$K^{n}$* is (smoothly) slice in a smooth rational homology $4$-ball; and*

• •

$K^{n}\#~{}K^{n}$* is a slice knot.*

Proof.

It was shown in [10, Theorem 7.1] that, for any any doubling operator $R$,

[TABLE]

Since any knot of Arf invariant zero is known to lie in $\mathcal{F}_{0}$ [11, Remark 8.14,Thm. 8.11], and since $K^{n}$ is the image of $K^{0}$ under a composition of $n$ doubling operators, it follows that $K^{n}\in\mathcal{F}_{n}$.

Note that $K^{n}$ is the connected sum of two knots each of which is of the form shown in Figure 2.6 (hence of the form of Figure 2.1). Thus, by Lemma 2.2, each such $K^{n}$ is slice in a rational homology $4$-ball. Moreover, by Lemma 2.1, $K^{n}$ is negative amphichiral so $K^{n}\#K^{n}$ is isotopic to $K^{n}\#r(\overline{K^{n}})$. But the latter is a ribbon knot and hence a slice knot. ∎

For specificity we define the following infinite families:

Definition 2.6**.**

Given an $n$-tuple $(m_{1},...,m_{n})$ of integers and an Arf invariant zero knot $K^{0}$, we define $\mathcal{K}^{n}(m_{1},\dots,m_{n},K^{0})$ to be the image of $K^{0}$ under the following composition of $n$ doubling operators. Specifically let

[TABLE]

as shown in Figure 2.11, where $\mathcal{K}^{n-1}$ is

[TABLE]

where the $\mathcal{R}^{m_{i}}$ are the operators of Figure 2.10. In other words, we recursively set:

[TABLE]

Even though $\mathcal{K}^{n}$ depends on $(m_{1},...,m_{n},K^{0})$, we will often suppress the latter from the notation.

3. Commutator Series and Filtrations of the knot concordance groups

To accomplish our goals, we must establish that many of the knots in the families given by Figure 2.11, and specifically those in Definition 2.6, are not in $\mathcal{F}_{n.5}$ and, indeed, are distinct from each other in $\mathcal{F}_{n}/\mathcal{F}_{n.5}$. To this end we review recent work of the authors that introduced refinements of the $n$-solvable filtration parameterized by certain classes of group series that generalized the derived series. In particular the authors defined specific filtrations of $\mathcal{C}$ that depend on a sequence of polynomials. These filtrations can then be used to distinguish between knots with different Alexander modules or different higher-order Alexander modules. All of the material in this section is a review of the relevant terminology of  [8, Sections 2,3].

Recall that the derived series, $\{G^{(n)}~{}|~{}n\geq 0\}$, of a group $G$ is defined recursively by $G^{(0)}\equiv G$ and $G^{(n+1)}\equiv[G^{(n)},G^{(n)}]$. The rational derived series [23], $\{G_{r}^{(n)}~{}|~{}n\geq 0\}$, is defined by $G_{r}^{(0)}\equiv G$ and

[TABLE]

More generally,

Definition 3.1** ([8, Definition 2.1]).**

A commutator series defined on a class of groups is a function, $*$, that assigns to each group $G$ in the class a nested sequence of normal subgroups

[TABLE]

such that $G^{\scriptscriptstyle(n)}_{*}/G^{\scriptscriptstyle(n+1)}_{*}$ is a torsion-free abelian group.

Proposition 3.2** ([8, Proposition 2.2]).**

For any commutator series $\{G^{\scriptscriptstyle(n)}_{*}\}$,

$\{x\in G^{\scriptscriptstyle(n)}_{*}~{}|~{}\exists k>0,~{}x^{k}\in[G^{\scriptscriptstyle(n)}_{*},G^{\scriptscriptstyle(n)}_{*}]\}\subset G^{\scriptscriptstyle(n+1)}_{*}$* (and in particular $[G^{\scriptscriptstyle(n)}_{*},G^{\scriptscriptstyle(n)}_{*}]\subset G^{\scriptscriptstyle(n+1)}_{*}$, whence the name commutator series);*

• 2.

$G^{\scriptscriptstyle(n)}_{r}\subset G^{\scriptscriptstyle(n)}_{*}$, that is, every commutator series contains the rational derived series;

• 3.

$G/G^{\scriptscriptstyle(n)}_{*}$* is a poly-(torsion-free abelian) group (abbreviated PTFA);*

• 4.

$\mathbb{Z}[G/G^{\scriptscriptstyle(n)}_{*}]$* and $\mathbb{Q}[G/G^{\scriptscriptstyle(n)}_{*}]$ are right (and left) Ore domains.*

Any commutator series that satisfies a weak functoriality condition induces a filtration, $\{\mathcal{F}^{*}_{n}\}$, of $\mathcal{C}$ by subgroups. These filtrations generalize and refine the ($n$)-solvable filtration $\{\mathcal{F}_{n}\}$ of  [11]. Let $M_{K}$ denote the closed $3$-manifold obtained by zero-framed surgery on $S^{3}$ along $K$.

Definition 3.3** ([8, Definition 2.3]).**

A knot $K$ is an element of $\boldsymbol{\mathcal{F}_{n}^{*}}$ if the zero-framed surgery $M_{K}$ bounds a compact, connected, oriented, smooth $4$-manifold $W$ such that

$H_{1}(M_{K};\mathbb{Z})\to H_{1}(W;\mathbb{Z})$ is an isomorphism;

• 2.

$H_{2}(W;\mathbb{Z})$ has a basis consisting of connected, compact, oriented surfaces, $\{L_{i},D_{i}~{}|~{}1\leq i\leq r\}$, embedded in $W$ with trivial normal bundles, wherein the surfaces are pairwise disjoint except that, for each $i$, $L_{i}$ intersects $D_{i}$ transversely once with positive sign;

• 3.

for each $i$, $\pi_{1}(L_{i})\subset\pi_{1}(W)^{(n)}_{*}$ and $\pi_{1}(D_{i})\subset\pi_{1}(W)^{(n)}_{*}$.

A knot $K\in\boldsymbol{\mathcal{F}_{n.5}^{*}}$ if in addition,

for each $i$, $\pi_{1}(L_{i})\subset\pi_{1}(W)^{(n+1)}_{*}$.

Such a $4$-manifold is called an $\boldsymbol{(n,*)}$-solution (respectively an $(n.5,*)$-solution) for $K$ and it is said that $K$ is $\boldsymbol{(n,*)}$-solvable (respectively $(n.5,*)$-solvable) via $W$. The case where the commutator series is the derived series (without the torsion-free abelian restriction) is denoted $\mathcal{F}_{n}$ and we speak of $W$ being an ($n$)-solution, and $K$ or $M_{K}$ being ($n$)-solvable via $W$  [11, Section 8].

Definition 3.4**.**

A commutator series $\{G^{(n)}_{*}\}$ is weakly functorial (on a class of $\{$groups, maps$\}$) if $f(G^{(n)}_{*})\subset\pi^{(n)}_{*}$ for each $n$ and for any homomorphism $f:G\to\pi$ (in the class) that induces an isomorphism $G/G^{(1)}_{r}\cong\pi/\pi^{(1)}_{r}$ (i.e. induces an isomorphism on $H_{1}(-;\mathbb{Q})$).

Proposition 3.5** ([8, Prop. 2.5]).**

Suppose $*$ is a weakly functorial commutator series defined on the class of groups with $\beta_{1}=1$. Then $\{\mathcal{F}_{n}^{*}\}_{n\geq 0}$ is a filtration by subgroups of the classical (smooth) knot concordance group $\mathcal{C}$:

[TABLE]

Moreover, for any $n\in\frac{1}{2}\mathbb{Z}$

[TABLE]

The case where the commutator series is the derived series (without the torsion-free abelian restriction) is the $\boldsymbol{(n)}$-solvable filtration  [11], denoted $\boldsymbol{\{\mathcal{F}_{n}\}}$.

3.1. The Derived Series Localized at $\boldsymbol{\mathcal{P}}$

Fix an $n$-tuple $\mathcal{P}=(p_{1}(t),...,p_{n}(t))$ of non-zero elements of $\mathbb{Q}[t,t^{-1}]$, such that $p_{1}(t)\doteq p_{1}(t^{-1})$. For each such $\mathcal{P}$ we now recall from [8] the definition of a partial commutator series that we call the (polarized) derived series localized at $\mathcal{P}$, that is defined on the class of groups with $\beta_{1}=1$.

Suppose $G$ is a group such that $G/G_{r}^{(1)}\cong\mathbb{Z}=\langle\mu\rangle$. Then we define the derived series localized at $\mathcal{P}$ recursively in terms of certain right divisor sets $S_{p_{n}}\subset\mathbb{Q}[G/G^{\scriptscriptstyle(n)}_{\mathcal{P}}]$.

Definition 3.6**.**

For $n\geq 0$, let

[TABLE]

and for $n\geq 1$

[TABLE]

To make sense of (3.1) one must realize that, for any $H\vartriangleleft G$, $H/[H,H]$ is a right $\mathbb{Z}[G/H]$-module where $g$ acts on $h$ by $h\mapsto g^{-1}hg$. One must also verify, at each stage, that $G^{\scriptscriptstyle(n)}_{\mathcal{P}}$ has been defined in such a way that $G^{(k)}_{\mathcal{P}}/G^{(k+1)}_{\mathcal{P}}$ is a torsion-free abelian group for each $k<n$, so $G/G^{\scriptscriptstyle(n)}_{\mathcal{P}}$ is a poly-(torsion-free-abelian) group (PTFA), from which it follows that $\mathbb{Q}[G/G^{\scriptscriptstyle(n)}_{\mathcal{P}}]$ is a right Ore domain. Therefore, for any right divisor set $S_{p_{n}}\subset\mathbb{Q}[G/G^{\scriptscriptstyle(n)}_{\mathcal{P}}]$ we may define the Ore localization $\mathbb{Q}[G/G^{\scriptscriptstyle(n)}_{\mathcal{P}}]S_{p_{n}}^{-1}$ as in (3.1) (see [8, Sections $3$,$4$]).

For the (polarized) derived series localized at $\mathcal{P}$ we use the following right divisor sets:

Definition 3.7**.**

The (polarized) derived series localized at $\mathcal{P}$ is defined as in Definition 3.6 by setting

[TABLE]

and for $n\geq 2$

[TABLE]

so $S_{p_{n}}\subset\mathbb{Q}[G^{(n-1)}_{\mathcal{P}}/G^{(n)}_{\mathcal{P}}]$.

Here $p_{i}(t)$ and $q_{j}(t)$ are in $\mathbb{Q}[t,t^{-1}]$. By $(p_{1},q_{j})=1$ we mean that $p_{1}$ is coprime to $q_{j}$ in $\mathbb{Q}[t,t^{-1}]$, as usual. But by $~{}\widetilde{(p_{n},q_{j})}=1$ we mean something slightly stronger.

Definition 3.8** ([8, Defn. 4.4]).**

Two non-zero polynomials $p(t),q(t)\in\mathbb{Q}[t,t^{-1}]$ are said to be strongly coprime, denoted $\widetilde{(p,q)}=1$ if, for every pair of non-zero integers, $n,k$, $p(t^{n})$ is relatively prime to $q(t^{k})$. Alternatively, $\widetilde{(p,q)}\neq 1$ if and only if there exist non-zero roots, $r_{p},r_{q}\in\mathbb{C}$*, of $p(t)$ and $q(t)$ respectively, and non-zero integers $k,n$, such that $r_{p}^{k}=r_{q}^{n}$. Clearly, $\widetilde{(p,q)}=1$ if and only if for each prime factor $p_{i}(t)$ of $p(t)$ and $q_{j}(t)$ of $q(t)$, $\widetilde{(p_{i},q_{j})}=1$.

Note that $\mathbb{Q}-\{0\}\subset S_{p_{n}}$ (take $q_{j}$ to be a non-zero constant). It is easy to see (and was proved in [8, Section 4]) that $S_{p_{n}}$ is closed (up to units) under the involution on $\mathbb{Q}[G/G^{\scriptscriptstyle(n)}_{\mathcal{P}}]$. Here we need $p_{1}(t)\doteq p_{1}(t^{-1})$.

Example 3.9**.**

Consider the family of quadratic polynomials

[TABLE]

whose roots are $\{m/(m+1),(m+1)/m\}$. The polynomial $q_{m}$ is the Alexander polynomial of the ribbon knot $\mathcal{R}^{m}$ shown in in Figure 2.10. It can easily be seen (and was proved in  [8, Example 4.10]) that $\widetilde{(q_{m},q_{n})}=1$ if $m\neq n$.

Theorem 3.10** ([8, Thm. 4.16]).**

The (polarized) derived series localized at $\mathcal{P}$ is a weakly functorial commutator series on the class of groups with $\beta_{1}=1$.

4. von Neumann signature defects as obstructions to $(n.5,*)$-solvability

To each commutator series there exist signature defects that offer obstructions to a given knot lying in a term of $\mathcal{F}^{*}$. Given a closed, oriented 3-manifold $M$, a discrete group $\Gamma$, and a representation $\phi:\pi_{1}(M)\to\Gamma$, the von Neumann $\boldsymbol{\rho}$-invariant, $\rho(M,\phi)\in\mathbb{R}$, was defined by Cheeger and Gromov  [5]. If $(M,\phi)=\partial(W,\psi)$ for some compact, oriented 4-manifold $W$ and $\psi:\pi_{1}(W)\to\Gamma$, then it is known that $\rho(M,\phi)=\sigma^{(2)}_{\Gamma}(W,\psi)-\sigma(W)$ where $\sigma^{(2)}_{\Gamma}(W,\psi)$ is the $L^{(2)}$-signature (von Neumann signature) of the equivariant intersection form defined on $H_{2}(W;\mathbb{Z}\Gamma)$ twisted by $\psi$, and $\sigma(W)$ is the ordinary signature of $W$  [40][13, Section 2]. Thus the $\rho$-invariants should be thought of as signature defects. They were first used to detect non-slice knots in  [11]. For a more thorough discussion see  [11, Section 5][13, Section 2][12, Section 2]. All of the coefficient systems $\Gamma$ in this paper will be of the form $\pi/\pi^{(n)}_{*}$ where $\pi$ is the fundamental group of a space. Hence all such $\Gamma$ will be PTFA. Aside from the definition, the properties that we use in this paper are:

Proposition 4.1**.**

If $\phi$ factors through $\phi^{\prime}:\pi_{1}(M)\to\Gamma^{\prime}$ where $\Gamma^{\prime}$ is a subgroup of $\Gamma$, then $\rho(M,\phi^{\prime})=\rho(M,\phi)$.

• 2.

If $\phi$ is trivial (the zero map), then $\rho(M,\phi)=0$.

• 3.

If $M=M_{K}$ is the zero-surgery on a knot $K$ and $\phi:\pi_{1}(M)\to\mathbb{Z}$ is the abelianization, then $\rho(M,\phi)$ is denoted $\boldsymbol{\rho_{0}(K)}$ and is equal to the integral over the circle of the Levine-Tristram signature function of $K$  **[12, Prop. 5.1]**. Thus $\rho_{0}(K)$ is the average of the classical signatures of $K$.

• 4.

If $K$ is a slice knot or link and $\phi:M_{K}\to\Gamma$ ($\Gamma$* PTFA) extends over $\pi_{1}$ of a slice disk exterior then $\rho(M_{K},\phi)=0$ by  [11, Theorem 4.2].*

• 5.

The von Neumann signature satisfies Novikov additivity, i.e. if $W_{1}$ and $W_{2}$ intersect along a common boundary component then $\sigma^{(2)}_{\Gamma}(W_{1}\cup W_{2})=\sigma^{(2)}_{\Gamma}(W_{1})+\sigma^{(2)}_{\Gamma}(W_{2})$  **[11, Lemma 5.9]**.

• 6.

For any $3$-manifold $M$, there is a positive real number $C_{M}$, called the Cheeger-Gromov constant **[5]**[13, Section 2]** of $M$ such that, for any $\phi$

[TABLE]

We will also need the following generalization of property ($4$).

Theorem 4.2** ([8, Theorem 5.2]).**

Suppose $*$ is a commutator series (no functoriality is required). Suppose $K\in\mathcal{F}_{n.5}^{*}$, so the zero-framed surgery $M_{K}$ is $(n.5,*)$-solvable via $W$ as in Definition 3.3. Let $G=\pi_{1}(W)$ and consider

[TABLE]

where $\Gamma$ is an arbitrary PTFA group. Then

[TABLE]

5. Statements of Main Results and the outline of the proof

We will show that for any $n\geq 2$, not only does there exist

[TABLE]

but there are also many distinct such classes

[TABLE]

distinguished by the sequence of orders of certain higher-order Alexander modules of the knots.

Given the sequence $\mathcal{P}=(p_{1}(t),...,p_{n}(t))$, we have defined (in Definitions 3.6 and  3.7) an associated commutator series called the derived series localized at $\mathcal{P}$.

Definition 5.1**.**

Let $\{\mathcal{F}_{n}^{\mathcal{P}}\}$ denote the filtration of $\mathcal{C}$ associated, by Definition 3.3, to the derived series localized at $\mathcal{P}$.

Since for any group $G$ and integer $n$ (or half-integer), $G^{(n)}\subset G^{(n)}_{\mathcal{P}}$, one sees that $\mathcal{F}_{n}\subset\mathcal{F}_{n}^{\mathcal{P}}$. In particular $\mathcal{F}_{n.5}\subset\mathcal{F}_{n.5}^{\mathcal{P}}$, so there is a surjection

[TABLE]

The point of the filtration $\{\mathcal{F}_{n}^{\mathcal{P}}\}$, is that any knot $K\in\mathcal{F}_{n}$, whose classical Alexander polynomial is coprime to $p_{1}(t)$, will lie in the kernel of $\pi$. Moreover, the idea of Theorem 5.3 below is that a knot will of necessity lie in the kernel of $\pi$, unless $p_{1}(t)$ divides its classical Alexander polynomial and, loosely speaking, the higher $p_{i}(t)$ are related to torsion in its $i^{th}$ higher-order Alexander module.

Definition 5.2**.**

Given $\mathcal{P}=(p_{1}(t),...,p_{n}(t))$ and $\mathcal{Q}=(q_{1}(t),...,q_{n}(t))$, we say that $\mathcal{P}$ is strongly coprime to $\mathcal{Q}$ if either $(p_{1},q_{1})=1$, or for some $k>1$, $\widetilde{(p_{k},q_{k})}=1$.

Theorem 5.3** ([8, Theorem 6.5]).**

For any $n\geq 1$, let $R_{\alpha_{n-1}}^{n-1},\dots,R_{\alpha_{1}}^{1}$ be any doubling operators and $K^{0}$ be any Arf invariant zero input knot. Consider the knot $K^{n}\equiv\mathfrak{R}^{m}_{\eta_{1},\eta_{2}}(K^{n-1},\overline{K^{n-1}})$, where $K^{n-1}=R_{\alpha_{n-1}}^{n-1}\circ\dots\circ R_{\alpha_{1}}^{1}(K^{0})$. Then

[TABLE]

for each $\mathcal{P}=(p_{1}(t),p_{2}(t),...,p_{n}(t))$, with $p_{1}(t)\doteq p_{1}(t^{-1})$, that is strongly coprime to $(\Delta_{m}(t),q_{n-1}(t),\dots,q_{1}(t))$, where $\Delta_{m}$ is the Alexander polynomial of $\mathbb{E}^{m}$ and $q_{i}$ is the Alexander polynomial of $R^{i}$.

This applies, in particular, to the families of Definition 2.6, constructed using the ribbon knots of Figures 2.9 and 2.10.

Corollary 5.4**.**

For any $(m_{1},\dots,m_{n})$ and any input knot $K^{0}$ with Arf invariant zero,

[TABLE]

for each $\mathcal{P}=(p_{1}(t),p_{2}(t),...,p_{n}(t))$ that is strongly coprime to $(\Delta_{m_{n}}(t),q_{n-1}(t)\dots,q_{1}(t))$ where $\Delta_{m_{n}}$ is the Alexander polynomial of $\mathbb{E}^{m_{n}}$ and $q_{i}$ is the Alexander polynomial of $\mathcal{R}^{m_{i}}$.

Now we need a non-triviality theorem to complement Theorem 5.3.

Theorem 5.5**.**

Suppose

[TABLE]

where $K^{n-1}$ is the result of applying any sequence of $n-1$ doubling operators, $R_{\alpha_{n-1}}^{n-1}\circ\dots\circ R_{\alpha_{1}}^{1}$ to an Arf invariant zero “input” knot $K^{0}$. Suppose additionally that $n\geq 2$ and

$m\neq 0$;

• 2.

for each $i$, $\alpha_{i}$ generates the rational Alexander module of $R^{i}$, and this module is non-trivial;

• 3.

$|\rho_{0}(K^{0})|$, the average Levine-Tristram signature of $K^{0}$, is greater than twice the sum of the Cheeger-Gromov constants of the ribbon knots $\mathfrak{R}^{m}$, $R^{1},\dots,R^{n-1}$ (see Section 4).

If $\mathcal{P}$ is the sequence of classical Alexander polynomials of the knots $(\mathbb{E}^{m},R^{n-1},\dots,R^{1})$, then

[TABLE]

This can be applied to the specific families of Definition 2.6.

Corollary 5.6**.**

Fix $n\geq 2$ and an $n$-tuple of positive integers $(m_{1},\dots,m_{n})$. Suppose $K^{0}$ is chosen so that $|\rho_{0}(K^{0})|$ is greater than twice the sum of the Cheeger-Gromov constants of the ribbon knots $\mathfrak{R}^{m_{n}}$, $\mathcal{R}^{m_{n-1}},\dots,\mathcal{R}^{m_{1}}$. If $\mathcal{P}$ is the $n$-tuple of Alexander polynomials of the knots $(\mathbb{E}^{m_{n}},\mathcal{R}^{m_{n-1}},\dots,\mathcal{R}^{m_{1}})$, then

[TABLE]

The proofs of Theorems 5.3 and  5.5 will constitute Sections 6 and 7. Assuming these theorems, we now derive our main results.

Theorem 5.7**.**

Fix $n\geq 2$. For any $n$-tuple of positive integers $(m_{1},...,m_{n})$ choose an Arf invariant zero knot $K^{0}(m_{1},...,m_{n})$ such that $|\rho_{0}(K^{0})|$ is greater than twice the sum of the Cheeger-Gromov constants of $\mathfrak{R}^{m_{n}}$, $\mathcal{R}^{m_{n-1}},\dots,\mathcal{R}^{m_{1}}$. Then the resulting set of knots

[TABLE]

as in Definition 2.6, represent linearly independent, order two elements of $\mathcal{F}_{n}/\mathcal{F}_{n.5}$. They also represent linearly independent order two elements in $\mathcal{C}$. In particular this gives

[TABLE]

where each class is represented by a negative amphichiral knot that is slice in a rational homology $4$-ball.

Proof of Theorem 5.7 assuming Theorems 5.3 and  5.5.

By Proposition 2.5, $\mathcal{K}^{n}$ is negative amphichiral, $\mathcal{K}^{n}\in\mathcal{F}_{n}$ and $\mathcal{K}^{n}\#\mathcal{K}^{n}$ is a slice knot. Thus $2[\mathcal{K}^{n}]=0$ in $\mathcal{F}_{n}/\mathcal{F}_{n.5}$. By Corollary 5.6, for a certain $\mathcal{P}$, $\mathcal{K}^{n}\notin\mathcal{F}_{n.5}^{\mathcal{P}}$, so in particular $\mathcal{K}^{n}\notin\mathcal{F}_{n.5}$ by Proposition 3.5. Therefore each $[\mathcal{K}^{n}]$ has order precisely two in $\mathbb{G}_{n}$.

Suppose there exists a nontrivial relation

[TABLE]

Set $\mathcal{P}=(p_{1},...,p_{n})=(\Delta_{1n},q_{1n-1},...,q_{11})$, the reverse of the sequence of Alexander polynomials of the operators corresponding to the first summand of $J$. For each of the other summands of $J$, the corresponding $n$-tuple $(m_{i1},...,m_{in})$ is assumed distinct from $(m_{11},...,m_{1n})$. Therefore, the (reversed) sequence of Alexander polynomials of the operators corresponding to this other summand is strongly coprime to $\mathcal{P}$ by Proposition 2.3 and Example 3.9. Thus, by Theorem 5.3, each summand of $J$, aside from the first, lies in $\mathcal{F}_{n+1}^{\mathcal{P}}$ and hence in $\mathcal{F}_{n.5}^{\mathcal{P}}$. Since $J\in\mathcal{F}_{n.5}$, $J\in\mathcal{F}_{n.5}^{\mathcal{P}}$, by Proposition 3.5. Since $\mathcal{F}_{n.5}^{\mathcal{P}}$ is a subgroup, it would follow that the first summand of $J$ also lay in $\mathcal{F}_{n.5}^{\mathcal{P}}$, contradicting Corollary 5.6. ∎

More generally,

Theorem 5.8**.**

Suppose $n\geq 2$. Let $\mathbb{P}_{n}$ be any set of $n$-tuples $\mathcal{P}=(\delta_{1}(t),\delta_{2}(t),\dots,\delta_{n}(t))$ of prime polynomials $\delta_{i}(t)\in\mathbb{Z}[t,t^{-1}]$ such that $\delta_{i}(1)=\pm 1$, $\delta_{1}(t)=\Delta_{m}=m^{2}t^{2}-(2m^{2}+1)t+m^{2}$ and with the property that, for any distinct $\mathcal{P}$, $\mathcal{P}^{\prime}\in\mathbb{P}_{n}$, $\mathcal{P}$ and $\mathcal{P}^{\prime}$ are strongly coprime. Then there exists a set of negative amphichiral $n$-solvable knots indexed by $\mathbb{P}_{n}$ that is linearly independent modulo $\mathcal{F}_{n.5}$, that is, that spans

[TABLE]

where the knot corresponding to the sequence $(\delta_{1}(t),\delta_{2}(t),\dots,\delta_{n}(t))$ admits a sequence of higher-order Alexander modules containing submodules whose orders are determined by the sequence $(\delta_{1}(t)\delta_{1}(t^{-1}),\dots,\delta_{n}(t)\delta_{n}(t^{-1}))$ with the classical Alexander polynomial being $\delta_{1}(t)\delta_{1}(t^{-1})$. Moreover each class is represented by a negative amphichiral knot that is slice in a rational homology $4$-ball.

Proof of Theorem 5.8 assuming Theorems 5.3 and  5.5.

By [49], for any prime $\delta(t)$ with $\delta(1)=\pm 1$ there exists a ribbon knot whose Alexander module is cyclic of order $\delta(t)\delta(t^{-1})$. Hence, given $\mathcal{P}=(\delta_{1}(t),\delta_{2}(t),\dots,\delta_{n}(t))$, choose such ribbon knots $R^{n-1},\dots,R^{1}$ whose Alexander polynomials are $\delta_{2}(t)\delta_{2}(t^{-1}),\dots,\delta_{n}(t)\delta_{n}(t^{-1})$ respectively, and choose curves $\alpha_{i}$ (unknotted in $S^{3}$), that generate the Alexander modules of the $R^{i}$. Thus doubling operators $R^{i}_{\alpha_{i}}$, $1\leq i\leq n-1$, are defined. Since $\delta_{1}(t)=\Delta_{m}=m^{2}t^{2}-(2m^{2}+1)t+m^{2}$, there is a ribbon knot, namely $\mathfrak{R}^{m}=\mathbb{E}^{m}\#\mathbb{E}^{m}$ of Figure 2.9, whose Alexander polynomial is $\delta_{1}(t)\delta_{1}(t^{-1})$. The hypotheses imply $m\neq 0$. Choose any Arf invariant zero knot $K^{0}$ such that $|\rho_{0}(K^{0})|$ is greater than twice the sum of the Cheeger-Gromov constants of $\mathfrak{R}^{m}$, $R^{n-1},\dots,R^{1}$. Then set

[TABLE]

where $K^{n-1}\equiv R_{\alpha_{n-1}}^{n-1}\circ\dots\circ R_{\alpha_{1}}^{1}(K^{0})$. To each $\mathcal{P}$ there is an associated $n$-tuple, $\mathcal{P}^{*}=(\delta_{1},\delta_{2}(t)\delta_{2}(t^{-1}),\dots,\delta_{n}(t)\delta_{n}(t^{-1}))$, that gives the sequence of Alexander polynomials of the knots $\mathbb{E}^{m},R^{n-1},\dots,R^{1}$ that define $K^{n}_{\mathcal{P}}$.

By Lemma 2.1 and Proposition 2.5, each $K^{n}_{\mathcal{P}}$ is negative amphichiral and $n$-solvable. By Theorem 5.5,

[TABLE]

so $K^{n}_{\mathcal{P}}\notin\mathcal{F}_{n.5}$. Thus $[K^{n}_{\mathcal{P}}]$ has order precisely two in $\mathbb{G}_{n}$. Suppose there were a non-trivial relation

[TABLE]

By hypothesis, if $i\neq 1$ then $\mathcal{P}_{i}$ is strongly coprime to $\mathcal{P}_{1}$. It follows that $\mathcal{P}_{i}^{*}$ is strongly coprime to $\mathcal{P}_{1}^{*}$. Thus, by Theorem 5.3, if $i\neq 1$ then

[TABLE]

Since $J\in\mathcal{F}_{n.5}$, $J\in\mathcal{F}_{n.5}^{\mathcal{P}_{1}^{*}}$. Since the latter is a subgroup,

[TABLE]

contradicting (5.2).

It remains only to relate the sequence $\mathcal{P}$ to the higher-order Alexander modules of the knots $K^{n}_{\mathcal{P}}$. Since this is not central to our results, we sketch the proof. Recall:

Definition 5.9** ([6, Def. 2.8][23, Def. 5.3]).**

The $i^{th}$, $i\geq 1$, higher-order (integral) Alexander module of a knot $K$ is

[TABLE]

where $G\equiv\pi_{1}(M_{K})$. Note: The case $i=0$ would give the classical Alexander module.

Thus $\mathcal{A}^{\mathbb{Z}}_{i}(K^{n}_{\mathcal{P}})$ is a module over $\Gamma_{i}\equiv G/G^{(i+1)}_{r}$, where $G\equiv\pi_{1}(M_{K^{n}_{\mathcal{P}}})$. The following lemma shows that the (two) images of the classical Alexander polynomial, $\delta_{i+1}(t)\delta_{i+1}(t^{-1})$, of the constituent operator $R^{n-i}$ under certain maps

[TABLE]

wherein $t\mapsto x_{1}$ and $t\mapsto x_{2}$, appear as the orders of cyclic submodules of $\mathcal{A}^{\mathbb{Z}}_{i}(K^{n}_{\mathcal{P}})$.

Lemma 5.10**.**

Fixing $\mathcal{P}=(\delta_{1}(t),\delta_{2}(t),\dots,\delta_{n}(t))$, for each $1\leq i\leq n-1$, the $i^{th}$ higher-order Alexander module of $K^{n}_{\mathcal{P}}$ (the knot defined in (5.1)) contains two non-trivial summands

[TABLE]

for certain $x_{1},x_{2}\in G^{(i)}/G^{(i+1)}_{r}$.

Proof.

Recall that $K^{n}_{\mathcal{P}}$ is defined as the image of $K^{0}$ under a composition of $n$ doubling operators. In particular $K^{n-1}\equiv R_{\alpha_{n-1}}^{n-1}\circ\dots\circ R_{\alpha_{1}}^{1}(K^{0})$. Sequences of satellite operations have a certain associativity property yielding, for each $i\geq 2$, an alternative description of $K^{n-1}$ as a single infection on single ribbon knot, $\tilde{R}^{i}$, along a curve lying in $\pi_{1}(S^{3}-\tilde{R}^{i})^{(i-1)}$, using the knot $K^{n-i}$ [7, Prop. 4.7][9, Prop. 5.10]. Specifically,

[TABLE]

where

[TABLE]

and $\beta_{i}$ is the image of $\alpha_{n-i+1}$. The specific nature of $\tilde{R}^{i}$ is not important to our present considerations. If $i=1$, let $\tilde{R}^{i}_{\beta_{i}}$ be the identity operator. Then, for any $i\geq 1$, it follows that

[TABLE]

This can be reformulated, by the same considerations as above, to yield

[TABLE]

where $\tilde{\mathfrak{R}}=\mathfrak{R}^{m}_{\eta_{1},\eta_{2}}(\tilde{R}^{i},\tilde{R}^{i})$ and $\{\gamma_{1},\gamma_{2}\}$ are the images of the two copies of $\beta_{i}$. These curves can inductively shown to lie in $\pi_{1}(S^{3}-\tilde{\mathfrak{R}})^{(i)}$ [7, Prop. 4.7][9, Prop. 5.10]. The latter computation is very similar to the computation we will perform in (7.15).

Now we can apply known results about the effect of single infection on the higher-order Alexander modules [33, Theorem 3.5][6, Theorem 8.2]:

[TABLE]

where $\mathcal{A}_{0}^{\mathbb{Z}}$ denotes the classical Alexander module and the first tensor product is given by $t\mapsto x_{1}=\gamma_{1}$ and the second by $t\mapsto x_{2}=\gamma_{2}$. But

[TABLE]

where $t\mapsto x_{1}$. The Alexander modules of $\overline{R^{n-i}}$ and $R^{n-i}$ are isomorphic. Thus $\mathcal{A}^{\mathbb{Z}}_{i}(K^{n})$ contains two cyclic summands as claimed. By [33, Theorem 3.5][6, Theorem 8.2] these summands are non-zero precisely when $x_{1}$ and $x_{2}$ are not zero in $\Gamma_{i}$. The verification of the latter requires further computation as in [7, Theorem 4.11][9, Propoposition 5.14]. These calculations are entirely similar to and easier than the ones we will do to verify our Proposition 7.4. They are not included.

This concludes what we will say about the connections between $\mathcal{P}$ and the orders of the higher-order Alexander modules of $K^{n}_{\mathcal{P}}$. ∎

This concludes the proof of Theorem 5.8.

∎

6. Sketch of Proof of Theorem 5.3

Theorem 5.3 is a consequence of [8, Theorem 6.5]. However, we shall sketch the proof since the basic idea is elementary and it also shows that $K^{n}\in\mathcal{F}_{n}$.

Theorem 5.3 ([8, Theorem 6.5]).

For any $n\geq 1$ and $m\in\mathbb{Z}$, let $R_{\alpha_{n-1}}^{n-1},\dots,R_{\alpha_{1}}^{1}$ be any doubling operators and $K^{0}$ be any Arf invariant zero input knot. Consider the knot $K^{n}\equiv\mathfrak{R}^{m}_{\eta_{1},\eta_{2}}(K^{n-1},\overline{K^{n-1}})$, where $K^{n-1}=R_{\alpha_{n-1}}^{n-1}\circ\dots\circ R_{\alpha_{1}}^{1}(K^{0})$. Then

[TABLE]

for each $\mathcal{P}=(p_{1}(t),p_{2}(t),...,p_{n}(t))$, with $p_{1}(t)\doteq p_{1}(t^{-1})$, that is strongly coprime to $(\Delta_{m}(t),q_{n-1}(t),\dots,q_{1}(t))$, where $\Delta_{m}$ is the Alexander polynomial of $\mathbb{E}^{m}$ and $q_{i}$ is the Alexander polynomial of $R^{i}$.

Proof of Theorem 5.3.

We set $K^{1}=R^{1}(K^{0}),\dots,K^{i}=R^{i}(K^{i-1})$ for $i=1,\dots,n-1$ and $K^{n}=\mathfrak{R}^{m}(K^{n-1},\overline{K^{n-1}})$. Recall from  [10, Lemma 2.3, Figure 2.1] that, whenever a knot $L$ is obtained from a knot $R$ by infection using knots $K_{1},K_{2},\dots$ there is a cobordism $E$ whose boundary is the disjoint union of the zero surgeries $M_{L}$, $-M_{R}$ and $-M_{K_{1}}$, $-M_{K_{2}}$ et cetera, as shown on the left-hand side of Figure 6.1.

Therefore, since $K^{n}=\mathfrak{R}^{m}(K^{n-1},\overline{K^{n-1}})$, there is a cobordism $E_{n}$ whose boundary is the disjoint union of the zero surgeries on $K^{n}$, $K^{n-1}$, $\overline{K^{n-1}}$ and $\mathfrak{R}^{m}$ as shown on the right-hand side of Figure 6.1 and schematically in Figure 6.2. Similarly there is a cobordism $E_{i}$, for $1\leq i<n$ whose boundary is the disjoint union of the zero surgeries on $K^{i}$, $K^{i-1}$ and $R^{i}$. Consider $X=E_{n}\cup E_{n-1}\cup\overline{E}_{n-1}\cup...\cup E_{1}\cup\overline{E}_{1}$, gluing $E_{i}$ to $E_{i-1}$ along their common boundary component $M_{K^{i-1}}$, and gluing $\overline{E}_{i}$ to $\overline{E}_{i-1}$ along their common boundary component $M_{\overline{K^{i-1}}}$, as shown schematically in Figure 6.2. The boundary of $X$ is a disjoint union of $M_{K^{n}}$, $-M_{\mathfrak{R}^{m}}$, $-M_{K^{0}}$, $-M_{\overline{K^{0}}}$ and two copies each of $\pm M_{R^{n-1}},...,\pm M_{R^{1}}$. For $1\leq i<n$, let $S_{i}$ denote the exterior of any ribbon disk in $B^{4}$ for the ribbon knot $R^{i}$. Let $S_{n}$ denote the exterior of any ribbon disk in $B^{4}$ for the ribbon knot $\mathfrak{R}^{m}$. Since Arf($K^{0}$)$=0$, $K^{0}\in\mathcal{F}_{0}$ via some $V$  [12, Section 5]. Gluing $V$, $\overline{V}=-V$ and all the $S_{i}$ and $\overline{S}_{i}$ to $X$, we obtain a $4$-manifold, $Z$ as shown in Figure 6.2. Note $\partial Z=M_{K^{n}}$.

We claim that,

[TABLE]

and if $\mathcal{P}$ is strongly coprime to $(\Delta_{m}(t),q_{n-1}(t),\dots,q_{1}(t))$, then

[TABLE]

First, simple Mayer-Vietoris sequences together with an analysis of the homology of the $E_{i}$ (as given by Lemma 6.1 below) imply that $H_{2}(Z)\cong H_{2}(V)\oplus H_{2}(\overline{V})$ since $H_{2}(S_{i})=0$. Since $V$ is a [math]-solution, $H_{2}(V)$ has a basis of connected compact oriented surfaces, $\{L_{j},D_{j}|1\leq j\leq r\}$, satisfying the conditions of Definition 3.3. Similarly for $H_{2}(\overline{V})$. We claim that,

[TABLE]

and if $\mathcal{P}$ is strongly coprime to $(\Delta_{m}(t),q_{n-1}(t),\dots,q_{1}(t))$ then

[TABLE]

Indeed equations (6.3) and (6.4) were shown inductively in the proof of [8, Theorem 6.2, Theorem 6.5] using the fact that, for each $i$, the doubling operator $R^{i}_{\alpha_{i}}$ satisfies $\ell k(\alpha_{i},R^{i})=0$ leading to the fact that

[TABLE]

Then,

[TABLE]

and if $\mathcal{P}$ is strongly coprime to $(\Delta_{m}(t),q_{n-1}(t),\dots,q_{1}(t))$,

[TABLE]

and similarly for $\pi_{1}(D_{j})$. The same holds for $\overline{V}$. This would complete the verification of claims  (6.1) and  (6.2) since $\{L_{j},D_{j}\}$ (together with their counterparts in $\overline{V}$ would then satisfy the criteria of Definition 3.3.

This concludes our sketch of the proof as given in [8, Theorem 6.5]. We include the relevant result about the elementary topology of the cobordism $E$. We will need several of these properties in later proofs.

Lemma 6.1** ([10, Lemma 2.5]).**

With regard to $E$ on the left-hand side of Figure 6.1, the inclusion maps induce

• (1)

an epimorphism $\pi_{1}(M_{L})\to\pi_{1}(E)$ whose kernel is the normal closure of the longitudes of the infecting knots $K_{i}$ viewed as curves $\ell_{i}\subset S^{3}-K_{i}\subset M_{L}$;

• (2)

isomorphisms $H_{1}(M_{L})\to H_{1}(E)$ and $H_{1}(M_{R})\to H_{1}(E)$;

• (3)

and isomorphisms $H_{2}(E)\cong H_{2}(M_{L})\oplus_{i}H_{2}(M_{K_{i}})\cong H_{2}(M_{R})\oplus_{i}H_{2}(M_{K_{i}})$.

• (4)

The meridian of $K$, $\mu_{K}\subset M_{K}$ is isotopic in $E$ to both $\alpha\subset M_{R}$ and to the longitudinal push-off of $\alpha$, often called $\alpha\subset M_{L}$ by abuse of notation.

• (5)

The longitude of $K$, $\ell_{K}\subset M_{K}$ is isotopic in $E$ to the reverse of the meridian of $\alpha$, $(\mu_{\alpha})^{-1}\subset M_{L}$ and to the longitude of $K$ in $S^{3}-K\subset M_{L}$ and to the reverse of the meridian of $\alpha$, $(\mu_{\alpha})^{-1}\subset M_{R}$ (the latter bounds a disk in $M_{R}$).

∎

7. Proof of Theorem 5.5

The proof of Theorem 5.5 will occupy the remainder of the paper.

Theorem 5.5.

Consider knots $K^{n}$, $n\geq 2$ as in Figure 2.11

[TABLE]

where $K^{n-1}$ is the result of applying a composition of $n-1$ doubling operators, $R_{\alpha_{n-1}}^{n-1}\circ\dots\circ R_{\alpha_{1}}^{1}$ to some Arf invariant zero input knot $K^{0}$. Suppose additionally that

$m\neq 0$;

• 2.

for each $i$, $\alpha_{i}$ generates the rational Alexander module of $R^{i}$, and this module is non-trivial;

• 3.

$|\rho_{0}(K^{0})|$, the average Levine-Tristram signature of $K^{0}$, is greater than twice the sum of the Cheeger-Gromov constants of the ribbon knots $\mathfrak{R}^{m}$, $R^{1},\dots,R^{n-1}$ (see Section 4).

If $\mathcal{P}$ is the $n$-tuple of Alexander polynomials of the knots $(\mathbb{E}^{m},R^{n-1},\dots,R^{1})$, then

[TABLE]

Proof of Theorem 5.5.

We assume that

[TABLE]

is the $n$-tuple of Alexander polynomials of the knots $(\mathbb{E}^{m},R^{n-1},\dots,R^{1})$. Suppose that $K^{n}\in\mathcal{F}_{n.5}^{\scriptscriptstyle{\mathcal{P}}}$. Let $V$ be the putative $(n.5,\mathcal{P})$-solution. We will derive a contradiction.

Let $W_{0}$ be the $4$-manifold (refer to Figure 7.1) obtained from $V$ by adjoining the cobordisms $E_{n}$, $E_{n-1}$, $\overline{E}_{n-1},\dots E_{1},\overline{E}_{1}$ as defined in the proof of Theorem 5.3. For specificity, set

[TABLE]

Note that, unlike in the manifold $Z$ of Figure 6.2, we do not cap off the zero surgeries on the various ribbon knots. Thus the boundary of $W_{0}$ is the disjoint union of the zero surgeries on the ribbon knots $\mathfrak{R}^{m}$, $R^{n-1},\dots,R^{1}$, $\overline{R}^{{n-1}},\dots,\overline{R}^{1}$, together with the zero surgeries on $K^{0}$, $\overline{K}^{0}$.

Below we will define a commutator series $\{\pi^{(n)}_{\mathcal{S}}\}$ that is slightly larger than the derived series localized at $\mathcal{P}$. In particular,

[TABLE]

Then we consider the coefficient system on $W_{0}$ given by the projection

[TABLE]

The bulk of the proof ($14$ pages!) will be to show that:

[TABLE]

[TABLE]

We now show that (7.1), (7.2) and (7.3) imply Theorem 5.5. Consider the von Neumann signature defect of ($W_{0},\phi$):

[TABLE]

By the additivity of these signatures (property ($5$) of Proposition 4.1), this quantity is the sum of the signature defects for $V$ and those of the $E_{i}$ and $\overline{E}_{i}$. Note that the coefficient system on $\pi_{1}(V)$ factors

[TABLE]

where we used Theorem 3.10 to establish the second map and we use (7.1) for the third map. Thus, since $V$ is an $(n.5,\mathcal{P})$-solution, the signature defect of $V$ vanishes by Theorem 4.2. All of the signature defects of the $E_{i}$ vanish by [10, Lemma 2.4] (essentially because $H_{2}(E)$ comes from $H_{2}(\partial E)$). Therefore the signature defect vanishes for $W_{0}$. On the other hand, by Section 4,

[TABLE]

Hence

[TABLE]

By (7.2) and properties ($1$) and ($3$) of Proposition 4.1,

[TABLE]

while by (7.3) and properties ($1$) and ($2$) of Proposition 4.1

[TABLE]

But, by choice, $|\rho_{0}(K^{0})|$ is greater than twice the sum of the Cheeger-Gromov constants of the $3$-manifolds $M_{\mathfrak{R}^{m}},\dots,M_{R^{1}}$, which is a contradiction (see property ($6$) of Proposition 4.1).

Therefore the proof of Theorem 5.5 is reduced to defining a commutator series $\{\pi^{(n)}_{\mathcal{S}}\}$ such that (7.1), (7.2) and (7.3) hold.

The commutator series $\pi^{(j)}_{\mathcal{S}}$ will be defined only for the groups $\pi=\pi_{1}(W_{i})$, because we need not be concerned with any other groups. It will be defined exactly as in Definition 3.6 except that the sequence of right divisor sets $S_{1},...,S_{n}$ will be slightly different than those of Definition 3.7. We now define $S_{1},...,S_{n}$. In these definitions $\pi$ is the fundamental group of one of the $W_{i}$.

We define

[TABLE]

(Note that $\pi^{(1)}=\pi^{(1)}_{r}=\pi^{(1)}_{\mathcal{P}}=\pi^{(1)}_{\mathcal{S}}$.) Before defining the other $S_{i}$ we make a few remarks. Since $p_{1}(t)$ is a knot polynomial, $p_{1}(t)\doteq p_{1}(t^{-1})$, so $S_{1}$ is closed (up to units) under the natural involution. In fact, since $p_{1}(t)=\Delta_{m}(t)$ is the Alexander polynomial of $\mathbb{E}^{m}$, $p_{1}(t)$ is prime. Hence one sees that

[TABLE]

Therefore for any $\mathbb{Q}[\mu^{\pm 1}]$-module $M$, $MS^{-1}_{1}=M_{\langle\Delta_{m}\rangle}$, the classical localization of $\boldsymbol{M}$ at the prime ideal $\boldsymbol{\langle\Delta_{m}\rangle}$.

Therefore, by (3.1),

[TABLE]

where $\boldsymbol{\mathcal{A}(W)_{(\Delta)}}$ is the classical localization of $\mathcal{A}(W)$ at the prime $\langle\Delta_{m}\rangle$. (If $W$ is any space with $\pi_{1}(W)=\pi$ and $H_{1}(W)\cong\mathbb{Z}$ then by its integral Alexander module, denoted $\mathcal{A}^{\mathbb{Z}}(W)$ we mean $H_{1}(W;\mathbb{Z}[\mu,\mu^{-1}])\cong\pi^{(1)}/\pi^{(2)}$. By its rational Alexander module, denoted $\mathcal{A}(W)$, we mean $H_{1}(W;\mathbb{Q}[\mu,\mu^{-1}])$.)

Now let $\Gamma\equiv\pi/\pi^{(2)}_{\mathcal{S}}\equiv\pi/\pi^{(2)}_{\mathcal{P}}$ and $A=\pi^{(1)}/\pi^{(2)}_{\mathcal{S}}\equiv\pi^{(1)}/\pi^{(2)}_{\mathcal{P}}\subset\Gamma$. Thus $\Gamma$ is the semidirect product of the abelian group $A$ with $\pi/\pi^{(1)}\cong\mathbb{Z}$. Note that the circle $\eta_{2}$ (see Figure 2.9) represents an element of $\pi_{1}(M_{K^{n}})^{(1)}$ and hence, under inclusion, an element of $\pi^{(1)}$ for each of the groups $\pi=\pi_{1}(W_{i})$ under consideration. Hence, for any $\pi$, $\eta_{2}$ has an unambiguous interpretation as an element of $A$. By abuse of notation we allow $\eta_{2}$ to stand for its image in any of the appropriate groups. Recall that a set $S\subset\Gamma$ is $\boldsymbol{\Gamma}$-invariant if $gsg^{-1}\in S$ for all $s\in S$ and $g\in\Gamma$. Note that the set $\{\mu^{i}\eta_{2}\mu^{-i}~{}|~{}i\in\mathbb{Z}\}$ is $\Gamma$-invariant where $\mu\in\Gamma$ generates $\pi/\pi^{(1)}$. Then we define the other $S_{n}$ as follows:

Definition 7.1**.**

Let $S_{2}=S_{2}(\pi)\subset\mathbb{Q}[\pi^{(1)}/\pi^{(2)}_{\mathcal{S}}]\subset\mathbb{Q}[\pi/\pi^{(2)}_{\mathcal{S}}]$ be the multiplicative set generated by

[TABLE]

and for $2<i\leq n$ let

[TABLE]

Since $S_{2}$ is a multiplicative subset of $\mathbb{Q}A$ that is $\Gamma$-invariant, it is a right divisor set of $\mathbb{Q}\Gamma$ by [8, Proposition 4.1]. Therefore Definition 3.6 applies to give a partially defined commutator series $\{\pi^{(i)}_{\mathcal{S}}\}$. Since $p_{2}(t)=q_{n-1}(t)$ is a knot polynomial, $p_{2}(t)\doteq p_{2}(t^{-1})$. Thus $S_{2}$ is closed (up to units) under the natural involution.

Lemma 7.2**.**

For each $\pi$ and each $0\leq i\leq n+1$

[TABLE]

Proof.

The proof is by induction on $i$. By Definition 3.6, $\pi^{(1)}_{\mathcal{P}}=\pi^{(1)}_{r}=\pi^{(1)}_{\mathcal{S}}$, so the Lemma is true for $i=0,1$. Suppose it is true for all values up to some fixed $i\geq 1$. Let $j:\pi\to\pi$ be the identity map. By  [8, Proposition 3.2], it suffices to show that the induced ring map

[TABLE]

has the property that $j_{*}(S_{p_{i}}(\pi))\subset S_{i}(\pi)$. For $i=1$, $j_{*}$ is the identity map and $S_{1}(\pi)$ is, by definition, identical to $S_{p_{1}}(\pi)$. It follows that $\pi^{(2)}_{\mathcal{P}}=\pi^{(2)}_{\mathcal{S}}$ as already observed in  (7.4). Thus, for $i=2$, $j_{*}$ is again the identity map and, by Definitions 7.1 and  3.7, $S_{2}(\pi)$ strictly contains $S_{p_{2}}(\pi)$. For $i>2$, the map $j_{*}$, although induced by the identity, will be a surjection with non-zero kernel. Nonetheless, by the inductive hypothesis, $j$ induces a homomorphism

[TABLE]

Recall from Definition 3.7 that

[TABLE]

which is the multiplicative set generated by the described set of polynomials $q(a)$. If $q(a)$ is any such polynomial then $j_{*}(q(a))=q(j_{*}(a))$ and since

[TABLE]

Thus, upon examining  (7.5), we see that $q(j_{*}(a))\in S_{i}(\pi)$