2.1. Classification of linking forms

A linking form will refer to a pair $(M,\lambda )$ , where $M$ is a torsion $\mathbb{C}[t^{\pm 1}]$ -module and $ \lambda \colon M \times M\to \mathbb{C}(t)/\mathbb{C}[t^{\pm 1}]$ is a non-degenerate sesquilinear Hermitian pairing. In order to state our classification of linking forms, we recall some terminology.

While it is helpful to work with arbitrary $\xi$ -positive polynomials, having a concrete example sometimes also proves useful: if $\xi \neq \pm 1$ , then $r(t)=1-\xi t$ is $\xi$ -positive if $\textrm{Im}(\xi )\gt 0$ and $-(1-\xi t)$ is $\xi$ -positive if $\textrm{Im}(\xi )\lt 0$ . Next, we refer to

\begin{equation*}\textsf {C}_{\xi }(t) = \begin {cases} (t-\xi ), & \text {if } |\xi |=1,\\[5pt] (t-\xi )(t^{-1}-\xi ), & \text {if } |\xi | \in (0,1), \end {cases}\end{equation*}

as basic polynomials and define the building blocks for our classification of linking forms.

Definition 2.2. Fix an integer $n\gt 0$ and $\epsilon = \pm 1$ . For $|\xi |=1$ , the basic linking form $\mathfrak{e}(n,\epsilon,\xi,\mathbb{C})$ is defined as

(2.1) \begin{align} \mathbb{C}[t^{\pm 1}]/ \textsf{C}_{\xi }(t)^n \times \mathbb{C}[t^{\pm 1}]/ \textsf{C}_{\xi }(t)^n &\to \mathbb{C}(t)/\mathbb{C}[t^{\pm 1}], \nonumber \\[5pt] (x,y) &\mapsto \frac{\epsilon x y^{\#}}{\textsf{C}_{\xi }(t)^{\frac{n}{2}}\textsf{C}_{\bar{\xi }}(t^{-1})^{\frac{n}{2}}}, \quad \ \ \ \ \ \text{if}\;n\;\text{is even}, \end{align}

(2.2) \begin{align} (x,y) &\mapsto \frac{\epsilon r(t) x y^{\#}}{\textsf{C}_{\xi }(t)^{\frac{n+1}{2}}\textsf{C}_{\bar{\xi }}(t^{-1})^{\frac{n-1}{2}}}, \quad \text{if } n \text{ is odd} \end{align}

where $r(t)$ is a $\xi$ -positive polynomial Footnote 1. For $|\xi |\lt 1$ , the basic linking form $\mathfrak{f}(n,\xi,\mathbb{C})$ is

(2.3) \begin{align} \mathbb{C}[t^{\pm 1}]/ \textsf{C}_{\xi }(t)^n \times \mathbb{C}[t^{\pm 1}]/ \textsf{C}_{\xi }(t)^n &\to \mathbb{C}(t)/\mathbb{C}[t^{\pm 1}], \nonumber \\[5pt] (x,y) &\mapsto \frac{x y^{\#}}{\textsf{C}_{\xi }(t)^{n}}. \end{align}

We can now state the classification of linking forms over $\mathbb{C}[t^{\pm 1}]$ that was proved in [Reference Borodzik, Conway and Politarczyk2].

Theorem 2.3. Every linking form $(M,\lambda )$ over $\mathbb{C}[t^{\pm 1}]$ can be presented as a direct sum

\begin{equation*}(M,\lambda )= \bigoplus _{\substack {n_i,\epsilon _i,\xi _i\\ i\in I}}\mathfrak {e}(n_i,\epsilon _i,\xi _i,\mathbb {C})\oplus \bigoplus _{\substack {\xi _j\\ j\in J}}\mathfrak {f}(n_j,\xi _j,\mathbb {C})\end{equation*}

for some finite set of indices $I$ and $J$ . Here, the $n_i\ge 0$ are integers, the $\epsilon _i$ equal $\pm 1$ and the $\xi _i$ and $\xi _j$ are non-zero complex numbers. The presentation is unique up to permuting summands.

2.2. Signatures of linking forms

Given $n \geq 0,\xi \in \mathbb{C}$ and $\epsilon =\pm 1$ , we define the Hodge number $\mathcal{P}(n,\epsilon,\xi,\mathbb{C})$ of a linking form $(M,\lambda )$ over $\mathbb{C}[t^{\pm 1}]$ as the number of times $\mathfrak{e}(n,\epsilon,\xi,\mathbb{C})$ enters the decomposition in Theorem 2.3.

Definition 2.4. The signature jump of a linking form $(M,\lambda )$ over $\mathbb{C}[t^{\pm 1}]$ at $\xi \in S^1$ is defined as the following integer:

(2.4) \begin{equation} \delta \sigma _{(M,\lambda )}(\xi )=\sum _{\substack{n \;\textrm{ odd}\\ \epsilon =\pm 1}} \epsilon \mathcal{P}(n,\epsilon,\xi,\mathbb{C}). \end{equation}

We collect some properties of signature jumps for later use; details can be found in [Reference Borodzik, Conway and Politarczyk2].

Next, we describe how to calculate signature jumps algorithmically. This underlies several of the steps of the algorithm from Subsection 1.5 of the introduction.

In the introduction, given a linking form $(M,\lambda )$ , we not only mentioned the signature jumps $\delta \sigma _{(M,\lambda )}$ but also the signature function $\sigma _{(M,\lambda )} \colon S^1 \to \mathbb{Z}$ . While the use of the latter is conceptually enlightening, this paper only makes use of the former.

3.1. Twisted Blanchfield pairings

Let $X$ be a space with universal cover $p \colon \widetilde{X} \to X$ . We assume that $X$ has the homotopy type of a finite CW complex. The left action of $\pi _1(X)$ on $\widetilde{X}$ endows the singular chain groups of $C_*(\widetilde{X})$ with the structure of left $\mathbb{Z}[\pi _1(X)]$ -modules. Given a representation $\beta \colon \pi _1(X)\to GL_d({\mathbb{F}[t^{\pm 1}]})$ , use ${\mathbb{F}[t^{\pm 1}]}_\beta ^d$ to denote the $({\mathbb{F}[t^{\pm 1}]},\mathbb{Z}[\pi _1(X)])$ -bimodule whose right $\mathbb{Z}[\pi _1(X)]$ -module structure is given by right multiplication by $\beta (\gamma )$ on row vectors. The chain complexes

\begin{align*} C_*(X;\;{\mathbb{F}[t^{\pm 1}]}_\beta ^d)&\;:\!=\;{\mathbb{F}[t^{\pm 1}]}_\beta ^d \otimes _{\mathbb{Z}[\pi _1(X)]} C_*(\widetilde{X}) \\[5pt] C^*(X;\;{\mathbb{F}[t^{\pm 1}]}_\beta ^d)&\;:\!=\;\textrm{Hom}_{\operatorname{right}-\mathbb{Z}[\pi _1(X)]}( C_*(\widetilde{X})^\#,{\mathbb{F}[t^{\pm 1}]}_\beta ^d) \end{align*}

of left $\mathbb{F}[t^{\pm 1}]$ -modules will be called the (co)chain complexes of $X$ twisted by $\beta$ . The corresponding homology of left $\mathbb{F}[t^{\pm 1}]$ -modules $H_*(X;\;{\mathbb{F}[t^{\pm 1}]}_\beta ^d)$ and $H^*(X;\;{\mathbb{F}[t^{\pm 1}]}_\beta ^d)$ will be called the (co)homology of $X$ twisted by $\beta$ . The representation $\beta$ is acyclic if the chain complex $\mathbb{F}(t) \otimes _{{\mathbb{F}[t^{\pm 1}]}} C_*(X;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta )$ is acyclic and unitary if $\beta (\gamma )=\beta (\gamma ^{-1})^{\#T}$ .

We now assume that $N$ is a closed, oriented $3$ -manifold. If $\beta \colon \pi _1(N) \to GL_d({\mathbb{F}[t^{\pm 1}]})$ is a representation that is both acyclic and unitary, then the $\mathbb{F}[t^{\pm 1}]$ -module $H_1(N;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta )$ is endowed with a linking form

(3.1) \begin{equation} \textrm{Bl}_{\beta }(N) \colon H_1(N;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta ) \times H_1(N;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta ) \to \mathbb{F}(t)/{\mathbb{F}[t^{\pm 1}]}, \end{equation}

called a twisted Blanchfield pairing. The definition of this pairing on $x,y \in H_1(N;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta )$ is as $\textrm{Bl}_\beta (N)(x,y)=\Theta (y)(x)$ where $\Theta$ is the composition

\begin{align*} \Theta \colon H_1(N;\;\mathbb{F}[t^{\pm 1}]^d_\beta ) \, & \xrightarrow{\operatorname{PD}^{-1}}\, H^2(N;\;\mathbb{F}[t^{\pm 1}]^d_\beta ) \\[5pt] & \xrightarrow{\operatorname{BS}^{-1}} H^1(N;\;(\mathbb{F}(t)/\mathbb{F}[t^{\pm 1}])^d_\beta ) \\[5pt] & \xrightarrow{\text{ev}} \textrm{Hom}_{\mathbb{F}[t^{\pm 1}]}(H_1(N;\;\mathbb{F}[t^{\pm 1}]^d_\beta ),\mathbb{F}(t)/\mathbb{F}[t^{\pm 1}])^\# \end{align*}

of the inverse of the Poincaré duality isomorphism, the inverse of a Bocktein isomorphism and an evaluation map.

While we do not give further details on the definition of this pairing (referring instead to [Reference Borodzik, Conway and Politarczyk1, Reference Miller and Powell33, Reference Powell36]), the next subsection instead describes an algorithm to compute its value on any pair of elements of $H_1(N;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta )$ .

3.2. A review of Powell’s algorithm

We briefly recall Powell’s algorithm to compute the Blanchfield pairing [Reference Powell35]. In [Reference Powell36] Powell defines twisted Blanchfield pairings for arbitrary $3$ -dimensional symmetric chain complexes. When $N$ is a closed, oriented $3$ -manifold and $\beta \colon \pi _1(N) \to GL_d({\mathbb{F}[t^{\pm 1}]})$ is a unitary acyclic representation, his definition yields a non-singular linking form

\begin{equation*} \textrm {Bl}^{\beta }(N) \colon H^2(N;\;{\mathbb {F}[t^{\pm 1}]}^d_\beta ) \times H^2(N;\;{\mathbb {F}[t^{\pm 1}]}^d_\beta ) \to \mathbb {F}(t)/{\mathbb {F}[t^{\pm 1}]} \end{equation*}

on the twisted cohomology of $N$ whose relation to $\textrm{Bl}_\beta (N)$ is described in Remark 3.3 below. We now focus on the algorithm described in [Reference Miller and Powell33, Reference Powell35] to compute $\textrm{Bl}^{\beta }(N)$ , and take the result as our definition of $ \textrm{Bl}^{\beta }(N)$ .

Fix a handle decomposition of $N$ and choose a chain representative $[N] \in C_3(N)$ for the fundamental class of $N$ . Here, as indicated in Remark 3.1, we are technically working in the chain complex of a space homotopy equivalent to $N$ . Use $\widetilde{N}$ to denote the universal cover of $N$ and let $(C^*(\widetilde{N}), \partial ^*)$ be the resulting cochain complex of left $\mathbb{Z}[\pi _1(N)]$ -modules. As explained in [Reference Miller and Powell33, Proposition 2.10] the choice of $[N]$ together with the symmetric construction [Reference Ranicki37] leads to a $\mathbb{Z}[\pi _1(N)]$ -chain homotopy equivalence

(3.2) \begin{equation} \Phi \colon C^{3-*}(\widetilde{N}) \to C_*(\widetilde{N}) \end{equation}

which should be thought of as a chain level version of Poincaré duality. We will not delve into the details of the symmetric construction, but instead note that passing to twisted chain complexes, $\partial ^*$ and $\Phi$ induce maps

\begin{align*} &\beta (\partial ^*) \colon C^*(N;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta ) \to C^{*+1}(N;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta ), \\[5pt] &\beta (\Phi ) \colon C^{3-*}(N;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta ) \to C_{*}(N;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta ). \end{align*}

For later use, we also recall that matrices are assumed to act on row vectors from the right and that the cohomological differentials are determined by the homological differentials via the formula $\beta (\partial ^{i}) = (-1)^{i} \beta (\partial _{i})^{\#T}$ .

The following definition is due to Powell [Reference Powell36] (see also [Reference Miller and Powell33]).

Definition 3.2. Let $N$ be a closed, oriented $3$ -manifold and let $\beta \colon \pi _1(N) \to GL_d({\mathbb{F}[t^{\pm 1}]})$ be a unitary acyclic representation. Fix a handle decomposition of $N$ and a chain representative $[N] \in C_3(N)$ of the fundamental class and let $\Phi \colon C^{3-*}(\widetilde{N}) \to C_*(\widetilde{N})$ be the chain homotopy equivalence resulting from the symmetric construction. The cohomological twisted Blanchfield is defined as

(3.3) \begin{align} \textrm{Bl}^{\beta }(N) \colon H^2(N;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta ) \times H^2(N;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta ) &\to \mathbb{F}(t)/{\mathbb{F}[t^{\pm 1}]} \nonumber \\[5pt] ([v],[w]) & \mapsto \frac{1}{s} \left (v \cdot \beta (\Phi ) \cdot Z^{\# T}\right )^{\# T}, \end{align}

where $v,w \in C^2(N;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta )$ and $Z \in C^1(N;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta )$ satisfies $Z\beta (\partial ^2)=sw$ for some $s \in \,{\mathbb{F}[t^{\pm 1}]} \setminus \lbrace 0 \rbrace$ . The fact that this pairing does not depend on any of the choices involved was proved in[ Reference Powell36].

The next remark summarises the relation between the cohomological pairing of Definition 3.2 and the homological pairing mentioned in Subsection 3.1.

Remark 3.3. The cohomological twisted Blanchfield pairing $\textrm{Bl}^{\beta }(N)$ is isometric to the twisted Blanchfield pairing $\textrm{Bl}_{\beta }(N)$ from Subsection 3.1 . Indeed [Reference Miller and Powell33 , Proposition 5.3] implies that for $x,y \in H_1(N;\;{\mathbb{F}[t^{\pm 1}]}^d_\beta )$ , the pairings are related by

\begin{equation*} \textrm {Bl}^{\beta }(N)(x,y) = \textrm {Bl}_{\beta }(N)(\Phi ^{-*}(x),\Phi ^{-*}(y)),\end{equation*}

where $\Phi ^{-*}=(\Phi ^*)^{-1}$ denotes the inverse of the homomorphism on (co)homology induced by $\Phi$ . We emphasise that the reader should consider Definition 3.2 as a computational device: as far as the definitions go, the approach outlined in Subsection 3.1 is more satisfactory.

3.3. Twisted signatures

We briefly recall the definition of the twisted signature invariants from [Reference Borodzik, Conway and Politarczyk1]. As we mentioned in the introduction, given a knot $K$ and a unitary acyclic representation $\beta \colon \pi _1(M_K) \to GL_d(\mathbb{C}[t^{\pm 1}])$ , the twisted signature jump of $(K,\beta )$ is the signature jump of the linking form $\textrm{Bl}_\beta (K)$ :

\begin{equation*} \delta \sigma _{K,\beta }(\xi )\;:\!=\;\delta _{(H_1(M_K;\;\mathbb {C}[t^{\pm 1}]^n_\beta ), \textrm {Bl}_\beta (K))}.\end{equation*}

Properties of $\delta \sigma _{K,\beta }$ can be deduced from the properties of $\delta _{(M,\lambda )}$ listed in Theorem 2.5. We record two additional remarks for later use:

4.1. Metabelian representations

The abelianisation homomorphism $\phi \colon \pi _1(M_K)\xrightarrow{\cong }\mathbb{Z}=\langle t_K\rangle$ endows $\mathbb{Z}[t_K^{\pm 1}]$ with a right $\mathbb{Z}[\pi _1(M_K)]$ -module structure. This gives rise to the twisted homology $\mathbb{Z}[t_K^{\pm 1}]$ -module $H_1(M_K;\;\mathbb{Z}[t_{K}^{\pm 1}])$ . As in [Reference Friedl and Powell13, Corollary 2.4], identify $H_1(\Sigma _n(K);\;\mathbb{Z})$ with the quotient module $H_1(M_K;\;\mathbb{Z}[t_K^{\pm 1}])/(t_K^n-1)$ . Consider the semidirect product $\mathbb{Z} \ltimes H_1(\Sigma _n(K);\;\mathbb{Z})$ , where the group law is given by $ (t_K^i,v)\cdot (t_K^j,w)=(t_K^{i+j},t_K^{-j}v+w)$ . Next, let $\gamma _K(n,\chi )$ be the homomorphism:

(4.1) \begin{align} \gamma _K(n,\chi ) \colon \mathbb{Z} \ltimes H_1(\Sigma _n(K);\;\mathbb{Z}) &\to \, \operatorname{GL}_n(\mathbb{C}[t^{\pm 1}]) \nonumber \\[5pt] (t_K^j,v) &\mapsto \begin{pmatrix} 0\;\;\;& 1\;\;\; & \cdots\;\;\; &0 \\[5pt] \vdots\;\;\; & \vdots\;\;\; & \ddots\;\;\; & \vdots \\[5pt] 0\;\;\; & 0\;\;\; & \cdots\;\;\; & 1 \\[5pt] t\;\;\; & 0\;\;\; & \cdots\;\;\; & 0 \end{pmatrix}^j \begin{pmatrix} \xi _{m}^{\chi (v)}\;\;\; & 0\;\;\; & \cdots\;\;\; &0 \\[5pt] 0\;\;\; & \xi _{m}^{\chi (t_K \cdot v)}\;\;\; & \cdots\;\;\; &0 \\[5pt] \vdots\;\;\; & \vdots\;\;\; & \ddots\;\;\; & \vdots \\[5pt] 0\;\;\; & 0\;\;\; & \cdots\;\;\; & \xi _{m}^{\chi (t_K^{n-1} \cdot v)} \end{pmatrix}. \end{align}

Identify the module $H_1(M_K;\;\mathbb{Z}[t_{K}^{\pm 1}])$ with the quotient $\pi _1(M_K)^{(1)}/\pi _1(M_K)^{(2)}$ and consider the following composition of canonical projections

(4.2) \begin{equation} q_K \colon \pi _{1}(M_{K})^{(1)} \to H_1(M_K;\;\mathbb{Z}[t_K^{\pm 1}]) \to H_1(\Sigma _n(K);\;\mathbb{Z}). \end{equation}

Fix an element $\mu _{K}$ in $\pi _1(M_K)$ such that $\phi _K(\mu _{K})=t_K$ . For every $g \in \pi _1(M_K)$ , we have $\phi _K(\mu _K^{-\phi _K(g)}g)=1$ and so $\mu _K^{-\phi _K(g)}g\in \pi _1(M_K)^{(1)}$ . As a consequence, we obtain the following map:

\begin{align*} \widetilde{\rho }_K \colon \pi _1(M_K) &\to \mathbb{Z} \ltimes H_1(\Sigma _n(K);\;\mathbb{Z}) \\[5pt] g &\mapsto (\phi _K(g),q_K(\mu _{K}^{-\phi _K(g)}g)). \end{align*}

The unitary representation $\alpha _K(n,\chi )$ is obtained as the composition

\begin{equation*} \alpha _K(n,\chi ) \colon \pi _1(M_K) \stackrel {\widetilde {\rho }_K}{\to } \mathbb {Z} \ltimes H_1(\Sigma _n(K);\;\mathbb {Z}) \stackrel {\gamma _K(n,\chi )}{\longrightarrow } \operatorname {GL}_n(\mathbb {C}[t^{\pm 1}]). \end{equation*}

This representation is unitary, and if $m$ is a prime power and $\chi \colon H_1(\Sigma _n(K);\;\mathbb{Z}) \to \mathbb{Z}_m$ is non-trivial, then $\alpha _K(n,\chi )$ is acyclic; see [Reference Friedl and Powell12] and [Reference Miller and Powell33, Lemma 6.6]. Furthermore, when the knot is clear from the context, we will often write $\alpha (n,\chi )$ instead of $\alpha _K(n,\chi )$ .

The relevance of $\textrm{Bl}_{\alpha (n,\chi )}(K)$ to knot concordance stems from the following theorem of Miller-Powell [Reference Miller and Powell33, Theorem 6.9] which itself builds on work of Casson and Gordon [Reference Casson and Gordon7].

The take-away from Theorem 4.2 is that if we can find enough representations $\chi$ for which $\textrm{Bl}_{\alpha (n,\chi _b)}(K)$ is not metabolic, then we can show that $K$ is not slice. To obstruct $\textrm{Bl}_{\alpha (n,\chi _b)}(K)$ from being metabolic, we will use signature jumps.

4.2. The satellite formula for the metabelian Blanchfield form

Let $P,K \subset S^{3}$ be knots, and let $\eta$ be a simple closed curve in the complement of $P$ . The satellite knot $P(K,\eta )$ with pattern $P$ , companion $K$ , and infection curve $\eta$ is the image of $P$ under the diffeomorphism $(S^{3}\setminus \mathcal{N}(\eta )) \cup _{\partial } (S^{3} \setminus \mathcal{N}(K)) \cong S^{3}$ , where the gluing of the exteriors of $\eta$ and $K$ identifies the meridian of $\eta$ with the zero-framed longitude of $K$ and vice versa. The zero-surgery on the satellite knot $P(K,\eta )$ can be obtained by the infection of $M_{P}$ by $K$ along $\eta \subset S^{3} \setminus P \subset M_{P}$ :

(4.3) \begin{equation} M_{P(K,\eta )}=M_P \setminus \mathcal{N}(\eta ) \cup _\partial S^3 \setminus \mathcal{N}(K). \end{equation}

Let $\mu _{\eta }$ denote a meridian of $\eta$ . A representation $\beta \colon \pi _{1}(M_{P(K,\eta )}) \to GL_{d}(\mathbb{F}[t^{\pm 1}])$ induces representations on $ \pi _{1}(M_{P} \setminus \mathcal{N}(\eta ))\,$ and $\pi _{1}(S^{3} \setminus \mathcal{N}(K))$ . In turn, as explained in [Reference Borodzik, Conway and Politarczyk1, Section 3.3], these representations extend to representations

\begin{equation*}\beta _{P} \colon \pi _{1}(M_{P}) \to GL_{d}(\mathbb {F}[t^{\pm 1}]), \quad \beta _{K} \colon \pi _{1}(M_K) \to GL_{d}(\mathbb {F}[t^{\pm 1}]).\end{equation*}

provided $\beta (\mu _{\eta }) = \textrm{id}$ and $\det\!(\textrm{id} - \beta (\eta )) \neq 0$ , in which case we say that $\beta$ is $\eta$ -regular.

Thus, given a character $\chi \colon H_1(\Sigma _n(P(K,\eta ));\;\mathbb{Z}) \to \mathbb{Z}_m$ , if the representation $\alpha (n,\chi )$ is $\eta$ -regular, then it gives rise to representations $\alpha (n,\chi )_P$ on $\pi _1(M_P)$ and $\alpha (n,\chi )_K$ on $\pi _1(M_K)$ . The representation $\alpha (n,\chi )_P$ can be shown to agree with $\alpha (n,\chi _P)$ , where $\chi _P$ is the character induced by $\chi$ on $H_1(\Sigma _n(P);\;\mathbb{Z})$ , as in [Reference Litherland29, Section 4]. The main step in the proof of the metabelian satellite formula is to decompose $\alpha (n,\chi )_K$ into $h\;:\!=\;\operatorname{gcd}(n,w)$ representations. To describe the outcome, for $i=1,\ldots,h$ , one considers the character

\begin{align*} \chi _i \colon H_1(\Sigma _{n/h}(K);\;\mathbb{Z}) &\to \mathbb{Z}_m \\[5pt] v &\mapsto \chi (t_Q(\iota _n(v))), \end{align*}

where $t_Q$ denotes the generator of the deck transformation group of the infinite cyclic cover of $M_{P(K,\eta )}$ and $\iota _n \colon H_1(\Sigma _{n/h}(K);\;\mathbb{Z}) \to H_1(\Sigma _{n}(P(K,\eta ));\;\mathbb{Z})$ is inclusion induced; we refer to [Reference Litherland29] and [Reference Borodzik, Conway and Politarczyk1] for further details on this later map.

Additionally, use $\mu _Q$ to denote a meridian of $P(K,\eta )$ and define the map

\begin{equation*} q_Q \colon \pi _1^{(1)}(M_{P(K,\eta )}) \to H_1(\Sigma _n(P(K,\eta ));\;\mathbb {Z})\end{equation*}

as in (4.2). The satellite formula for the metabelian Blanchfield form now reads as follows in the winding number $w\;:\!=\;\ell k(\eta,P) \neq 0$ case; we refer to [Reference Borodzik, Conway and Politarczyk1] for the general statement.

Theorem 4.3 takes a particularly simple form for connected sums. In this case, we have $w=1$ (so $h=1$ ) as well as $\eta =\mu _P$ .

5.1. Crossed modules

In this section, we give a brief summary of the theory of crossed modules. For more details refer to the Part 1 of [Reference Brown, Higgins and Sivera5]. The reason for considering crossed modules is that for $2$ -complexes, they provide a convenient formalism to relate identities of presentations (which we need to calculate the third differential and the symmetric structure of our chain complex) and $\pi _2$ ; see Proposition 5.9.

Let us note the following facts whose proofs are left to the reader.

Example 5.4. Let $\mathcal{P} = \langle \textbf{x} \mid \textbf{r} \rangle$ be a finite presentation of a group $\pi$ , let $X$ denote the $2$ -dimensional presentation CW complex of $\mathcal{P}$ and denote by $X^{1}$ the $1$ -skeleton of $X$ . It is known (see e.g. [Reference Whitehead40]) that the pair $ (\pi _{2}(X,X^{1}),\partial )$ is a crossed $\pi _{1}(X^{1})$ -module, where

\begin{equation*}\partial \colon \pi _{2}(X,X^{1}) \to \pi _{1}(X^{1}),\end{equation*}

is the connecting homomorphism from the long exact sequence of the pair $(X,X^{1})$ .

Whitehead [Reference Whitehead40 , Section 16] proved that $(\pi _{2}(X,X^{1}),\partial )$ is isomorphic to the free crossed $\pi _{1}(X^{1})$ -module generated by the set $R= \{f^{2}_{r} \colon r \in \textbf{r}\}$ of characteristic maps $f^{2}_{r} \colon (D^{2},\partial D^{2}) \to (X,X^{1})$ of the $2$ -cells $\{e^{2}_{r} \colon r \in \textbf{r}\}$ of $X$ . Furthermore, the map $ m \colon R \to \pi _{1}(X^{1})$ is given by the formula $m(f^{2}_{r}) = r$ .

5.2. Identities of presentations

This section is concerned with identities of group presentations. As will become apparent in the next section, this is the algebra that underlies the calculation of the third differential in the chain complex of the universal cover of a $3$ -dimensional CW complex.

Let $\mathcal{P} = \langle \textbf{x} \mid \textbf{r} \rangle$ be a presentation of a group $G$ , let $F$ be the free group generated by $\textbf{x}$ , and let $P=\langle \rho _{r} \colon r \in \textbf{r} \rangle$ be the free group generated by symbols $\rho _{r}$ , for $r \in \textbf{r}$ . Moreover, following Trotter [Reference Trotter39, Section 2.1], consider the homomorphism

\begin{equation*}\psi \colon F \ast P \to F\end{equation*}

defined on generators by $\psi (x)=x$ , for $x \in \textbf{x}$ , and $\psi (\rho _{r}) = r$ for $r \in \textbf{r}$ .

Definition 5.5. Let $\mathcal{P} = \langle \textbf{x} \mid \textbf{r} \rangle$ be a presentation of a group $G$ . Denote by $N(\mathcal{P})$ the normal subgroup of $F \ast P$ generated by $P$ :

\begin{equation*} N(\mathcal {P})\;:\!=\;\langle \langle P \rangle \rangle \trianglelefteq F \ast P.\end{equation*}

An identity of the presentation $\mathcal{P}$ is an element of $\ker\!(\psi ) \cap N(\mathcal{P})$ . We write $I(\mathcal{P})$ for the group of identities of $\mathcal{P}$ :

\begin{equation*} I(\mathcal {P})\;:\!=\;\ker\!(\psi ) \cap N(\mathcal {P}).\end{equation*}

More explicitly, an identity is an element of $\ker\!(\psi )$ , which can be written as a product of words of the form $w \rho ^{\epsilon }_{r} w^{-1}$ , where $w$ lies in $F$ , $\epsilon = \pm 1$ , and $r \in \textbf{r}$ .

Construction 5.6. Given a group $G$ , following Trotter [Reference Trotter39 , page 473], we outline the definition of a 3-dimensional chain complex of free left $\mathbb{Z}[G]$ -modules

\begin{equation*}C_\bullet (\textbf {x},\textbf {r},\textbf {s})=\left ( C_3 \xrightarrow {\partial _3} C_2 \xrightarrow {\partial _2} C_1 \xrightarrow {\partial _1} C_0 \right )\end{equation*}

associated with a presentation $\mathcal{P}=\langle \textbf{x} \mid \textbf{r}\rangle$ of $G$ and a set $\textbf{s}$ of identities of $\mathcal{P}$ . The chain complex $C_\bullet (\textbf{x},\textbf{r},\textbf{s})$ satisfies $H_{0}(C_\bullet (\textbf{x},\textbf{r},\textbf{s})) \cong \mathbb{Z}$ (where $\mathbb{Z}$ is endowed with the $\mathbb{Z}[G]$ -module structure induced by augmentation), $H_{1}(C_\bullet (\textbf{x},\textbf{r},\textbf{s})) = 0$ .

The chain module $C_0$ is free of rank one on an element $v$ , $C_1$ is free on elements $\{a_{x} \colon x \in \textbf{x}\}$ , $C_2$ is free on elements $\{b_{r} \colon r \in \textbf{r}\}$ , and $C_3$ is free on the set $\{c_{s} \colon s \in \textbf{s}\}$ . The differentials are defined in terms of Fox derivatives

\begin{align*} \partial _{1}(a_{x}) &= (x-1)v, \quad \text{ for\ } x \in \textbf{x}, \\[5pt] \partial _{2}(b_{r}) &= \sum _{x \in \textbf{x}} \frac{\partial r}{\partial x} a_{x}, \quad \text{ for\ } r \in \textbf{r}, \\[5pt] \partial _{3}(c_{s}) &= \sum _{r \in \textbf{r}} \frac{\partial s}{\partial \rho _{r}} b_{r}, \quad \text{ for\ } s \in \textbf{s}, \end{align*}

where the Fox derivatives $\frac{\partial s}{\partial \rho _{r}}$ are computed in $F \ast P$ . The fact that $H_{0}(C_\bullet (\textbf{x},\textbf{r},\textbf{s})) \cong \mathbb{Z}$ and $H_{1}(C_\bullet (\textbf{x},\textbf{r},\textbf{s})) = 0$ follows because Fox derivatives calculate the differentials in the cellular chain complex of the universal cover.

When $\textbf{s}$ is a complete set of identities for a presentation $\mathcal{P}$ , we can think of $C_\bullet (\textbf{x},\textbf{r},\textbf{s})$ as a truncation of a free resolution of the trivial $\mathbb{Z}[G]$ -module $\mathbb{Z}$ .

5.3. Identities and 3-dimensional CW complexes

We use identities to describe the chain complex of the universal cover of a $3$ -dimensional CW complex.

Let $\mathcal{P} = \langle \textbf{x} \mid \textbf{r} \rangle$ be a presentation of a group $G$ . As in the previous section, we write $F$ for the free group on the set $\textbf{x}$ , $P$ for the free group generated by symbols $\rho _{r}$ with $r \in \textbf{r}$ , and

\begin{equation*} \psi \colon F \ast P \to F \end{equation*}

for the homomorphism defined on generators by $\psi (x)=x$ , for $x \in \textbf{x}$ , and $\psi (\rho _{r}) = r$ for $r \in \textbf{r}$ . We also recall that

\begin{equation*} N(\mathcal {P})\;:\!=\;\langle \langle P \rangle \rangle \trianglelefteq F \ast P.\end{equation*}

Observe that $F$ acts on $N(\mathcal{P})$ by conjugation.

Let $X$ denote a $2$ -dimensional presentation CW complex of $\mathcal{P}$ . In particular, $X$ is built of $1$ -cells $e^{1}_{x}$ , for $x \in \textbf{x}$ , and $2$ -cells $e^{2}_{r}$ , for $r \in \textbf{r}$ .

Construction 5.8. There is a group homomorphism $\kappa _{N} \colon N(\mathcal{P}) \to \pi _{2}(X,X^{1})$ given by the formula

\begin{equation*}\kappa _{N}(w \rho _{r}^{\epsilon } w^{-1}) = (w \cdot f^{2}_{r})^{\epsilon },\end{equation*}

where $f^{2}_{r} \colon (D^{2},\partial D^{2}) \to (X,X^{1})$ is the characteristic maps of the $2$ -cell $e^{2}_{r}$ .

This uniquely defines $\kappa _{N}$ because, as we now assert, $N(\mathcal{P})$ is freely generated by words of the form $w \rho _{r} w^{-1}$ , for $r \in \textbf{r}$ and $w \in F$ . Consider the canonical projection $p \colon F \ast P \to F$ with $\ker\!(p) = N(\mathcal{P})$ and $(F \ast P)/ N(\mathcal{P}) \cong F$ . Therefore, $N(\mathcal{P}) = \pi _{1}(Y)$ , where $Y \to K(F \ast P,1)$ is the covering associated with the projection $p$ . The space $Y$ can be constructed as a pullback of the universal cover $\widetilde{K(F,1)} \to K(F,1)$ along the map $K(F \ast P,1) \to K(F,1)$ induced by the projection $p$ . Both $K(F,1)$ and $K(F \ast P,1)$ are bouquets of circles and we write $(W_f)_{f \in F}$ for the lifts of the single $0$ -cell of $K(F,1)$ . The aforementioned pullback of $\widetilde{K(F,1)} \to K(F,1)$ can be build by attaching a copy of $K(P,1)$ at every $0$ -cell of $\widetilde{K(F,1)}$ , that is, at every $w_f$ . It follows that $Y$ is homotopy equivalent to a bouquet of $K(P_{w},1)$ , where $P_{w} = w P w^{-1} \subset F \ast P$ and $w \in F$ , that is, $Y \simeq \bigvee _{w \in F} K(P_{w},1)$ . Hence, $N(\mathcal{P})$ is freely generated by words of the form $w \rho _{r} w^{-1}$ , asserted and $\kappa _N$ is defined.

The following proposition, though not explicitly stated, is a consequence of the results of Section 16 of [Reference Whitehead40] and of Section 2.3 in [Reference Trotter39].

Proposition 5.9. Let $X$ be a $2$ -dimensional CW complex realising a presentation $\mathcal{P}$ of a group $G$ . The map $\kappa _{N} \colon N(\mathcal{P}) \to \pi _{2}(X,X^{1})$ is surjective and descends to an isomorphism of crossed $F$ -modules

\begin{equation*}\kappa _{N} \colon \left (N(\mathcal {P})/ [[N(\mathcal {P}),N(\mathcal {P})]]_{\psi },\psi \right ) \xrightarrow {\cong } \left ( \pi _{2}(X,X^{1}), \partial \right ).\end{equation*}

The restriction of $\kappa _{N}$ to $I(\mathcal{P})$ induces an isomorphism of left $\mathbb{Z}[G]$ -modules

\begin{equation*}\kappa _{I} \colon I(\mathcal {P})/[[N(\mathcal {P}),N(\mathcal {P})]]_{\psi } \to \pi _2(X),\end{equation*}

where, by abuse of notation, we denote by $\psi$ the restriction of the map $\psi \colon F \ast P \to F$ to $N(\mathcal{P})$ .

Proof. First, we argue that $\kappa _{N}$ is surjective and that $\ker\!( \kappa _{N}) = [[N(\mathcal{P}),N(\mathcal{P})]]_{\psi }$ . As in Definition 5.2, we denote by $H$ the free group generated by the set $S \times F$ , where $S = \{e^{2}_{r} \colon r \in \textbf{r}\}$ is the collection of $2$ -cells of $X$ . It follows that the map

\begin{equation*}\theta \colon N(\mathcal {P}) \owns w \rho _{r} w^{-1} \mapsto (f^{2}_{r},w) \in H\end{equation*}

defines a group isomorphism. Indeed, this follows from the fact that $N(\mathcal{P})$ is freely generated by words of the form $w \rho _{r} w^{-1}$ , where $w \in F$ .

This group homomorphism fits into the commutative diagram

(5.1)

where $q \colon H \to \pi _2(X,X^{1})$ is given on generators by the formula $q(f_{r},w) = w \cdot f_{r}$ . Set $F\;:\!=\;\pi _1(X^1)$ . As noted in Example 5.4, $\pi _2(X,X^{1})$ is a free $F$ -crossed module, meaning that the homomorphism $q$ is surjective with kernel $\ker\!(q) = [[H,H]]_{\partial _{H}}$ , where $\partial _{H}$ denotes the map

\begin{equation*}\partial _{H} \colon S \times F \to F=\pi _{1}(X^{1}), \quad \partial _{H}(f^{2}_{r},w) = w r w^{-1}.\end{equation*}

Since $\theta$ and $q$ are both surjective, the commutativity of (5.1) implies that so is $\kappa _{N}$ .

We now show that $\ker\!( \kappa _{N}) = [[N(\mathcal{P}),N(\mathcal{P})]]_{\psi }$ . Since $\ker\!(q)=[[H,H]]_{\partial _{H}}$ , the commutativity of (5.1) implies that

\begin{equation*}\ker\!(\kappa _{N}) = \theta ^{-1}(\!\ker\!(q)) = \theta ^{-1}([[H,H]]_{\partial _{H}}).\end{equation*}

In order to understand the subgroup $\theta ^{-1}([[H,H]]_{\partial _{H}})$ , consider the diagram

(5.2)

whose commutativity implies the required equality:

\begin{equation*}\ker\!(\kappa _{N}) = \theta ^{-1}(\!\ker\!(q)) = \theta ^{-1}([[H,H]]_{\partial _{H}}) = [[N(\mathcal {P}),N(\mathcal {P})]]_{\psi }.\end{equation*}

In particular, $\kappa _{N}$ induces an isomorphism

\begin{equation*}N(\mathcal {P})/ [[N(\mathcal {P}),N(\mathcal {P})]]_{\psi } \cong \pi _{2}(X,X^{1}).\end{equation*}

It remains to show that the restriction of $\kappa _{N}$ to $I(\mathcal{P})$ gives rise to the claimed isomorphism. Consider the following diagram with exact rows:

(5.3)

The right-hand side square of (5.3) is commutative because for any $w \rho _{r}^{\epsilon } w^{-1} \in N(\mathcal{P})$ , we have

\begin{equation*}(\partial \circ \kappa _{N})(w \rho _{r}^{\epsilon } w^{-1}) = \partial ((f^{2}_{r},w)^{\epsilon }) = w r^{\epsilon } w^{-1} = \psi (w \rho _{r}^{\epsilon } w^{-1}).\end{equation*}

The commutativity of this diagram and the exactness of its rows gives the required isomorphism:

\begin{equation*}\pi _{2}(X) = \ker\!(\partial ) \cong \ker\!(\psi )/ \ker\!(\kappa _{N}) = I(\mathcal {P})/ [[N(\mathcal {P}),N(\mathcal {P})]]_{\psi }.\end{equation*}

This concludes the proof of the proposition.

Roughly speaking, the next proposition describes how a set of identities $\textbf{s}$ for a group presentation $\mathcal{P}$ determines a $3$ -dimensional CW complex that realises the data of $\mathcal{P}$ and $\textbf{s}$ .

Proof. Let $X$ be a presentation $2$ -complex of $\mathcal{P}$ . By Proposition 5.9, the set $\textbf{s}$ determines a collection of elements $\{[g_{s}]\}_{s \in \textbf{s}}$ of $\pi _{2}(X)$ . Construct $Z$ by adjoining $3$ -cells $\{e^{3}_{s}\}_{s \in \textbf{s}}$ , where $e^{3}_{s}$ is attached using $g_{s} \in \pi _{2}(X)$ . For later use, we denote the characteristic map of the $3$ -cell $e^{3}_{s}$ by

\begin{equation*} f^{3}_{s} \colon (D^{3},\partial D^{3}) \to (Z,Z^{2}).\end{equation*}

The first three points of the proposition now follow immediately from the construction of $Z$ . We prove the fourth point. Since $Z$ is a $3$ -complex, the relative homotopy group $\pi _{3}(Z,Z^{2})$ is a free $\mathbb{Z}[G]$ -module generated by homotopy classes of characteristic maps $\{f^{3}_{s}\}_{s \in \textbf{s}}$ [Reference Whitehead41, Chapter V.1, Theorem 1.1]. Using the exact sequence

\begin{equation*}\pi _{3}(Z,Z^{2}) \xrightarrow {\partial } \pi _{2}(Z^{2}) \to \pi _{2}(Z) \to 0\end{equation*}

we conclude that $\pi _{2}(Z) \cong \pi _{2}(Z^{2})/ \operatorname{im} \partial$ . Observe that the boundary map in the exact sequence maps the homotopy class of the characteristic map $[f_{s}]$ to the homotopy class of the corresponding attaching map $[g_{s}]$ , for $s \in \textbf{s}$ . Observe that for any $s \in \textbf{s}$ , $\kappa _{I}([g_{s}]) = s$ , where $\kappa _{I}$ is the isomorphism from Proposition 5.9. Therefore, $\kappa _{I}^{-1}$ maps $\operatorname{im} \partial$ isomorphically onto the $\mathbb{Z}[G]$ -submodule of $I(\mathcal{P})/ [[N(\mathcal{P}),N(\mathcal{P})]]_{\psi }$ generated by identities $s \in \textbf{s}$ , as desired. This concludes the proof of the fourth point.

We prove the fifth and last point. For each $k$ , the chain groups underlying the chain complex $C^{\text{cell}}_\bullet (\widetilde{Z})$ are $C^{\text{cell}}_{k}(\widetilde{Z}) = H_{k}(\widetilde{Z}^{k},\widetilde{Z}^{k-1})$ and the boundary maps are given by the composition

\begin{equation*}\partial ^{\text {cell}}_{k+1} \colon C_{k+1}^{\text {cell}}(\widetilde {Z}) = H_{k+1}(\widetilde {Z}^{k+1},\widetilde {Z}^{k}) \xrightarrow {\partial } H_{k}(\widetilde {Z}^{k}) \xrightarrow {j} H_{k}(\widetilde {Z}^{k},\widetilde {Z}^{k-1}) = C_{k}^{\text {cell}}(\widetilde {Z}),\end{equation*}

where the maps $\partial$ and $j$ come from the exact sequence of the pairs $(\widetilde{Z}^{k+1},\widetilde{Z}^{k})$ and $(\widetilde{Z}^{k},\widetilde{Z}^{k-1})$ , respectively. The expressions for $\partial _{1}^{\text{cell}}$ and $\partial _{2}^{\text{cell}}$ are well-known (see e.g. [Reference Trotter39, page 473]) and agree with the formulas given in Construction 5.6. Therefore, we focus only on identifying the differential $\partial _{3}^{\text{cell}}$ .

Consider the following commutative diagram

where the vertical map $\kappa$ comes from Proposition 5.9, the vertical maps between the third and the second row are induced by the covering map $q\colon \widetilde{Z} \to Z$ , and the vertical maps between the third and fourth row come from the (relative) Hurewicz Theorem. In particular, (relative) Hurewicz Theorem implies that $h_{3}$ and $h_{2}$ are isomorphisms and that $h_{2,\text{rel}}$ induces an isomorphism

\begin{equation*}\pi _{2}(\widetilde {Z}^{2},\widetilde {Z}^{1})^{ab} \cong H_{2}(\widetilde {Z}^{2},\widetilde {Z}^{1}).\end{equation*}

Note that the map $\partial _3^{\text{cell}} \colon H_3(\widetilde{Z},\widetilde{Z}^{2}) \to H_2(\widetilde{Z}^{2},\widetilde{Z}^{1})$ appears as the composition of the maps on the last row of this diagram. Since $H_{3}(\widetilde{Z},\widetilde{Z}^{2}) \cong \pi _{3}(\widetilde{Z},\widetilde{Z}^{2}) \cong \pi _{3}(Z,Z^{2})$ is freely generated as a $\mathbb{Z}[G]$ -module by the characteristic maps $f^{3}_{s} \colon (D^{3},\partial D^{3}) \to (Z,Z^{2})$ , it remains to understand

\begin{equation*}\partial _3^{\text {cell}}([f^{3}_{s}])=j \circ \partial ([f^{3}_{s}]).\end{equation*}

For $[f^{3}_{s}] \in H_{3}(\widetilde{Z},\widetilde{Z}^{2})$ , we have $\partial ([f^{3}_{s}]) = [f^{3}_{s}|_{\partial D^{2}}] = [g_{s}]$ , where $g_s \in \pi _2(Z^2)=\pi _2(X)$ are the maps along which we glued the $3$ -cells. It follows that $s \in \kappa ^{-1}( q_{\ast } (h_{2}^{-1}(\{[g_{s}]\})))$ and therefore the commutativity of the diagram gives

\begin{equation*}\partial _{3}^{\text {cell}}([f^{3}_{s}]) = (j \circ \partial ) ([f^{3}_{s}]) = j([g_{s}]) = h_{2,\text {rel}}(q_{\ast }^{-1}(\kappa (j(s)))).\end{equation*}

We calculate this last expression. Observe that for any $x = w \rho _{r_{0}} w^{-1} \in N(\mathcal{P})$ with $r_{0} \in \textbf{r}$ we have

\begin{equation*}h_{2,\text {rel}}(q_{\ast }^{-1}(\kappa (x))) = w \cdot f^{2}_{r_{0}} = \frac {\partial x}{\partial \rho _{r_{0}}} \cdot f^{2}_{r_{0}} = \sum _{r \in \textbf {r}} \frac {\partial x}{\partial \rho _{r}} \cdot f^{2}_{r}.\end{equation*}

Here the Fox derivatives $\frac{\partial x}{\partial \rho _{r}}$ are calculated in $F \ast P$ and we used that $ \frac{\partial x}{\partial \rho _{r_{0}}}=w$ and $ \frac{\partial x}{\partial \rho _{r}}=0$ for $r \neq r_0$ .

One then verifies that for any $y \in N(\mathcal{P})$ , we have

\begin{equation*}h_{2,\text {rel}}(q_{\ast }^{-1}(\kappa (y))) = \sum _{r \in \textbf {r}} \frac {\partial y}{\partial \rho _{r}} \cdot f_{r}^{2}.\end{equation*}

Plugging this into the previous calculation, we obtain

\begin{equation*}\partial _{3}^{\text {cell}}([f^{3}_{s}]) = h_{2,\text {rel}}(q_{\ast }^{-1}(\kappa (j(s)))) = \sum _{r \in \textbf {r}} \frac {\partial s}{\partial \rho _{r}} \cdot [f^{2}_{r}],\end{equation*}

This concludes the proof of the proposition.

Proof. The $2$ -skeleton of $Y$ determines a presentation $\mathcal{P} = \langle \textbf{x} \mid \textbf{r} \rangle$ of $\pi _{1}(Y)$ . Using the isomorphism $\pi _{2}(Y^{2}) \cong I(\mathcal{P})/ [[N(\mathcal{P}),N(\mathcal{P})]]_{\psi }$ from Proposition 5.9, we see that the attaching maps of the $3$ -cells of $Y$ determine identities $s \in I(\mathcal{P})$ , which are well-defined modulo Peiffer commutators.

The homotopy equivalence $Y \simeq Z(\textbf{x},\textbf{r},\textbf{s})$ follows from the fact that there is a bijection between cells of $Y$ and $Z(\textbf{x},\textbf{r},\textbf{s})$ in each dimension, and the corresponding cells are attached via homotopic maps.

6.1. The handle chain complex

Let $M$ be a connected $n$ -manifold that admits a handle decomposition. Assume that the handles are attached in increasing order of index. In what follows we denote the $n$ -handles of $M$ by $h_i^n$ and, for $d \geq 0$ , write $M^{(d)}$ for the submanifold of $M$ obtained by taking the union of all handles of indices $\leq d$ .

For $d \geq 0$ , the associated handle chain complex has chain groups

\begin{equation*}C_{d}^{\text {hnd}}(M) = H_{d}(M^{(d)},M^{(d-1)})\end{equation*}

and differentials

\begin{equation*}\partial ^{\text {hnd}}_{d+1} \colon C_{d+1}^{\text {hnd}}(M) = H_{d+1}(M^{(d+1)},M^{(d)}) \xrightarrow {\partial } H_{d}(M^{(d)}) \xrightarrow {j} H_{d}(M^{(d)},M^{(d-1)}) = C_{d}^{\text {hnd}}(M),\end{equation*}

where $\partial$ is the connecting homomorphism in the long exact homology sequence of the pair $(M^{(d+1)},M^{(d)})$ and $j$ is induced by inclusion of pairs $(M^{(d)},\emptyset ) \subset (M^{(d)},M^{(d-1)})$ .

Let $p \colon \widetilde{M} \to M$ be the universal covering projection. The handle decomposition of $M$ lifts to a handle decomposition of $\widetilde{M}$ and we write $\widetilde{M}^{(d)} = p^{-1}(M^{(d)})$ . We can now consider the handle chain complex of $\widetilde{M}$ :

\begin{equation*}C_{\ast }^{\text {hnd}}(M;\;\mathbb {Z}[\pi _{1}(M)]) \;:\!=\; C_{\ast }^{\text {hnd}}(\widetilde {M}).\end{equation*}

Using the action of $\pi _{1}(M)$ on $\widetilde{M}$ , each chain group $C^{\text{hnd}}_{d}(M;\;\mathbb{Z}[\pi _{1}(M)])$ becomes a free left $\mathbb{Z}[\pi _{1}(M)]$ -module. These chain $\mathbb{Z}\pi _{1}(M)$ -modules are generated by a collection of lifts of the $d$ -dimensional handles of $M$ where we choose one lift for each handle. In what follows, we implicitly choose (arbitrarily) such lifts and use them as a basis for the relevant chain groups.

Lemma 6.2. The inclusion map $\iota \colon X \hookrightarrow M$ induces chain isomorphisms

\begin{equation*}\iota _{\ast } \colon C^{\text {cell}}_{\bullet }(X) \cong C^{\text {hnd}}_{\bullet }(M), \quad \iota _{\ast } \colon C^{\text {cell}}_{\bullet }(X;\;\mathbb {Z}[\pi _{1}(X)]) \cong C^{\text {hnd}}_{\bullet }(M;\;\mathbb {Z}[\pi _{1}(M)]).\end{equation*}

Every cell $e^{d}_{j}$ of $X$ gives rise to a generator $[e^{d}_{j}]$ in $C^{\text{cell}}_{d}(M;\;\mathbb{Z}[\pi _{1}(M)])$ , and every handle $h^{d}_{j}$ of $M$ gives rise to a generator $h^{d}_{j} \in C^{\text{hnd}}_{d}(M;\;\mathbb{Z}[\pi _{1}(M)])$ . The map induced by $\iota$ , maps each generator $e^{d}_{j}$ to the corresponding generator $h^{d}_{j}$ .

Proof. Since $\iota$ preserves the respective filtrations $\lbrace X^{(d)} \rbrace _d$ of $X$ and $\lbrace M^{(d)} \rbrace _d$ of $M$ , it follows that $\iota$ induces isomorphisms of respective chain groups. The naturality of the exact sequence of a pair in homology implies that $\iota$ is a chain map, and hence, the lemma follows.

6.2. Aspherical $3$ -manifolds and identities of presentation

This short section records a technical lemma concerning identities. A $1$ - and $2$ - handles in the handle decomposition $\mathcal{H}$ of a connected manifold $M$ determine a presentation $\mathcal{P}_{\mathcal{H}}$ of $\pi _1(M)$ . The next lemma notes that the $3$ -handles of $M$ determine identities for this presentation.

Lemma 6.3. If $M$ is a closed, connected, oriented, aspherical $3$ -manifold admitting a handle decomposition $\mathcal{H}$ with $k+1$ $3$ -handles, for some $k \geq 0$ , then the presentation $\mathcal{P}_{\mathcal{H}}$ of $\pi _{1}(M)$ admits a complete set of identities

\begin{equation*}s_{1},\ldots,s_{k+1} \in I(\mathcal {P}_{\mathcal {H}})\end{equation*}

such that, modulo Peiffer commutators, any other identity can be written (uniquely up to order of factors) as a product of conjugates of the $s_i$ .

Proof. Since $M$ is aspherical, it follows that $H_i(\widetilde{M})=0$ , for $i \geq 1$ . The Hurewicz theorem and the long exact sequence of the pair $(\widetilde{M},\widetilde{M}^{(2)})$ gives rise to isomorphisms

\begin{equation*}\pi _2(M^{(2)}) \cong H_{2}(\widetilde {M}^{(2)}) \cong H_{3}(\widetilde {M},\widetilde {M}^{(2)}) \cong \mathbb {Z}[\pi _{1}(M)]^{k+1} \end{equation*}

Let $f_{1},\ldots,f_{k+1} \in \pi _{2}(M^{(2)})$ be a $\mathbb{Z}[\pi _{1}(M)]$ -basis and consider the isomorphism

\begin{equation*}\kappa _{I} \colon I(\mathcal {P_{\mathcal {H}}})/ [[N(\mathcal {P}_{\mathcal {H}}),N(\mathcal {P}_{\mathcal {H}})]]_{\psi } \cong \pi _{2}(M^{(2)})\end{equation*}

from Proposition 5.9. A complete set of identities with the required properties is now given by considering $s_{i} = \kappa ^{-1}_{I}(f_{i})$ , for $i=1,2,\ldots,k+1$ .

Taking $k=0$ in the previous lemma gives the following result.

Proposition 6.4. If $M$ is a closed, connected, oriented, aspherical $3$ -manifold admitting a handle decomposition $\mathcal{H}$ with a single $3$ -handle, then the presentation $\mathcal{P}_{\mathcal{H}}$ of $\pi _{1}(M)$ admits a unique (up to conjugation and modulo Peiffer commutators) identity

\begin{equation*} s \in I(\mathcal {P}_{\mathcal {H}}).\end{equation*}

6.3. Explicit formulas for the symmetric structure of a closed aspherical $3$ -manifold

Given a $3$ -manifold $M$ , this section collects the work from Section 5 to describe the differentials and symmetric structure on chain complex $C_*^{\text{hnd}}(M,\mathbb{Z}[\pi _1(M)]).$

Given a closed, connected, oriented, aspherical $3$ -manifold $M$ , with a handle decomposition $\mathcal{H}$ , we would like to obtain an explicit formula for the map

\begin{equation*}\Phi \colon C^{2}_{\text {hnd}}(M;\;\mathbb {Z}[\pi _{1}(M)]) \to C_{1}^{\text {hnd}}(M;\;\mathbb {Z}[\pi _{1}(M)])\end{equation*}

induced by the symmetric structure on $C^{3-*}(M;\;\mathbb{Z}[\pi _{1}(M)])$ .

For the $3$ -complex $X$ obtained from $M$ by deformation retracting each handle to its core (as in Construction 6.1), the most important ingredient that goes into the calculation of the map $\Phi$ (namely the diagonal chain map) was worked out by Trotter [Reference Trotter39, Section 2.4], see also [Reference Miller and Powell33, Section 3.3]. Building on this result we obtain the following proposition.

Proposition 6.5. Let $M$ be a closed, connected, oriented, aspherical $3$ -manifold that admits a handle decomposition $\mathcal{H}$ . Let

\begin{equation*}\mathcal {P}_{\mathcal {H}} = \langle \textbf {x} \mid \textbf {r} \rangle \end{equation*}

denote the presentation of $\pi _{1}(M)$ determined by $\mathcal{H}$ and let $\textbf{s}$ denote the set of identities from Lemma 6.3 . Each $s \in \textbf{s}$ can be written in the form

(6.1) \begin{equation} s = \prod _{j=1}^{l_{s}} w_{s,j} \rho _{r_{s,j}}^{\epsilon _{s,j}} w_{s,j}^{-1}, \end{equation}

where $r_{s,j} \in \textbf{r}$ and $\epsilon _{s,j} \in \{\pm 1\}$ , for all $s$ and $j$ .

Endow $C_{2}^{\text{hnd}}(M;\;\mathbb{Z}[\pi _{1}(M)])$ with a basis $h_{r}^{2}$ , for $r \in \textbf{r}$ , induced by the handle structure $\mathcal{H}$ . Denote by $(h^{2}_{r})^{\#}$ , for $r \in \textbf{r}$ , the dual basis of $C^{2}_{\text{hnd}}(M;\;\mathbb{Z}[\pi _{1}(M)])$ . With respect to these bases, the coefficients of the matrix of the map $\Phi$ are given by the formula

\begin{equation*}\Phi ((h^{2}_{r})^{\#}) = \sum _{(s,j) \in \mathcal {I}(r)} \sum _{x \in \textbf {x}} \epsilon _{s,j} w_{s,j}^{-1} \frac {\partial w_{s,j}}{\partial x} h^{1}_{x},\end{equation*}

where $\mathcal{I}(r) = \{(s,j) \in \textbf{s} \times \mathbb{Z}_{\gt 0} \colon r_{s,j} = r\}$ , for $r \in \textbf{r}$ . In other words, for a specified $r \in \textbf{r}$ , in the interior sum includes only the terms $w_{s,j} \rho _{r_{s,j}} w_{s,j}^{-1}$ coming from (6.1) which are conjugates of $\rho _{r}^{\pm 1}$ .

Proof. This formula can be deduced from [Reference Miller and Powell33, Section 3.3].

For later use, we summarise the considerations of this section and the previous one in the case of an aspherical $3$ -manifold endowed with a handle decomposition with a single $3$ -handle.

Proposition 6.6. Let $M$ be a closed, connected, oriented, aspherical $3$ -manifold admitting a handle decomposition $\mathcal{H}$ with a single $0$ -handle and a single $3$ -handle, and let

\begin{equation*}s = \prod _{j=1}^{l} w_{j} \rho _{r_{j}}^{\epsilon _{j}} w_{j}^{-1}.\end{equation*}

be the identity of the presentation $\mathcal{P}_{\mathcal{H}}=\langle \textbf{x} \mid \textbf{r} \rangle$ of $\pi _{1}(M)$ from Proposition 6.4 .

• • The handle chain module $C_0^{\text{hnd}}(M;\;\mathbb{Z}[\pi _{1}(M)])$ is freely generated by the class of a lift the $0$ -handle $h^{0}$ .

• • The handle chain module $C_1^{\text{hnd}}(M;\;\mathbb{Z}[\pi _{1}(M)]$ is freely generated by classes of lifts of the $1$ -handles $\{h^{1}_{x} \colon x \in \textbf{x}\}.$

• • The handle chain module $C_2^{\text{hnd}}(M;\;\mathbb{Z}[\pi _{1}(M)]$ is freely generated by the classes of lifts of the $2$ -handles $\{h^{2}_{r} \colon r \in \textbf{r}\}$ ,

• • The handle chain module $C_3^{\text{hnd}}(M;\;\mathbb{Z}[\pi _{1}(M)]$ is freely generated by the class of a lift of the $3$ -handle $h^{3}_{s}$ .

The differentials can be expressed in terms of Fox derivatives as

\begin{align*} \partial _{1}(h^{1}_{x}) &= (x-1)h^{0}, \quad \text{ for\ } x \in \textbf{x}, \\[5pt] \partial _{2}(h^{2}_{r}) &= \sum _{x \in \textbf{x}} \frac{\partial r}{\partial x} h^{1}_{x}, \quad \text{ for\ } r \in \textbf{r}, \\[5pt] \partial _{3}(h^{3}_{s}) &= \sum _{r \in \textbf{r}} \frac{\partial s}{\partial \rho _{r}} h^{2}_{r}, \end{align*}

where the Fox derivatives $\frac{\partial s}{\partial \rho _{r}}$ are computed in $F \ast P$ .

With respect to these bases, the map $\Phi$ is given by the formula

\begin{equation*}\Phi ((h^{2}_{r})^{\#}) = \sum _{x \in \textbf {x}} \sum _{j \in \mathcal {I}(r)} \epsilon _{j} w_{j}^{-1} \frac {\partial w_{j}}{\partial x} h^{1}_{x},\end{equation*}

where $\mathcal{I}(r) = \{j \in \mathbb{Z}_{\gt 0} \colon r_{j} = r\}$ , for $r \in \textbf{r}$ .

Proof. The calculation of the handle chain complex follows from Corollary 5.11 and Lemma 6.2. The formula for the symmetric structure comes from Proposition 6.5.

7.1. The chain complex of the universal cover

In this subsection, we explicitly describe the chain complex of the universal cover of the $0$ -framed surgery $M_{T_{2,2k+1}}$ .

We start by providing a presentation for the fundamental group of $M_{T_{2,2k+1}}$ . Such a presentation can be computed from the knot group once a word for the longitude of $T_{2,2k+1}$ is known. Using Figure 1, a Wirtinger presentation of the fundamental group of the complement of $T_{2,2k+1}$ is

\begin{equation*}\pi \;:\!=\; \pi _{1}(S^{3} \setminus T_{2,2k+1}) = \langle x_{1}, x_{2}, \ldots, x_{2k+1} \mid r_{1}, r_{2}, \ldots, r_{2k} \rangle,\end{equation*}

where $r_{i} = x_{i} x_{i+1} x_{i+2}^{-1} x_{i+1}^{-1}$ and the indices are taken mod $2k+1$ . We will now simplify the presentation. A standard argument is recalled, because it is used to describe the presentation of the fundamental group of $M_{T_{2,2k+1}}$ .

Lemma 7.1. There exists an isomorphism $\phi \colon \pi \to G\;:\!=\;\langle a, b \mid a^{2k+1}b^{2} \rangle$ such that

\begin{align*} \phi (x_{2(k-s)+1}) &= a^{s} (a^{k}b)^{-1} a^{-s}, \quad 0 \leq s \leq k, \\[5pt] \phi (x_{2(k-s)}) &= a^{s} (a^{k+1}b) a^{-s}, \quad 0 \leq s \leq k-1. \end{align*}

Proof. We first prove that $\pi$ is generated by $x_{2k}$ and $x_{2k+1}$ . Notice that the relation $r_{i}$ implies that the equality $x_{i}x_{i+1} = x_{i+1}x_{i+2}$ holds. As a consequence, we obtain $x_{i} x_{i+1} = x_{i+s}x_{i+s+1}$ for any $i$ and $s$ and, applying this formula recursively, we obtain

\begin{align*} x_{2k-1} &\stackrel{r_{2k-1}}{=} x_{2k}x_{2k+1}x_{2k}^{-1}, \\[5pt] x_{2k-2} &\stackrel{r_{2k-2}}{=} x_{2k-1}x_{2k}x_{2k-1}^{-1} = (x_{2k} x_{2k+1}) x_{2k} (x_{2k}x_{2k+1})^{-1}, \\[5pt] x_{2k-3} &\stackrel{r_{2k-3}}{=} x_{2k-2}x_{2k-1}x_{2k-2}^{-1} \\[5pt] &= (x_{2k}x_{2k+1}x_{2k}x_{2k+1}^{-1}x_{2k}^{-1})(x_{2k}x_{2k+1}x_{2k}^{-1})(x_{2k}x_{2k+1}x_{2k}^{-1}x_{2k+1}^{-1}x_{2k}^{-1}) \\[5pt] &= (x_{2k}x_{2k+1})x_{2k}x_{2k+1}x_{2k}^{-1}(x_{2k}x_{2k+1})^{-1}. \end{align*}

As a consequence, we eliminate the generators $x_{1}, x_{2}, \ldots, x_{2k-1}$ and the relations $r_{1}, r_{2}, \ldots, r_{2k-1}$ since we have the following equalities:

\begin{align*} x_{2(k-s)+1} &= (x_{2k}x_{2k+1})^{s-1} x_{2k}x_{2k+1}x_{2k}^{-1} (x_{2k}x_{2k+1})^{-s+1}, \quad 1 \leq s \leq k, \\[5pt] x_{2(k-s)} &= (x_{2k}x_{2k+1})^{s} x_{2k+1} (x_{2k}x_{2k+1})^{s}, \quad 1 \leq s \leq k-1. \end{align*}

Reformulating, we have proved that $\pi$ is indeed generated by $x_{2k}$ and $x_{2k+1}$ and we have obtained the presentation $\pi = \langle x_{2k}, x_{2k+1} \mid r'_{\!\!2k} \rangle$ , where

\begin{align*} r'_{\!\!2k} &= x_{2k} x_{2k+1} x_{1}^{-1} x_{2k+1}^{-1} = x_{2k} x_{2k+1} (x_{2k} x_{2k+1})^{k-1} x_{2k} x_{2k+1}^{-1} x_{2k}^{-1} (x_{2k}x_{2k+1})^{-k+1} x_{2k+1}^{-1} \\[5pt] &= (x_{2k} x_{2k+1})^{k} x_{2k} (x_{2k} x_{2k+1})^{-k} x_{2k+1}^{-1}. \end{align*}

Using this presentation of $\pi$ , it is now straightforward to verify that $\phi \colon \pi \to G$ is a group isomorphism, where the inverse is defined by setting $\phi ^{-1}(a) = x_{2k}x_{2k+1}$ and $\phi ^{-1}(b) = (x_{2k+1}^{-1}x_{2k}^{-1})^{k} x_{2k+1}^{-1}$ . This concludes the proof of the lemma.

As a consequence, we can describe the fundamental group of the 0-framed surgery on $T_{2,2k+1}$ .

Lemma 7.2. The fundamental group of the $0$ -surgery on $T_{2,2k+1}$ admits the following presentation:

(7.1) \begin{equation} G_{0} = \langle a, b \mid a^{2k+1}b^{2}, (a^{k}b)^{2k+1} a^{2k+1} (a^{k}b)^{2k+1}\rangle. \end{equation}

Proof. Let $ (\mu,\lambda )$ be the meridian-longitude pair of $T_{2,2k+1}$ expressed in words of $\pi$ . Using van Kampen’s theorem, to prove the proposition, it suffices to show that $\phi (\lambda ) = (a^{k}b)^{2k+1}a^{2k+1}(a^{k}b)^{2k+1}$ , where $\phi$ is the isomorphism described in Lemma 7.1. Working with the Wirtinger presentation arising from Figure 1, we choose $(\mu,\lambda )$ as follows:

\begin{equation*}\mu = x_{2k+1}, \quad \lambda = x_{2k+1}^{-2k-1} x_{1} x_{3} x_{5} \cdots x_{2k+1} x_{2} x_{4} \cdots x_{2k}.\end{equation*}

Next, the definition of $\phi$ gives $\phi (x_{1} x_{3} \cdots x_{2k+1}) = a^{k} (b^{-1} a^{-k-1})^{k+1} a = a^{2k+1} (a^{-k-1}b^{-1})^{k+1} a^{-k}$ and $\phi (x_{2} x_{4} \ldots x_{2k}) = a^{k} (a^{k}b)^{k}$ . Since the relation $a^{2k+1}b^{2}=1$ holds, we have $a^{-k-1}b^{-1} = a^{k}b$ , we deduce that $\phi (x_{1}x_{3} \cdots x_{2k+1} x_{2} \cdots x_{2k}) = a^{2k+1} (a^{k}b)^{2k+1}$ , and the lemma is now concluded by recalling the definition of $\lambda$ .

We now describe a handle decomposition for $M_{T_{2,2k+1}}$ . Recall from Remark 3.4 that it is possible to obtain a handle decomposition of $M_{T_{2,2k+1}}$ from a reduced diagram for $T_{2,2k+1}$ and that this decomposition can be used to calculate twisted Blanchfield forms. While this handle decomposition is easy to describe, it has one serious disadvantage: the number of handles grows with $k$ . To be able to perform computations for the whole family of torus knots $T_{2,2k+1}$ , for $k \geq 1$ , we use a handle decomposition with far fewer handles.

Figure 2. Left frame: the standard genus $1$ Heegaard decomposition of $S^3$ . Central frame: the knots $K_1$ and $K_2$ lying in the Heegaard torus $T$ . Right frame: the neighbourhoods $\overline{\nu }(K_1)$ and $\overline{\nu }(K_2)$ of $K_1$ and $K_2$ that satisfy $T = \overline{\nu }(K_1) \cup \overline{\nu }(K_2)$ and $\overline{\nu }(K_1) \cap \overline{\nu }(K_2) = \partial \overline{\nu }(K_1) = \partial \overline{\nu }(K_2)$ .

Construction 7.3. We describe an explicit handle decomposition for the exterior $X_{T_{2,2k+1}}$ . Our strategy is to first describe $X_{T_{2,2k+1}}$ as a union of three solid tori $U_1,U_2$ and $V_1$ . We will then read off a handle decomposition for $X_{T_{2,2k+1}}$ from a handle decomposition of the solid tori.

Consider the standard genus one Heegaard decomposition $S^{3} = H_{1} \cup _{\partial } H_{2}$ with Heegaard surface the torus $T = H_{1} \cap H_{2}$ . The solid tori $H_1$ and $H_2$ are sketched on the left frame of Figure 2 . Pick two parallel copies $K_1,K_2 \subset T$ of the torus knot $T_{2,2k+1}$ that lie on the torus $T$ as illustrated in the central frame of Figure 2 . The right-hand side of this figure shows closed neighbourhoods $\overline{\nu }(K_1),\overline{\nu }(K_2) \subset T$ that satisfy

\begin{equation*}T = \overline {\nu }(K_1) \cup \overline {\nu }(K_2) \text { and } \overline {\nu }(K_1) \cap \overline {\nu }(K_2) = \partial \overline {\nu }(K_1) = \partial \overline {\nu }(K_2).\end{equation*}

Shrink the solid tori $H_1 \cong S^1 \times D^2$ and $H_2 \cong S^1 \times D^2$ to obtain half-sized solid tori $S^1 \times D^2_{\frac{1}{2}} \cong U_i \subset H_i$ . It follows that $V \;:\!=\; \overline{S^{3} \setminus (U_{1} \cup U_{2})}$ is a tubular neighbourhood of $T \subset S^3$ and can be identified with $V \cong T \times I \cong (\overline{\nu }(K_1) \cup \overline{\nu }(K_2)) \times I$ . Since we shrank the solid tori $H_1$ and $H_2$ but expanded the torus $T=\overline{\nu }(K_1) \cup \overline{\nu }(K_2)$ , we obtain

\begin{equation*}S^3=V \cup U_1 \cup U_2=(\overline {\nu }(K_1) \times I) \cup (\overline {\nu }(K_2) \times I) \cup U_1 \cup U_2.\end{equation*}

Recall that $K_2 \subset T$ is a copy of the torus knot $T_{2,2k+1}$ . Thus when we remove $\nu (K_2) \cong K_2 \times^{^{^{\;\;\;\circ\!\!\!\!\!\!\!\!}}} {D}^2$ from $S^3$ , and write $V_1\;:\!=\;\overline{\nu }(K_1) \times I$ , we obtain the following decomposition of $X_{T_{2,2k+1}}$ :

(7.2) \begin{equation} X_{T_{2,2k+1}} = U_{1} \cup V_{1} \cup U_{2}. \end{equation}

The intersection $V_1 \cap U_i$ consists a normal push-off of the neighbourhood $\overline{\nu }(K_{1}) \subset T$ into $U_i$ . Put differently, $V_1 \cap U_i$ is diffeomorphic to $\overline{\nu }(K_i')\cong K_i' \times I$ , where $K_i' \subset U_i$ is a normal push-off of the knot $K_i\subset T$ .

The solid tori $U_i$ have handle decompositions with a single $0$ -handle and a single $1$ -handle, say $U_{i} = h_{i}^{0} \cup h_{i}^{1}$ , and we will obtain a handle decomposition for $X_{T_{2,2k+1}}$ by decomposing $V_1$ as a union of two $3$ -balls $B_1$ and $B_2$ and showing that adding $V_1$ to $U_1 \sqcup U_2$ is the same as adding a $1$ -handle followed by a $2$ -handle.

Figure 3. The solid torus $V_1 \cong S^1 \times I^2$ and its decomposition as a union of two $3$ -balls, $B_1$ (red) and $B_2$ (the remaining part of $V_{1}$ ). The attaching region $\partial _+ B_1=\partial _{+,1} B_{1} \sqcup \partial _{+,2} B_{1}$ of $B_1$ , thought of as a $1$ -handle, is also shown.

Fix a diffeomorphism $\psi \colon S^{1} \times I^{2} \xrightarrow{\cong } V_{1}$ such that $N_{1} = V_{1} \cap T = \psi (S^{1} \times I \times \{1/2\})$ , and $V_{1} \cap U_{i} = \psi (S^{1} \times I \times \{i-1\})$ . Define $B_{1} = \psi ([0,\pi ] \times I^{2}) \subset V_{1}$ and $B_{2} = \psi ([\pi,2 \pi ] \times I^{2}) \subset V_{1}$ . Note that both $B_1$ and $B_2$ are diffeomorphic to $3$ -balls and that $V_1=B_1 \cup B_2$ , as illustrated in Figure 3 . In what follows, we will be thinking of $B_1$ as a $3$ -dimensional $1$ -handle and of $B_2$ as a $3$ -dimensional $2$ -handle. The attaching region $\partial _+ B_1=\partial _{+,1} B_{1} \sqcup \partial _{+,2} B_{1}$ of $B_1$ can be also be seen in Figure 3 .

• • When $V_1$ is glued to $U_1 \sqcup U_2$ , the $1$ -handle $h_{3}^{1}=B_{1}$ is attached to the union $U_{1} \sqcup U_{2}$ as illustrated in the top frame of Figure 4 . In more detail, we consider the aforementioned normal push offs $K'_{\!\!1} \subset U_1$ and $K'_{\!\!2} \subset U_2$ of $K_1,K_2 \subset T$ , and the attaching regions of $B_1$ and $B_2$ are identified with a small $2$ -disc neighbourhood of an unknotted portion of $K'_{\!\!1}$ and $K'_{\!\!2}$ .

  

  Figure 4. The space $Z$ obtained from $U_1 \sqcup U_2$ by attaching $B_1$ is diffeomorphic to a genus two handlebody. This top figure shows the torus knots $K'_{\!\!1} \subset U_1$ and $K'_{\!\!2} \subset U_2$ , expressed using a symplectic basis for $\partial U_1$ and $\partial U_2$ . Taking the connected sum of these knots (as depicted in the bottom figure) leads to the knot $J$ , which serves as the attaching circle for the attachment of the $2$ -handle $B_2$ .

  The effect of the handle attachment, $Z \;:\!=\; U_{1} \cup B_{1} \cup U_{2},$ is diffeomorphic to a genus two handlebody and admits $h_{0}^{1} \cup h_{2}^{0} \cup h_{1}^{1} \cup h_{2}^{1} \cup h_{3}^{1}$ as a handle decomposition. By performing isotopies of handle decompositions of $U_{1}$ and $U_{2}$ , we can assume that $h_{3}^{1}$ cancels geometrically with either $0$ -handle. Thus, there is a handle decomposition of $Z$ of the form

  \begin{equation*}Z = h_{1}^{0} \cup h_{2}^{0} \cup h_{1}^{1} \cup h_{2}^{1} \cup h_{3}^{1} \cong h_{1}^{0} \cup h_{1}^{1} \cup h_{1}^{2},\end{equation*}
  where the right-hand side diffeomorphism results from cancelling $h_{2}^{0}$ and $h_{3}^{1} = B_{1}$ .

• • When $V_1$ is glued to $U_1 \sqcup U_2$ , the $2$ -handle $h_r^{2}=B_{2}$ is attached to $Z$ as illustrated in the bottom frame of Figure 4 . In more detail, the $2$ -handle $B_2$ is attached to $Z$ along the curve $J \subset \partial Z$ given by the connected sum of $K'_{\!\!1}$ and $K'_{\!\!2}$ that is also illustrated in the bottom frame of Figure 4 .

As a result, we have obtained a handle decomposition of $X_{T_{2,2k+1}}$ of the form

\begin{equation*}X_{T_{2,2k+1}} = h^{0}_{1} \cup h^{0}_{2} \cup h^{1}_{1} \cup h^{1}_{2} \cup h^{1}_{3} \cup h^{2}_{r}.\end{equation*}

When we cancel $h_{2}^{0}$ with $h_{3}^{1}$ , we obtain a handle decomposition

(7.3) \begin{equation} X_{T_{2,2k+1}} = h^{0} \cup h_{1}^{1} \cup h_{2}^{1} \cup h_{r}^{2}. \end{equation}

Thus, $X_{T_{2,2k+1}}$ can be obtained from the genus two handlebody $Z$ by attaching a $2$ -handle $h_{r}^{2}$ along a simple closed curve representing the element $\alpha = a^{2k+1} c^{2} b^{2} d^{2k+1}$ where $a,c,b,d$ is a standard basis of $\pi _{1}(\partial Z)$ , with $c$ and $d$ representing meridians of $Z$ . In particular, $\pi _{1}(Z)$ is a rank-two free group generated by $a$ and $b$ , and $J$ , viewed as a curve in $Z$ , represents the element $a^{2k+1}b^{2}$ , as desired. Observe that there is no framing ambiguity in this case, see [Reference Gompf and Stipsicz18 , Example 4.1.4.(c)].