2.1 Twisted homology and cohomology

In this subsection, we briefly review twisted (co)homology and twisted intersection forms. While standard references include [Reference Kirk and Livingston30, Reference Friedl and Kim20], we also refer the reader to [Reference Cha, Miller and Powell7, Section 3] for a thorough discussion of the twisted homology of disconnected manifolds and the resulting long exact sequence of pairs. Additionally, for ease of exposition, we will focus on CW complexes. This is not cause a cause for concern, when working with 3-manifolds and smooth 4-manifolds, both of which admit the structure of finite CW complexes of the appropriate dimension; we refer to [Reference Friedl, Nagel, Orson and Powell25, Section 4] for the precise statements and references. However, while (topological) $4$ -manifolds need not admit CW structures, arguments involving twisted homology and Poincaré duality can nevertheless be carried out as in the smooth category (using singular chain complexes instead of cellular chain complexes): this is nontrivial, and we refer the interested reader to (Reference Friedl, Nagel, Orson and Powell25, Appendix A) for many more details and references.Footnote ²

Let X be a path connected CW complex and let $Y \subset X$ be a possibly empty subcomplex. Use $p \colon \widetilde {X} \to X$ to denote the universal cover of X and set $\widetilde {Y}:=p^{-1}(Y)$ . The left action of $\pi _1(X)$ on $\widetilde {X}$ endows the chain complex $C_*(\widetilde {X},\widetilde {Y})$ with the structure of a left $\mathbb {Z}[\pi _1(X)]$ -module. Moreover, let R be a ring and let M be a $(R,\mathbb {Z}[\pi _1(X)])$ -module. The chain complexes

$$ \begin{align*} & C_*(X,Y;M):=M \otimes_{\mathbb{Z}[\pi_1(X)]}C_*(\widetilde{X},\widetilde{Y}) \\ & C^*(X,Y;M):=\operatorname{Hom}_{\text{right-}\mathbb{Z}[\pi_1(X)]}(C_*(\widetilde{X}, \widetilde{Y} )^\#,M) \end{align*} $$

of left R-modules will be called the twisted (co)chain complex of $(X,Y)$ with coefficients in M. The corresponding homology left R-modules $H_*(X,Y;M)$ and $H^*(X,Y;M)$ will be called the twisted (co)homology modules of $(X,Y)$ with coefficients in M.

Assume that R is endowed with an involution $x \mapsto x^\#$ . Let $M,M'$ be $(R,\mathbb {Z}[\pi _1(X)])$ -bimodules and let S be a $(R,R)$ -bimodule. Furthermore, let $\langle -,-\rangle \colon M' \times M \to S$ be a non-singular $\pi _1(X)$ -invariant sesquilinear pairing, in the sense that $\langle m\gamma , n\gamma \rangle =\langle m,n\rangle $ and $\langle rm,sn \rangle =r \langle m,n \rangle s^\#$ for all $\gamma \in \pi _1(X)$ , all $r,s \in R$ and all $m\in M, n \in M'$ . Non-singularity means that the induced map $M\to \operatorname {\mathrm {Hom}}_{\text {left-}R}(M',S)^\#$ is an isomorphism. In this setting, as recalled in the appendix (specifically Construction A.3), there is an evaluation map

$$ \begin{align*}\operatorname{\mathrm{ev}} \colon H^i(X,Y;M) \to \operatorname{Hom}_{\text{left-}R}(H_i(X,Y;M'),S)^\#.\end{align*} $$

The evaluation map need not be an isomorphism. If R is a principal ideal domain, then the universal coefficient theorem implies that the cohomology group $H^i(X,Y;M)$ decomposes as the direct sum of $\operatorname {Hom}_{\text {left-}R}(H_i(X,Y;M'),S)^\#$ and $\operatorname {Ext}^1_{\text {left-}R}(H_i(X,Y;M'),S)^\#$ . In general, the evaluation map can be studied using the universal coefficient spectral sequence [Reference Levine34, Theorem 2.3]. Returning to $\pi _1(X)$ -invariant pairings, we now discuss two examples which we shall use throughout this section.

Example 2.1. Let R be an integral domain with field of fractions Q. Given a positive integer d, let $\beta \colon \pi _1(X) \to GL_d(R)$ be a representation and use $R^d_\beta $ to denote the $(R,\mathbb {Z}[\pi _1(X)])$ -module whose right $\mathbb {Z}[\pi _1(X)]$ -module structure is given by right multiplication by $\beta (\gamma )$ on row vectors for $\gamma \in \pi _1(X)$ . Use to denote the representation defined by and consider the pairings

Note that the use of the representation guarantees that these pairings are indeed $\pi _1(X)$ -invariant. In particular, if $\beta $ is a unitary representation (so that ), then the two pairings displayed above give rise to evaluation maps $H^i(X,Y;Q/R \otimes _R R_\beta ^d) \to \operatorname {Hom}_{\text {left-}R}(H_i(X,Y;R_{\beta }^d),Q/R)^\#$ and $H^i(X,Y; R_\beta ^d) \to \operatorname {Hom}_{\text {left-}R}(H_i(X,Y; R_{\beta }^d),R)^\#$ .

Next, we briefly review the definition of twisted intersection forms. Let W be a compact oriented n-manifold and let M be a $(R,\mathbb {Z}[\pi _1(W)])$ -bimodule. As we mentioned above, if W does not admit a CW structure, we can use singular homology; see (Reference Friedl, Nagel, Orson and Powell25, Appendix A). The cap product with the fundamental class $[W, \partial W] \in H_{4}(W, \partial W)$ induces twisted Poincaré duality isomorphisms

$$ \begin{align*}H_k(W,\partial W;M) \cong H^{n-k}(W;M) \quad H_k(W;M) \cong H^{n-k}(W,\partial W;M),\end{align*} $$

both of which we denote by $\text {PD}$ ; for more details, we refer to (Reference Friedl, Nagel, Orson and Powell25, Appendix A).

In order to describe twisted intersection pairings, fix an $(R,R)$ -bimodule S and a non-singular $\pi _1(W)$ -invariant sesquilinear pairing $M \times M \to S$ . Compose the homomorphism induced by the inclusion $(W,\emptyset ) \to (W,\partial W)$ with Poincaré duality and the evaluation homomorphism described above. The result is the following homomorphism of left R-modules:

$$ \begin{align*}\Phi \colon H_k(W;M) \to H_k(W,\partial W;M) \xrightarrow{\text{PD}} H^{n-k}(W;M) \xrightarrow{\operatorname{\mathrm{ev}}}\operatorname{Hom}_{\text{left-}R}(H_{n-k}(W;M),S)^\#.\end{align*} $$

Restricting to the case where W is $2n$ -dimensional, the twisted intersection pairing

$$ \begin{align*}\lambda_{M,W} \colon H_n(W;M) \times H_n(W;M) \to S\end{align*} $$

is defined by $\lambda _{M,W}(x,y)=\Phi (y)(x)$ . Note that while $\lambda _{M,W}$ is $(-1)^n$ -Hermitian, it may be singular: the submodule $\operatorname {im} (H_{n}(\partial W;M) \to H_{n}(W;M)) $ is annihilated by $\lambda _{M,W}$ , and the evaluation map need not be an isomorphism.

2.2 The twisted Blanchfield pairing

In this subsection, we review twisted Blanchfield pairings. While the construction can be performed over non-commutative rings [Reference Powell42, Reference Cochran, Orr and Teichner9, Reference Cochran8], we focus first on the case $R=\mathbb {F}[t^{\pm 1}]$ . References on twisted Blanchfield pairings include [Reference Miller and Powell40, Reference Powell42].

Let N be a closed $3$ -manifold. Just as in Example 2.1, let $\beta \colon \pi _1(N) \to GL_d({\mathbb {F}}[t^{\pm 1}])$ be a unitary representation and use $\mathbb {F}[t^{\pm 1}]^d_\beta $ to denote the $({\mathbb {F}}[t^{\pm 1}],\mathbb {Z}[\pi _1(X)])$ -bimodule whose right $\mathbb {Z}[\pi _1(N)]$ -module structure is given by right multiplication by $\beta (\gamma )$ on row vectors. In what follows, we shall think of $\mathbb {F}[t^{\pm 1}], \mathbb {F}(t)$ and $\mathbb {F}(t)/\mathbb {F}[t^{\pm 1}]$ as $(\mathbb {F}[t^{\pm 1}],\mathbb {F}[t^{\pm 1}])$ -bimodules. Since

$$\begin{align*}\mathbb{F}(t)^{d}_{\beta} := \mathbb{F}(t^{\pm 1}) \otimes_{\mathbb{F}[t^{\pm 1}]}~\mathbb{F}[t^{\pm 1}]^d_\beta \quad \text{and} \quad (\mathbb{F}(t)/\mathbb{F}[t^{\pm1}])^{d}_{\beta} := \mathbb{F}(t)/\mathbb{F}[t^{\pm 1}] \otimes_{\mathbb{F}[t^{\pm 1}]}~\mathbb{F}[t^{\pm 1}]^d_\beta\end{align*}$$

are $(\mathbb {F}[t^{\pm 1}],\mathbb {Z}[\pi _1(N)])$ -bimodules, there is a short exact sequence

$$ \begin{align*}0 \to C^*(N; \mathbb{F}[t^{\pm 1}]^d_\beta) \to C^*(N,\mathbb{F}(t)^{d}_{\beta}) \to C^*(N;(\mathbb{F}(t)/ \mathbb{F}[t^{\pm1}])^{d}_{\beta}) \to 0\end{align*} $$

of cochain complexes of left $\mathbb {F}[t^{\pm 1}]$ -modules, where we identified $\mathbb {F}[t^{\pm 1}]^d_\beta $ with $\mathbb {F}[t^{\pm 1}] \otimes _{{\mathbb {F}}[t^{\pm 1}]} \mathbb {F}[t^{\pm 1}]^d_\beta $ . Passing to cohomology, we obtain a long exact sequence in which the connecting map

$$ \begin{align*}\operatorname{BS} \colon H^{k}(N;(\mathbb{F}(t)/\mathbb{F}[t^{\pm 1}])^{d}_{\beta}) \to H^{k+1}(N;\mathbb{F}[t^{\pm 1}]^d_\beta)\end{align*} $$

is called the Bockstein homomorphism. We say that $\beta $ is $H_1$ -null if $H_1(N;\mathbb {F}(t)^{d}_{\beta })$ vanishes (i.e., if the ${\mathbb {F}}[t^{\pm 1}]$ -module $H_1(N;{\mathbb {F}}[t^{\pm 1}]_\beta ^d)$ is torsion). Observe that if $\beta $ is $H_1$ -null, then the corresponding Bockstein homomorphism is an isomorphim: indeed, Poincaré duality and the universal coefficient theorem respectively imply that $H^2(N;\mathbb {F}(t)^d_\beta )$ and $H^1(N;(\mathbb {F}(t) /\mathbb {F}[t^{\pm 1}])^d_\beta )$ vanish.

Consider the composition

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

of the three following $\mathbb {F}[t^{\pm 1}]$ -homomorphisms: Poincaré duality, the inverse Bocktein and the evaluation map described in Example 2.1. The main definition of this section is the following.

Definition 2.2. 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 $H_1$ -null representation. The twisted Blanchfield pairing associated to $\beta $

$$ \begin{align*}\operatorname{\mathrm{Bl}}_\beta(N) \colon H_1(N;\mathbb{F}[t^{\pm 1}]^d_\beta) \times H_1(N;\mathbb{F}[t^{\pm 1}]^d_\beta) \to \mathbb{F}(t)/\mathbb{F}[t^{\pm 1}]\end{align*} $$

is defined as $\operatorname {\mathrm {Bl}}_\beta (N)(x,y)=\Theta (y)(x)$ . If $N=M_K$ is the $0$ -framed surgery along a knot K, then we write $\operatorname {\mathrm {Bl}}_\beta (K)$ instead of $\operatorname {\mathrm {Bl}}_\beta (M_K)$ .

We refer to [Reference Powell42] for a more general treatment of twisted Blanchfield pairings but make two remarks nonetheless. First, Powell proved that $\operatorname {\mathrm {Bl}}_\beta (N)$ is Hermitian [Reference Powell42]. Second, since $\mathbb {F}[t^{\pm 1}]$ is a PID, it follows that $\operatorname {\mathrm {Bl}}_\beta (N)$ is non-singular: indeed, $\mathbb {F}(t)/\mathbb {F}[t^{\pm 1}]$ is an injective $\mathbb {F}[t^{\pm 1}]$ -module, and thus, the evaluation map is an isomorphism.

Finally, although we mostly focus on closed 3-manifolds, we conclude this subsection with a remark on manifolds with boundary. To describe this case, we say that a representation $\beta \colon \pi _1(N) \to ~GL_d({\mathbb {F}}[t^{\pm 1}])$ is acyclic if $H_i(N;\mathbb {F}[t^{\pm 1}]_\beta ^d)$ is a torsion $\mathbb {F}[t^{\pm 1}]$ -module for each i.

2.3 Relation to Blanchfield pairings over noncommutative rings

Twisted Blanchfield pairing are also defined in the non-commutative setting, using Ore rings [Reference Cochran, Orr and Teichner9, Reference Friedl, Leidy, Nagel and Powell21, Reference Powell42]. In this subsection, after briefly reviewing some basics on these rings, we relate the resulting non-commutative Blanchfield pairings to the twisted Blanchfield pairings of Subsection 2.2. The reason for this incursion into the non-commutative setting is that the proof of Theorem 3.11 requires that we import a result from [Reference Friedl, Leidy, Nagel and Powell21] (which uses non-commutative Blanchfield forms) into the setting of twisted Blanchfield forms.

We start with some brief recollections on Ore rings, referring to [Reference Lam31, Chapter 4, Section 10] for further details. Let $\mathcal {R}$ be a ring. A multiplicative subset $S \subset \mathcal {R}$ is right permutable if, for any $r \in \mathcal {R}$ and $s \in S$ , we have $r S \cap s\mathcal {R} \neq \emptyset $ . An element $r \in \mathcal {R}$ is regular if it is neither a left zero divisor nor a right zero divisor. We say that $\mathcal {R}$ is a right Ore ring if the set S of regular elements is right permutable. In this case, $\mathcal {R}$ can be localized at S, and $\mathcal {R}S^{-1}$ is called the right ring of quotients of $\mathcal {R}$ . The left analogues of these notions are defined similarly. If $\mathcal {R}$ is both a left Ore ring and a right Ore ring, then $\mathcal {R}$ is called an Ore ring. In this case, the right and left rings of quotients agree, and we simply write $\mathcal {Q}$ .

Commutative rings are Ore rings [Reference Lam31, Chapter 4, Section 10, 10.18]. In what follows, however, we shall mostly be concerned with rings of matrices.

Next, suppose that R is a commutative ring with unit. Right multiplication on row vectors endows $R^{d}$ with a right $\mathcal {M}_{d}(R)$ -module structure. We write $R^{d}_{\mathcal {M}}$ for emphasis. An alternate description of $R^{d}_{\mathcal {M}}$ is obtained by considering the elementary matrix $E_{i,j}$ whose only nonzero coefficient is a $1$ in its $(i,j)$ -entry. Indeed, one has the isomorphim $ E_{i,i}\mathcal {M}_d(R) \cong R^{d}_{\mathcal {M}}$ of right $\mathcal {M}_d(R)$ -modules.

The next lemma shows that $\mathcal {M}_d(R)$ decomposes as a direct sum of d copies of $R^{d}_{\mathcal {M}}$ .

Lemma 2.5. If R is a commutative ring with unit, then $R^{d}_{\mathcal {M}}$ is a projective right $\mathcal {M}_{d}(R)$ -module. In fact, we have the following isomorphism of right $\mathcal {M}_{d}(R)$ -modules:

(2.1) $$ \begin{align} \mathcal{M}_{d}(R) \cong \bigoplus_{i=1}^d E_{i,i}\mathcal{M}_{d}(R) \cong \bigoplus_{i=1}^d R^{d}_{\mathcal{M}}. \end{align} $$

Next, assume that X is a CW complex together with a homomorphism $\gamma \colon \mathbb {Z}[\pi _1(X)] \to \mathcal {R}$ to an Ore ring $\mathcal {R}$ . Just as in Subsection 2.1, this endows $\mathcal {R}$ with a right $\mathbb {Z}[\pi _1(X)]$ -module structure which we denote by $\mathcal {R}_\gamma $ for emphasis. Given a commutative ring R with unit, note that any representation $\gamma \colon \pi _{1}(X) \to GL_d(R)$ canonically extends to a ring homomorphism $\gamma \colon \mathbb {Z}[\pi _{1}(X)] \to \mathcal {M}_{d}(R)$ . In particular, $\gamma $ endows $R^d$ and $\mathcal {R}=\mathcal {M}_d(R)$ with right $\mathbb {Z}[\pi _{1}(X)]$ -module structures.

The following proposition is a topological application of Lemma 2.5.

Proposition 2.6. Let R be a commutative ring with unit. If $\gamma \colon \pi _1(X) \to GL_d(R)$ is a representation, then we have the following chain isomorphisms of chain complexes of left R-modules:

$$ \begin{align*} & C_*(X;\mathcal{M}_d(R)_\gamma) \cong \bigoplus_{i=1}^d C_*(X;R_\gamma^d), \\ &C^*(X;\mathcal{M}_d(R)_\gamma) \cong \bigoplus_{i=1}^d C^*(X;R_\gamma^d). \end{align*} $$

Furthermore, one also has the following isomorphisms of left R-modules:

$$ \begin{align*} & H_*(X;R_\gamma^d) \cong R^d_{\mathcal{M}} \otimes_{\mathcal{M}_d(R)} H_*(X;\mathcal{M}_d(R)_\gamma) , \\ & H^*(X;R_\gamma^d) \cong R^d_{\mathcal{M}} \otimes_{\mathcal{M}_d(R)} H^*(X;\mathcal{M}_d(R)_\gamma). \end{align*} $$

Proof. The chain isomorphisms follow from the decomposition of $\mathcal {M}_d(R)$ displayed in (2.1). To deal with the second assertion, we once again use Lemma 2.5 to deduce that $\mathcal {M}_d(R)$ is a projective (and thus flat) $R_{\mathcal {M}}^d$ -module. We therefore obtain the following isomorphisms:

$$ \begin{align*}H_*(X;R_\gamma^d) \cong H_*(X; R_{\mathcal{M}}^d \otimes_{\mathcal{M}_d(R)} \mathcal{M}_d(R)_\gamma) \cong R_{\mathcal{M}}^d \otimes_{\mathcal{M}_d(R)} H_*(X; \mathcal{M}_d(R)_\gamma).\end{align*} $$

The proof for cohomology is analogous. This concludes the proof of the lemma.

Next, we deal with non-commutative Blanchfield pairings. Let N be a closed oriented $3$ -manifold. Let $\mathcal {R}$ be an Ore ring with involution and use $\mathcal {Q}$ to denote its ring of quotients. Given a homomorphism $\gamma \colon \mathbb {Z}[\pi _1(N)] \to \mathcal {R}$ , we can view $\mathcal {R},\mathcal {Q}$ and $\mathcal {Q}/\mathcal {R}$ as right $\mathbb {Z}[\pi _1(N)]$ -modules. In particular, as in Subsections 2.1 and 2.2, there are Poincaré duality isomorphisms and a Bockstein homomorphism on cohomology. Thus, if $H_1(N;\mathcal {Q}_\gamma )=0$ , then one can consider the composition

$$ \begin{align*}\Phi \colon H_1(N;\mathcal{R}_\gamma) \xrightarrow{\operatorname{PD}} H^2(N;\mathcal{R}_\gamma) \xrightarrow{BS^{-1}} H^1(N;\mathcal{Q}/\mathcal{R}_\gamma) \xrightarrow{\operatorname{ev} \circ \kappa} \operatorname{Hom}_{\mathcal{R}}(H_1(N;\mathcal{R}_\gamma);\mathcal{Q}/\mathcal{R})^\#.\end{align*} $$

We refer to [Reference Friedl, Leidy, Nagel and Powell21, Section 2] for further details, but the upshot is that there is a Blanchfield pairing in the non-commutative setting which is defined exactly as in Subsection 2.2.

Definition 2.7. Let N be a closed oriented $3$ -manifold together with a map $\gamma \colon \mathbb {Z}[\pi _1(N)] \to \mathcal {R}$ to an Ore ring with involution. Assuming that $H_1(N;\mathcal {Q}_\gamma )=0$ , there is a non-commutative Blanchfield pairing

$$ \begin{align*}\operatorname{\mathrm{Bl}}_{\mathcal{R},\gamma}(N) \colon H_1(N;\mathcal{R}_\gamma) \times H_1(N;\mathcal{R}_\gamma) \to \mathcal{Q}/\mathcal{R}\end{align*} $$

which is defined by setting $\operatorname {\mathrm {Bl}}_{\mathcal {R},\gamma }(N)(x,y)=\Phi (y)(x)$ .

Note that in the literature $\mathcal {R}$ , is usually assumed to be an Ore domain; however, the definition extends verbatim to the case of Ore rings. The next proposition relates the non-commutative Blanchfield pairing to the twisted Blanchfield pairing.

Proof. The first assertion follows immediately from the first assertion of Proposition 2.6. Next, we deal with the second assertion. Using once again the first assertion of Proposition 2.6, the linking form $\operatorname {\mathrm {Bl}}_{\mathcal {M}_d({\mathbb {F}}[t^{\pm 1}]),\gamma }(N)$ can canonically be thought of as a linking form

$$ \begin{align*}\operatorname{\mathrm{Bl}}_{\mathcal{M}_d({\mathbb{F}}[t^{\pm 1}]),\gamma}(N) \colon \bigoplus_{i=1}^d H_1(N;{\mathbb{F}}[t^{\pm 1}]_\gamma^d) \times \bigoplus_{i=1}^d H_1(N;{\mathbb{F}}[t^{\pm 1}]_\gamma^d) \to \mathcal{M}_d(\mathbb{F}(t))/\mathcal{M}_d({\mathbb{F}}[t^{\pm 1}]),\end{align*} $$

where the above decomposition is obtained by using the matrices $E_{i,i}$ , for $1 \leq i \leq d$ . The result now follows by tensoring with ${\mathbb {F}}[t^{\pm 1}]_{\mathcal {M}}^d$ and using the second assertion of Proposition 2.6. Indeed, both Blanchfield pairings are defined by composing Poincaré duality isomorphisms, inverse Bockstein isomorphisms and evaluation maps. The first two maps preserve the direct sum decomposition displayed above. After applying the adjoint of evaluation map, and using the identification from Proposition 2.6, $\operatorname {\mathrm {Bl}}_{\mathcal {M}_d({\mathbb {F}}[t^{\pm 1}])}$ is given by

$$\begin{align*}(E_{i,i} \cdot x, E_{j,j} \cdot y) \mapsto\operatorname{\mathrm{Bl}}_{\gamma}(x,y) E_{i,j} \in \mathcal{M}_{d}(\mathbb{F}(t) / {\mathbb{F}}[t^{\pm 1}]),\end{align*}$$

for $x,y \in H_{1}(N;{\mathbb {F}}[t^{\pm 1}]^{d}_{\gamma })$ . This concludes the proof of the proposition.

Informally, Proposition 2.8 implies that several statements that hold for non-commutative Blanchfield pairings also hold for twisted Blanchfield pairings. This will be used in Theorem 3.11.

3.1 Definition of the twisted signatures

In this short subsection, we define our twisted signature invariants for knots and 3-manifolds. To this effect, recall that Proposition 1.6 associated to a linking form $(M,\lambda )$ a locally constant functions $\sigma _{(M,\lambda )} \colon S^1 \to \mathbb {Z} $ and $\sigma _{(M,\lambda )}^{av} \colon S^1 \to \mathbb {Z}$ with desirable properties.

The corresponding invariants for a knot K are obtained by applying Definition 3.1 to the 0-framed surgery $M_K$ .

Before proving properties, we show how Definition 3.2 recovers the Levine-Tristram signature. This establishes the fourth property of Theorem 1.1 from the introduction.

Contrarily to $\sigma _K$ , the twisted signature function $\sigma _{K,\beta }$ is not known to be symmetric (i.e., $\sigma _{K,\beta }(\overline {\omega })$ is not known to equal $\sigma _{K,\beta }(\omega )$ ). More precisely, in [Reference Borodzik, Conway and Politarczyk3], we show that signature functions associated to abstract linking forms over ${\mathbb {C}}[t^{\pm 1}]$ are not symmetric, but it remains unknown whether such linking forms can be realized as twisted Blanchfield forms. Despite this difference, the next proposition shows that the twisted signature shares several properties with its classical counterpart. This establishes the first item of Theorem 1.1 from the introduction.

The order of $H_1(M_K;\mathbb {F}[t^{\pm 1}]_\beta ^d)$ can be related to the twisted Alexander polynomial of K which is usually defined on the exterior $X_K$ of K; see [Reference Kirk and Livingston30, Lemma 6.3] and [Reference Friedl and Vidussi24, Lemma 3]. In the abelian case, since $H_1(M_K;{\mathbb {F}}[t^{\pm 1}]) \cong H_1(X_K;{\mathbb {F}}[t^{\pm 1}])$ , the first item of Proposition 3.4 (and Remark 3.3) recovers the well-known fact that the Levine-Tristram signature function $\sigma _K$ is constant on the complement of the zero set of the Alexander polynomial $\Delta _K(t)$ . The second item of Proposition 3.4 is also a direct generalization of the corresponding property for the Levine-Tristram signature.

Motivated by Remark 3.3 and Proposition 3.4, the next subsections investigate to what extent the properties of the Levine-Tristram signature generalize to our twisted signatures.

3.2 Bordism and concordance invariance

In this subsection, we give conditions under which the signatures of 3-manifolds (resp. knots) are bordism (resp. concordance) invariants.

The proof of the next lemma can be found in [Reference Letsche32, Proposition 2.8] or [Reference Cochran, Orr and Teichner9, Theorem 4.4].

The following result uses Lemma 3.6 to provide (slightly less restrictive) assumptions under which the twisted Blanchfield pairing is metabolic.

Next, suppose that K and $K'$ are two concordant knots in $S^3$ and let $M_K$ and $M_{K'}$ denote their respective $0$ -framed surgeries. Given a locally flat concordance $\Sigma \subset S^{3} \times I$ from K to $K'$ , we can construct a cobordism $W_{\Sigma }$ from $M_{K}$ to $M_{K'}$ . Indeed, let $\mathcal {N}(\Sigma )$ denote a tubular neighborhood of $\Sigma $ (which can be identified with $\mathcal {N}(\Sigma ) \cong S^{1} \times D^{2} \times I$ ).Footnote ³ The desired cobordism is then obtained by setting

$$\begin{align*}W_{\Sigma} = ((S^{3} \times I) \setminus \mathcal{N}(\Sigma)) \cup_{h} S^{1} \times D^{2} \times I,\end{align*}$$

where $h \colon S^{1} \times S^{1} \times I \to S^{1} \times S^{1} \times I$ is the homeomorphism defined by $h(x,y,z) = (y,x,z)$ . The next result shows that the averaged twisted signature is a concordance invariant as long as the corresponding representations extend through the complement of the concordance to an acyclic representation.

The same proof yields a result for slice knots with a representation which extends to an acylic representation over the slice disk exterior. We record this result for completeness, as it was mentioned in the second item of Theorem 1.1 from the introduction.

3.3 Satellite formulas for twisted signatures

In this subsection, we use a result of Friedl-Leidy-Nagel-Powell [Reference Friedl, Leidy, Nagel and Powell21] to understand the behavior of the twisted Blanchfield pairings and signatures under various satellite operations. In particular, these results provide twisted generalizations of well-known theorems for the classical Blanchfield pairing and Levine-Tristram signature.

Let Y be a $3$ -manifold with empty or toroidal boundary. Suppose that we are given a simple closed curve $\eta $ in Y. If $K \subset S^{3}$ is a knot, then an infection of Y by K is a $3$ -manifold $Y_{K}$ obtained by gluing $Y_{K} = (Y \setminus \mathcal {N}(\eta )) \cup _{\partial } S^{3} \setminus \mathcal {N}(K)$ , where we glue a zero-framed longitude of K to the meridian of $\eta $ and some longitude of $\eta $ to the meridian of K. Notice that the ambiguity in the choice of the longitude of $\eta $ may lead to different outcomes; however, as pointed out in [Reference Friedl, Leidy, Nagel and Powell21], the resulting twisted Blanchfield pairing does not depend on this choice. Note that if Y has a boundary, then so does $Y_K$ . In this case, we use Remark 2.3 which describes twisted Blanchfield pairings for $3$ -manifolds with boundary.

Let $P \subset S^{3}$ be a knot and let $M_{P}$ be the zero-surgery on P. If $K \subset S^{3}$ is another knot and $\eta $ is a simple closed curve in the complement of P, we can form the satellite knot $P(K,\eta )$ with pattern P, companion C and infection curve $\eta $ by looking at 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.

From this description, one can see that the zero-surgery on the satellite knot $P(K,\eta )$ can be obtained by infection of $M_{P}$ by K along $\eta \subset S^{3} \setminus P \subset M_{P}$ :

(3.2) $$ \begin{align} M_{P(K,\eta)}=M_P \setminus \mathcal{N}(\eta) \cup_\partial S^3 \setminus \mathcal{N}(K). \end{align} $$

Let $\mu _{\eta }$ denote a meridian of $\eta $ . Our goal in this section is to express the Blanchfield form $\operatorname {\mathrm {Bl}}_\gamma (M_{P(K,\eta )})$ in terms of the twisted Blanchfield forms on $M_{P}$ and $M_{K}$ . Using (3.2), notice that given a representation $\gamma \colon \pi _{1}(M_{P(K,\eta )}) \to GL_d(\mathbb {F}[t^{\pm 1}])$ , there are induced representations

$$\begin{align*}\gamma_{P} \colon \pi_{1}(M_{P} \setminus \mathcal{N}(\eta)) \to GL_d(\mathbb{F}[t^{\pm1}]), \quad \gamma_{K} \colon \pi_{1}(S^{3} \setminus \mathcal{N}(K)) \to GL_d(\mathbb{F}[t^{\pm1}]).\end{align*}$$

In general, however, there is no guarantee that these representations can be extended to representations of the fundamental groups of $M_{P}$ and $M_{K}$ .

We say that a representation $\gamma \colon \pi _{1}(M_{P(K,\eta )}) \to GL_d(\mathbb {F}[t^{\pm 1}])$ is $\eta $ -regular if $\gamma (\mu _{\eta }) = \operatorname {id}$ and $\det (\operatorname {id} - \gamma (\eta )) \neq 0$ . As explained in Remark 3.10, if $\gamma $ is $\eta $ -regular, then it induces well-defined representations $\gamma _P$ and $\gamma _K$ on $\pi _1(M_P)$ and $\pi _1(M_K)$ .

Before proceeding to the next theorem, it is necessary to recall some terminology from [Reference Borodzik, Conway and Politarczyk3, Section 4.1]. If $(M,\lambda )$ is a linking form, then, for any submodule $L \subset M$ , the orthogonal complement of L is defined as $L^{\perp } = \{x \in M \mid \ \lambda (x,y) = 0 \ \forall {y \in L} \}$ . We say that L is a sublagrangian (or isotropic) submodule if $L \subset L^{\perp }$ . For a sublagrangian submodule $L \subset M$ , the sublagrangian reduction of $M $ by $ L $ is defined as $(L^{\perp }/L, \lambda _{L})$ , where $\lambda _{L}(x+L,y+L) := \lambda (x,y)$ . Note that $\lambda $ and $\lambda _L$ are Witt equivalent, and therefore, by Proposition 1.6, the averaged signature functions of $(M,\lambda )$ and its sublagrangian reduction are equal. The following theorem describes the twisted Blanchfield pairing of satellite knots and proves Theorem 1.5 from the introduction.

Theorem 3.11. Let $K,P$ be knots and let $\eta $ be a simple closed curve in $S^3 \setminus P \subset M_P$ . Let $\gamma \colon \pi _{1}(M_{P(K,\eta )}) \to GL_d(\mathbb {F}[t^{\pm 1}])$ be a unitary acyclic $\eta $ -regular representation. Let $T \subset M_{P(K,\eta )}$ denote the common boundary of $M_{P}\setminus \mathcal {N}(\eta )$ and $S^{3} \setminus \mathcal {N}(K)$ . Define

$$\begin{align*}L = \operatorname{im}(H_{1}(T;{\mathbb{F}}[t^{\pm 1}]_{\gamma}^{d}) \to H_{1}(M_{P(K,\eta)};{\mathbb{F}}[t^{\pm 1}]^{d}_{\gamma})).\end{align*}$$

The following assertions hold:

1. (1) The representations $\gamma _P$ and $\gamma _K$ are unitary and acyclic.

2. (2) The module L is isotropic and $\operatorname {\mathrm {ord}} L$ divides $\det (1-\gamma (\eta ))$ .

3. (3) The sublagrangian reduction of $\operatorname {\mathrm {Bl}}_\gamma (P(K,\eta ))$ with respect to L is isometric to the direct sum $\operatorname {\mathrm {Bl}}_{\gamma _{P}}(P) \oplus \operatorname {\mathrm {Bl}}_{\gamma _K}(K)$ . In particular, $\operatorname {\mathrm {Bl}}_\gamma (P(K,\eta ))$ is Witt equivalent to $\operatorname {\mathrm {Bl}}_{\gamma _{P}}(P) \oplus \operatorname {\mathrm {Bl}}_{\gamma _K}(K)$ .

The proof relies on the following result from [Reference Borodzik, Conway and Politarczyk3, Theorem 4.13] whose statement we recall for the reader’s convenience.

Theorem 3.12. Let $(M',\lambda ')$ and $(M",\lambda ")$ be two non-singular linking forms over ${\mathbb {F}}[t^{\pm 1}]$ , let M be a ${\mathbb {F}}[t^{\pm 1}]$ -module and let $\iota \colon M\to M'$ be a monomorphism. Write $\lambda $ for the form on M induced by $\iota $ . Suppose that $\pi \colon (M,\lambda )\to (M",\lambda ")$ is a surjective morphism of linking forms and set $L:=\ker (\pi )$ and $L':=\iota (L)$ . Then $\operatorname {\mathrm {ord}}(M)\operatorname {\mathrm {ord}}(M)^\#$ divides $\operatorname {\mathrm {ord}}(M')\operatorname {\mathrm {ord}}(M")$ . Moreover, if

(3.3) $$ \begin{align}\operatorname{\mathrm{ord}}(M)\operatorname{\mathrm{ord}}(M)^\#\doteq \operatorname{\mathrm{ord}}(M")\operatorname{\mathrm{ord}}(M'), \end{align} $$

then the following statements hold:

1. (1) $\iota (M)={L'}^{\perp }$ ;

2. (2) The linking form $(M",\lambda ")$ is isometric to the sublagrangian reduction of $M'$ with respect to $L'$ . In particular, $(M',\lambda ')$ and $(M",\lambda ")$ are Witt equivalent.

Proof of Theorem 3.11

During this proof, in order to simplify the notation, we will omit coefficients. As we mentioned in the statement, the torus T can be described either as $\partial (M_{P} \setminus \mathcal {N}(\eta ))$ or as $ \partial (S^{3} \setminus \mathcal {N}(K))$ . As a consequence, we have $M_{P(K,\eta )} = (M_{P} \setminus \mathcal {N}(\eta ))\cup _{ T}(S^{3} \setminus \mathcal {N}(K))$ . Applying [Reference Friedl, Leidy, Nagel and Powell21, Theorem 1.1], we know that if $H_*(T)$ and $H_*(M_{P(K,\eta )})$ are ${\mathbb {F}}[t^{\pm 1}]$ -torsion modules, then $H_*(M_{P} \setminus \mathcal {N}(\eta ))$ and $H_*(S^{3} \setminus \mathcal {N}(K))$ are also ${\mathbb {F}}[t^{\pm 1}]$ -torsion modules, and the direct sum of the canonical inclusions induces the following morphism of linking forms:

(3.4) $$ \begin{align} \psi \colon \operatorname{\mathrm{Bl}}_{\gamma_P}(M_{P} \setminus \mathcal{N}(\eta)) \oplus \operatorname{\mathrm{Bl}}_{\gamma_K}(S^{3} \setminus \mathcal{N}(K)) \to \operatorname{\mathrm{Bl}}_\gamma(M_{P(K,\eta)}). \end{align} $$

Note that although [Reference Friedl, Leidy, Nagel and Powell21, Theorem 1.1] is stated for Ore domain coefficients, Proposition 2.8 implies that it also holds in the twisted setting. The next paragraph is devoted to checking that $H_*(M_{P(K,\eta )})$ and $H_*(T)$ are indeed ${\mathbb {F}}[t^{\pm 1}]$ -torsion modules. Since the representation $\gamma $ was assumed to be acyclic, $H_*(M_{P(K,\eta )})$ is indeed torsion. We must only show that $H_*(T)$ is torsion. We assert that

(3.5) $$ \begin{align} H_{i}(T) = \begin{cases} \mathbb{F}[t^{\pm1}]^{d} / (1-\gamma(\eta)) & i = 0,1, \\ 0 & \text{otherwise}. \end{cases} \end{align} $$

Indeed, arguing as in [Reference Powell41, Proposition 3.11], we see that the twisted chain complex $C_{\ast }(T)$ is chain homotopy equivalent to

$$ \begin{align*}\mathbb{F}[t^{\pm1}]^{d} \xrightarrow{\left(1-\gamma(\eta) \quad\gamma(\mu_{\eta})-1\right)^{T}} \mathbb{F}[t^{\pm1}]^{2d} \xrightarrow{\left(1-\gamma(\mu_{\eta}) \quad 1-\gamma(\eta)\right)} \mathbb{F}[t^{\pm1}]^{d}\end{align*} $$

with a basis of $C_{1}(T) \cong \mathbb {F}[t^{\pm 1}]^{2d} $ given by the $\mu _{\eta } \otimes e_i $ and $\eta \otimes e_i$ for $i=1,\ldots ,d$ . Since $\gamma $ is $\eta $ -regular, $\gamma (\mu _{\eta })=1$ and the assertion follows. For later use, note that this argument also proves that the $\mathbb {F}[t^{\pm 1}]$ -module $H_{1}(T)$ is generated by the homology classes $[e_i \otimes \mu _{\eta }]$ for $i=1,\ldots ,d$ . Since $\gamma $ is $\eta $ -regular, (3.5) shows that $H_*(T)$ is torsion, concluding our verification of the assumptions of [Reference Friedl, Leidy, Nagel and Powell21, Theorem 1.1] and thus establishing the existence of the morphism $\psi $ in (3.4).

Our goal is now to relate the Blanchfield pairings on $M_P,M_K$ and $M_{P(K,\eta )}$ . Observe that the ${\mathbb {F}}[t^{\pm 1}]$ -modules $H_1(M_{P(K,\eta )}), H_1(M_{P} \setminus \mathcal {N}(\eta ),T)$ and $H_1(S^{3} \setminus \mathcal {N}(K),T)$ are torsion: for $H_1(M_{P(K,\eta )})$ , this follows from the $\eta $ -regularity of $\gamma $ . For the two latter modules, this can be seen by looking at the long exact sequence of the pairs $(S^{3} \setminus \mathcal {N}(K),T)$ and $(M_{P} \setminus \mathcal {N}(\eta ),T)$ together with the fact that $H_1(M_{P} \setminus \mathcal {N}(\eta )),H_1(S^{3} \setminus \mathcal {N}(K))$ and $H_*(T)$ are all ${\mathbb {F}}[t^{\pm 1}]$ -torsion.

Combining this observation with $ H_{0}(M_{P} \setminus \mathcal {N}(\eta ),T) = 0=H_{0}(M_{P} \setminus \mathcal {N}(\eta ),T)$ , Poincaré duality and the universal coefficient theorem (where the underlying ring is the PID ${\mathbb {F}}[t^{\pm 1}]$ ), we obtain the following isomorphisms:

(3.6) $$ \begin{align} H_{2}(M_{P(K,\eta)}) &\cong H^{1}(M_{P(K,\eta)}) \cong \operatorname{Ext}^{1}_{\text{left-}{\mathbb{F}}[t^{\pm 1}]}(H_{0}(M_{P(K,\eta)}),{\mathbb{F}}[t^{\pm 1}])^{\#} \cong H_{0}(M_{P(K,\eta)})^{\#},\end{align} $$

(3.7) $$ \begin{align} H_{2}(M_{P} \setminus \mathcal{N}(\eta)) &\cong H^{1}(M_{P}\setminus \mathcal{N}(\eta),T) \cong \operatorname{Ext}^{1}_{{\mathbb{F}}[t^{\pm 1}]}(H_{0}(M_{P} \setminus \mathcal{N}(\eta),T),{\mathbb{F}}[t^{\pm 1}]) = 0, \end{align} $$

(3.8) $$ \begin{align} H_{2}(S^{3} \setminus \mathcal{N}(K)) &\cong H^{1}(S^{3}\setminus \mathcal{N}(K),T) \cong \operatorname{Ext}^{1}_{{\mathbb{F}}[t^{\pm 1}]}(H_{0}(S^{3}\setminus \mathcal{N}(K),T),{\mathbb{F}}[t^{\pm 1}]) = 0. \end{align} $$

In more details, in each of the three lines, in our application of the universal coefficient theorem, we used our observation that the relevant first homology modules are ${\mathbb {F}}[t^{\pm 1}]$ -torsion. In the third isomorphism of (3.6), we also used the fact that for any torsion ${\mathbb {F}}[t^{\pm 1}]$ -module T, we have $\operatorname {Ext}^{1}_{{\mathbb {F}}[t^{\pm 1}]}(T,{\mathbb {F}}[t^{\pm 1}]) \cong ~T$ .

Our strategy to relate the various Blanchfield pairings is to apply Theorem 3.12. We now introduce the notation necessary to its application as well as verify one of its key assumptions.

Lemma 3.13. The inclusions $\iota _{P} \colon M_{P} \setminus \mathcal {N}(\eta ) \hookrightarrow M_{P}$ and $\iota _{K} \colon S^{3} \setminus \mathcal {N}(K) \hookrightarrow M_{K}$ induce isomorphisms on twisted homology in degree $j \neq 1,2$ and epimorphisms $ (\iota _{P})_{\ast }, (\iota _{K})_{\ast }$ in degree one. Moreover, $\ker (\iota _{P})_{\ast }$ and $\ker (\iota _{K})_{\ast }$ are generated by the $[e_i \otimes \mu _{\eta }]$ for $i=1,\ldots , d$ . Additionally, the modules $H_1(M_K)$ and $H_1(M_P)$ are ${\mathbb {F}}[t^{\pm 1}]$ -torsion. Finally, we have the following equalities:

$$ \begin{align*} \operatorname{\mathrm{ord}} H_{1}(M_{P} \setminus \mathcal{N}(\eta)) &\doteq \frac{\operatorname{\mathrm{ord}}(H_{1}(M_{P})) \cdot \operatorname{\mathrm{ord}}(H_{1}(T))}{\operatorname{\mathrm{ord}}(H_{0}(M_{P}))^\#}, \\ \operatorname{\mathrm{ord}} H_{1}(S^{3} \setminus \mathcal{N}(K)) &\doteq \frac{\operatorname{\mathrm{ord}}(H_{1}(M_{K})) \cdot \operatorname{\mathrm{ord}}(H_{1}(T))}{\operatorname{\mathrm{ord}}(H_{0}(M_{K}))^\# }. \end{align*} $$

Proof. We only prove the lemma for $M_{P} \setminus \mathcal {N}(\eta )$ : the proof for $S^{3} \setminus \mathcal {N}(K)$ is completely analogous. Using (3.5) and the Mayer-Vietoris sequence of $M_{P} = (M_{P} \setminus \mathcal {N}(\eta )) \cup _T \mathcal {N}(\eta )$ , we deduce that $i_P$ does indeed induce an isomorphism on twisted homology in degree $j> 2$ . In degree zero, notice that the inclusion $T \hookrightarrow \mathcal {N}(\eta )$ induces an isomorphism on $H_{0}$ , and therefore, with the aid of the Mayer-Vietoris sequence, we obtain an isomorphism $H_{0}(M_{P} \setminus \mathcal {N}(\eta )) \cong H_{0}(M_{P})$ .

To get the result in degrees $j=1,2$ , consider the relevant part of the Mayer-Vietoris sequence:

(3.9) $$ \begin{align} 0 \xrightarrow{} H_{2}(M_{P}) \xrightarrow{} H_1(T) \xrightarrow{j_{P}} H_1(M_{P} \setminus \mathcal{N}(\eta)) \xrightarrow{(\iota_{P})_{\ast}} H_1(M_P) \to 0. \end{align} $$

Notice that the leftmost part of (3.9) is exact by (3.7). Since $H_1(M_P \setminus \mathcal {N}(\eta ))$ is ${\mathbb {F}}[t^{\pm 1}]$ -torsion, (3.9) implies that this is also the case for $H_1(M_P)$ . Next, (3.9) also implies that the map $(\iota _{P})_{\ast }$ is an epimorphism with kernel generated by the $[e_i \otimes \mu _{\eta }]$ (or technically by $ j_P([e_i \otimes \mu _{\eta }])$ , because $H_{1}(T)$ is generated by these elements. The last assertion of the lemma also follows from (3.9): we use the multiplicativity of the orders and the same isomorphisms as in (3.6) to establish the isomorphism $H_{2}(M_{P}) \cong H_0(M_{P})^{\#}$ (this is possible since we now know that $H_1(M_P)$ is torsion). This concludes the proof of the lemma.

We now prove the first assertion of the theorem – namely, that $\gamma _P$ and $\gamma _K$ are acyclic. We only check this for $\gamma _P$ : the proof for $\gamma _K$ is identical. Since $H_*(M_P \setminus \mathcal {N}(\eta ))$ is ${\mathbb {F}}[t^{\pm 1}]$ -torsion, Lemma 3.13 implies that $H_i(M_P)$ is ${\mathbb {F}}[t^{\pm 1}]$ -torsion for $i \neq 2$ . For $i=2$ , we use the isomorphism $H_2(M_P)\cong H_0(M_P)^\#$ (that we established in the proof of Lemma 3.13), and the conclusion follows. The first statement of Theorem 3.11 is now proved.

As a next step, consider the decomposition $M_{P(K,\eta )} = \left (M_{P}\setminus \mathcal {N}(\eta )\right ) \cup _T \left (S^{3} \setminus \mathcal {N}(K)\right )$ . The associated Mayer-Vietoris sequence can be decomposed into the following three exact sequences:

(3.10) $$ \begin{align} &0 \xrightarrow{} H_{2}(M_{P(K,\eta)}) \xrightarrow{} H_{1}(T) \xrightarrow{} \mathcal{K} \xrightarrow{} 0,\end{align} $$

(3.11) $$ \begin{align} &0 \xrightarrow{} \mathcal{K} \xrightarrow{} H_{1}(M_{P}\setminus \mathcal{N}(\eta)) \oplus H_{1}(S^{3}\setminus \mathcal{N}(K)) \xrightarrow{\psi} H_{1}(M_{P(K,\eta)}) \xrightarrow{} \mathcal{C} \xrightarrow{} 0, \end{align} $$

(3.12) $$ \begin{align} &0 \xrightarrow{} \mathcal{C} \xrightarrow{} H_{0}(T) \xrightarrow{} H_{0}(M_{P}\setminus \mathcal{N}(\eta)) \oplus H_{0}(S^{3}\setminus \mathcal{N}(K)) \xrightarrow{} H_{0}(M_{P(K,\eta)}) \xrightarrow{} 0 , \end{align} $$

where $\mathcal {K}=\ker (\psi )$ and $\mathcal {C}=\operatorname {\mathrm {coker}}(\psi )$ . Therefore, applying the multiplicativity of the orders, the use of (3.10) and (3.12) (as well as (3.6)) implies that

$$ \begin{align*} \operatorname{\mathrm{ord}} (\mathcal{C}) &\doteq \frac{\operatorname{\mathrm{ord}}(H_{0}(T)) \cdot \operatorname{\mathrm{ord}}(H_{0}(M_{P(K,\eta)}))}{\operatorname{\mathrm{ord}}(H_{0}(M_{P}\setminus \mathcal{N}(\eta))) \cdot \operatorname{\mathrm{ord}}(H_{0}(S^{3} \setminus \mathcal{N}(K)))} \doteq \frac{\operatorname{\mathrm{ord}}(H_{0}(T)) \cdot \operatorname{\mathrm{ord}}(H_{0}(M_{P(K,\eta)}))}{\operatorname{\mathrm{ord}}(H_{0}(M_{P})) \cdot \operatorname{\mathrm{ord}}(H_{0}(M_{K}))}, \\ \operatorname{\mathrm{ord}} (\mathcal{K} )&\doteq \frac{\operatorname{\mathrm{ord}}(H_{1}(T))}{\operatorname{\mathrm{ord}}(H_{2}(M_{P(K,\eta)}))} \doteq \frac{\operatorname{\mathrm{ord}}(H_{1}(T))}{\operatorname{\mathrm{ord}}(H_{0}(M_{P(K,\eta)}))^\#}. \end{align*} $$

The key ${\mathbb {F}}[t^{\pm 1}]$ -module required in this proof is defined as

$$\begin{align*}A := \frac{H_{1}(M_{P}\setminus \mathcal{N}(\eta)) \oplus H_{1}(S^{3}\setminus \mathcal{N}(K))}{\mathcal{K}}. \end{align*}$$

The module A will be used both in the proof of the second and third statements of Theorem 3.11. As a preliminary, however, we first compute $ \operatorname {\mathrm {ord}} (A) \operatorname {\mathrm {ord}} (A)^\# $ . Combining (3.11) with our computation of $\operatorname {\mathrm {ord}} (\mathcal {C})$ , we obtain

$$\begin{align*}\operatorname{\mathrm{ord}} (A) \doteq \frac{\operatorname{\mathrm{ord}}(H_{1}(M_{P(K,\eta)}))}{\operatorname{\mathrm{ord}} (\mathcal{C})} \doteq \frac{\operatorname{\mathrm{ord}}(H_{1}(M_{P(K,\eta)})) \cdot \operatorname{\mathrm{ord}}(H_{0}(M_{P})) \cdot \operatorname{\mathrm{ord}} H_{0}(M_{K})}{\operatorname{\mathrm{ord}} (H_{0}(T)) \cdot \operatorname{\mathrm{ord}}(H_{0}(M_{P(K,\eta)}))}.\end{align*}$$

On the other hand, Lemma 3.13 and our computation of $\operatorname {\mathrm {ord}} (\mathcal {K})$ imply that

$$ \begin{align*} \operatorname{\mathrm{ord}} (A) &\doteq \frac{\operatorname{\mathrm{ord}}(H_{1}(M_{P} \setminus \mathcal{N}(\eta))) \cdot \operatorname{\mathrm{ord}}(H_{1}(S^{3}\setminus \mathcal{N}(K)))}{\operatorname{\mathrm{ord}} (\mathcal{K})} \\ &\doteq \frac{\operatorname{\mathrm{ord}}(H_{1}(M_{P})) \cdot \operatorname{\mathrm{ord}}(H_{1}(M_{K})) \cdot \operatorname{\mathrm{ord}}(H_{1}(T))^{2} \cdot \operatorname{\mathrm{ord}}(H_{0}(M_{P(K,\eta)}))^\# }{\operatorname{\mathrm{ord}}(H_{0}(M_{P}))^\# \cdot \operatorname{\mathrm{ord}}(H_{0}(M_{K}))^\# \cdot \operatorname{\mathrm{ord}}(H_{1}(T))} \\ &\doteq \frac{\operatorname{\mathrm{ord}}(H_{1}(M_{P})) \cdot \operatorname{\mathrm{ord}}(H_{1}(M_{K})) \cdot \operatorname{\mathrm{ord}}(H_{1}(T)) \cdot \operatorname{\mathrm{ord}}(H_{0}(M_{P(K,\eta)}))^\# }{\operatorname{\mathrm{ord}}(H_{0}(M_{P}))^\# \cdot \operatorname{\mathrm{ord}}(H_{0}(M_{K}))^\# }. \end{align*} $$

Additionally, observe that $\operatorname {\mathrm {ord}}(H_1(T))=\operatorname {\mathrm {ord}}(H_0(T))^\#$ (this follows by duality and the universal coefficient theorem) and $\operatorname {\mathrm {ord}}(H_1(M_{P(K,\eta )})=\operatorname {\mathrm {ord}}(H_1(M_{P(K,\eta )})^\#$ (since $H_1(M_{P(K,\eta )})$ supports a non-singular linking form). Combining these observations with the above formulas for $\operatorname {\mathrm {ord}} (A)$ , we obtain

(3.13) $$ \begin{align} \operatorname{\mathrm{ord}} (A) \operatorname{\mathrm{ord}} (A)^\# \doteq \operatorname{\mathrm{ord}}(H_{1}(M_{P})) \cdot \operatorname{\mathrm{ord}}(H_{1}(M_{K})) \cdot \operatorname{\mathrm{ord}}(H_{1}(M_{P(K,\eta)})). \end{align} $$

We now prove the second statement of Theorem 3.11 which asserts that L is isotropic and that $\operatorname {\mathrm {ord}} (L)$ divides $\det (1-\gamma (\eta ))$ . Observe that $(\iota _{P})_{\ast } \oplus (\iota _{K})_{\ast }$ and $\psi $ vanish on $\mathcal {K}$ . As a consequence, we let $(\iota _{P})_{\ast } \oplus (\iota _{K})_{\ast } / \mathcal {K}$ and $\psi / \mathcal {K}$ denote the maps induced on A. Additionally, using $\widetilde {L}$ to denote the kernel of $(\iota _{P})_{\ast } \oplus (\iota _{K})_{\ast } / \mathcal {K}$ and $\operatorname {Cok}$ to denote the cokernel of $\psi / \mathcal {K}$ , we obtain the following diagram of exact sequences:

(3.14)

To check the second assertion of the theorem, we will show that $(\psi /\mathcal {K})(\widetilde {L}) = L$ , where L was defined as $\operatorname {im} (H_1(T) \to H_1(M_{P(K,\eta )}))$ . Use $j_P$ (resp. $j_K$ ) to denote the map induced by the inclusion $T \to M_P \setminus \mathcal {N}(\eta )$ (resp. $T \to S^3 \setminus \mathcal {N}(K)$ ). The exact sequence in (3.11) implies that $\mathcal {K}$ , treated as a submodule of $H_{1}(M_{P}\setminus \mathcal {N}(\eta )) \oplus H_{1}(S^{3}\setminus \mathcal {N}(K))$ , is generated by $(j_{P}([e_{i} \otimes \mu _{\eta }]), -j_{K}([e_{i}\otimes \mu _{\eta }]))$ , for $i=1,\ldots , d$ . Therefore, for any i, we have $(j_{P}([e_{i} \otimes \mu _{\eta }]),0) = (0,j_{K}([e_{i} \otimes \mu _{\eta }]))$ in A. Moreover,

$$\begin{align*}\widetilde{L} =\ker \left((i_P)_{\ast} \oplus (i_K)_{\ast}\right) = \langle (j_{P}[e_{i}\otimes \mu_{\eta}], j_{K}[e_{i}\otimes \mu_{\eta}]) \mid 1 \leq i \leq d\rangle / \langle j_{P}[e_{i} \otimes \mu_{\eta}] = j_{K}[e_{i} \otimes \mu_{\eta}] \rangle,\end{align*}$$

and thus, $(\psi /\mathcal {K})(\widetilde {L}) = L$ . Since $\psi /\mathcal {K}$ is injective, it follows that L and $\widetilde {L}$ are isomorphic. In order to prove that L is isotropic, observe that the homology classes $[e_{i}\otimes \mu _{\eta }]$ belong to the radical of $\operatorname {\mathrm {Bl}}_{\gamma _{P}}(M_{P} \setminus \mathcal {N}(\eta ))$ and $\operatorname {\mathrm {Bl}}_{\gamma _{K}}(S^{3} \setminus \mathcal {N}(K))$ because they come from the boundary of the respective manifolds. Since $\psi $ and $\psi / \mathcal {K}$ are morphisms of linking forms, it follows that L is isotropic. The constraint on the order of L comes from the fact that L is a quotient of $H_{1}(T)$ . This establishes the second statement of Theorem 3.11.

We now move on to the third statement which describes the relation between $\operatorname {\mathrm {Bl}}_\gamma (P(K,\eta ))$ and $\operatorname {\mathrm {Bl}}_{\gamma _P}(P) \oplus \operatorname {\mathrm {Bl}}_{\gamma _K}(K)$ . First observe that $\operatorname {\mathrm {Bl}}_{\gamma _P}(M_P \setminus \mathcal {N}(\eta )) \oplus \operatorname {\mathrm {Bl}}_{\gamma _K}(S^3 \setminus \mathcal {N}(K)) $ vanishes on $\mathcal {K}$ : indeed, $\mathcal {K}$ only contains homology classes which come from the boundary torus T. As a consequence, $\operatorname {\mathrm {Bl}}_{\gamma _P}(M_P \setminus \mathcal {N}(\eta )) \oplus \operatorname {\mathrm {Bl}}_{\gamma _K}(S^3 \setminus \mathcal {N}(K))$ descends to A.

We wish to apply Theorem 3.12 to $(M",\lambda ")=\operatorname {\mathrm {Bl}}_\gamma (P(K,\eta )),(M',\lambda ')=\operatorname {\mathrm {Bl}}_{\gamma _P}(P) \oplus \operatorname {\mathrm {Bl}}_{\gamma _K}(K)$ with $M=A,\iota =\psi /\mathcal {K}$ and $\pi =(\iota _{P})_{\ast } \oplus (\iota _{K})_{\ast } / \mathcal {K}$ . First, it is clear that the maps $(\iota _{P})_{\ast } \oplus (\iota _{K})_{\ast } / \mathcal {K}$ and $\psi / \mathcal {K}$ are morphisms of linking forms. Next, by definition of $\mathcal {K}$ , the map $\psi / \mathcal {K}$ is injective, while $(\iota _{P})_{\ast } \oplus (\iota _{K})_{\ast } / \mathcal {K}$ is surjective thanks to (3.9). Since $M_P, M_K$ and $M_{P(K,\eta )}$ are closed manifolds, the corresponding Blanchfield forms are non-singular. Finally, we already showed in (3.13) that $\operatorname {\mathrm {ord}} (A) \cdot (\operatorname {\mathrm {ord}} (A))^\# \doteq \operatorname {\mathrm {ord}}(H_{1}(M_{P})) \cdot \operatorname {\mathrm {ord}}(H_{1}(M_{K})) \cdot \operatorname {\mathrm {ord}}(H_{1}(M_{P(K,\eta )}))$ . We can therefore apply Theorem 3.12 to the above diagram to obtain the third assertion of the theorem, concluding the proof.

The next result shows that Blanchfield pairings need not be additive under satellite operations.

Corollary 3.14. Using the same assumptions as in Theorem 3.11, if $\operatorname {\mathrm {ord}} (L) $ is relatively prime to $\operatorname {\mathrm {ord}}(H_{1}(M_{P};{\mathbb {F}}[t^{\pm 1}]^{d}_{\gamma _{P}})) \cdot \operatorname {\mathrm {ord}}(H_{1}(M_{K};{\mathbb {F}}[t^{\pm 1}]^{d}_{\gamma _{K}}))$ , then there is an isometry of linking forms

$$\begin{align*}\operatorname{\mathrm{Bl}}_\gamma(P(K,\eta)) \cong \operatorname{\mathrm{Bl}}_{\gamma_P}(P) \oplus \operatorname{\mathrm{Bl}}_{\gamma_K}(K) \oplus (X,\lambda_{X}),\end{align*}$$

where $(X,\lambda _{X})$ is a metabolic linking form and $L \subset X$ is a metabolizer for $\lambda _{X}$ .

In particular, if the map $H_{1}(T;{\mathbb {F}}[t^{\pm 1}]^{d}_{\gamma }) \to H_{1}(M_{P(K,\eta )};{\mathbb {F}}[t^{\pm 1}]^{d}_{\gamma })$ vanishes, then we obtain the following isometry of linking forms:

$$\begin{align*}\operatorname{\mathrm{Bl}}_{\gamma}(P(K,\eta)) \cong \operatorname{\mathrm{Bl}}_{\gamma_P}(P) \oplus \operatorname{\mathrm{Bl}}_{\gamma_K}(K).\end{align*}$$

Proof. We first recall some notation from the proof of Theorem 3.11. Recall that we set $\mathcal {K}=~\ker (\psi )$ , where $\psi \colon \operatorname {\mathrm {Bl}}_{\gamma _P}(M_{P} \setminus \mathcal {N}(\eta )) \oplus \operatorname {\mathrm {Bl}}_{\gamma _K}(S^{3} \setminus \mathcal {N}(K)) \to \operatorname {\mathrm {Bl}}_\gamma (M_{P(K,\eta )})$ is the morphism of linking forms described in (3.4). We once again omit coefficients and recall that $A:=(H_1(M_P \setminus \mathcal {N}(\eta ) )\oplus H_1(S^3 \setminus \mathcal {N}(K)))/\mathcal {K}$ . Since we saw that $\operatorname {\mathrm {Bl}}_{\gamma _P}(P) \oplus \operatorname {\mathrm {Bl}}_{\gamma _K}(K)$ descends to A; we let $\operatorname {\mathrm {Bl}}_A$ denote the resulting pairing.

Claim. There is a linking form $(X,\lambda _X)$ such that $ \operatorname {\mathrm {Bl}}_\gamma (P(K,\eta )) \cong \operatorname {\mathrm {Bl}}_{\gamma _{P}}(P) \oplus \operatorname {\mathrm {Bl}}_{\gamma _{K}}(K) \oplus (X,\lambda _X)$ .

Proof. Using the multiplicativity of orders on the vertical exact sequence in (3.14), we see that $\operatorname {\mathrm {ord}} (A) = \operatorname {\mathrm {ord}}(\widetilde {L}) \cdot \operatorname {\mathrm {ord}}(H_{1}(M_{P})) \cdot \operatorname {\mathrm {ord}}(H_{1}(M_{K})).$ Using the primary decomposition theorem (since ${\mathbb {F}}[t^{\pm 1}]$ is a PID), the Chinese remainder theorem and $\gcd (\operatorname {\mathrm {ord}} (\widetilde {L}), \operatorname {\mathrm {ord}}(H_{1}(M_{P})) \cdot \operatorname {\mathrm {ord}}(H_{1}(M_{K}))) = 1$ , we obtain an isomorphism $A \cong \widetilde {L} \oplus B,$ where B is isomorphic to $H_{1}(M_{P}) \oplus H_{1}(M_{K})$ . Observe that the surjectivity of $(\iota _{P})_{\ast } \oplus (\iota _{K})_{\ast }/\mathcal {K}$ implies that

(3.15) $$ \begin{align} \operatorname{\mathrm{Bl}}_{A}|_{B} \cong \operatorname{\mathrm{Bl}}_{\gamma_{P}}(P) \oplus \operatorname{\mathrm{Bl}}_{\gamma_{K}}(K). \end{align} $$

Let $B' = (\psi /\mathcal {K})(B) \subset H_{1}(M_{P(K,\eta )})$ . Since $\psi /\mathcal {K}$ is injective (recall (3.14)), $B \cong B'$ . By our assumptions, $\operatorname {\mathrm {ord}}(L)$ is relatively prime with $\operatorname {\mathrm {ord}}(B')$ . Furthermore, since $\operatorname {\mathrm {ord}}(B')$ is symmetric, it follows that $\gcd (\operatorname {\mathrm {ord}}(L)^{\#},\operatorname {\mathrm {ord}}(B')) = 1$ . Consequently,

$$\begin{align*}\gcd(\operatorname{\mathrm{ord}} (H_{1}(M_{P(K,\eta)})/B'), \operatorname{\mathrm{ord}} (B')) = \gcd(\operatorname{\mathrm{ord}}(L) \cdot \operatorname{\mathrm{ord}}(L)^{\#}, \operatorname{\mathrm{ord}}(B')) = 1,\end{align*}$$

where the first equality follows from the fact that $\operatorname {\mathrm {ord}}(H_{1}(M_{P(K,\eta )}) / B') = \operatorname {\mathrm {ord}}(L) \cdot \operatorname {\mathrm {ord}}(L)^{\#}$ . Hence, there exists a submodule $X \subset H_{1}(M_{P(K,\eta )})$ , such that $H_{1}(M_{P(K,\eta )}) = X \oplus B'$ . By [Reference Borodzik, Conway and Politarczyk3, Proposition 2.25], there is a corresponding splitting of linking forms $\operatorname {\mathrm {Bl}}_{\gamma }(P(K,\eta )) = \operatorname {\mathrm {Bl}}_{\gamma }(P(K,\eta ))|_{X} \oplus \operatorname {\mathrm {Bl}}_{\gamma }(P(K,\eta ))|_{B'}$ . Since, $\psi /\mathcal {K} \colon A \to H_1(M_{P(K,\eta )})$ is an injective map of linking forms, we obtain identifications

$$\begin{align*}\operatorname{\mathrm{Bl}}_{\gamma}(P(K,\eta))|_{B'} \cong \operatorname{\mathrm{Bl}}_{A}|_{B} \cong \operatorname{\mathrm{Bl}}_{\gamma_{P}}(P) \oplus \operatorname{\mathrm{Bl}}_{\gamma_{K}}(K),\end{align*}$$

where the first isomorphism is given by $(\psi / \mathcal {K})|_{B}$ and the last isomorphism follows from equation (3.15). Hence, we can take $\lambda _{X} = \operatorname {\mathrm {Bl}}(P(K,\eta ))|_{X}$ , which is non-singular by [Reference Borodzik, Conway and Politarczyk3, Proposition 2.25]. This concludes the proof of the claim.

Since Theorem 3.11 shows that $\operatorname {\mathrm {Bl}}_\gamma (M_{P(K,\eta )})$ is Witt equivalent to $\operatorname {\mathrm {Bl}}_{\gamma _P}(P) \oplus ~\operatorname {\mathrm {Bl}}_{\gamma _K}(K)$ , the claim implies that $(X,\lambda _X)$ is metabolic. It remains to show that a metabolizer for $(X,\lambda _X)$ is in fact given by $L:= (\psi /\mathcal {K})(\widetilde {L})$ . As $(\psi /\mathcal {K}) \colon A \to H_1(M_{P(K,\eta )}) $ is an injective morphism of linking forms, the isomorphisms $A \cong \widetilde {L} \oplus B$ and $H_1(M_{P(K,\eta )})~\cong \psi /\mathcal {K}(B) \oplus ~X$ imply that $L = (\psi /\mathcal {K})(\widetilde {L})$ is an isotropic submodule of X. Since Theorem 3.11 shows that $\operatorname {\mathrm {Bl}}_{\gamma _P}(P) \oplus \operatorname {\mathrm {Bl}}_{\gamma _K}(K)$ is isometric to the isotropic reduction of $\operatorname {\mathrm {Bl}}_\gamma (M_{P(K,\eta )})$ with respect to L, the multiplicativity of orders implies that $\operatorname {\mathrm {ord}} (X)=\operatorname {\mathrm {ord}}(L)\operatorname {\mathrm {ord}}(L)^{\#}$ . As ${\mathbb {F}}[t^{\pm 1}]$ is a PID, we deduce that L is a metabolizer in X, concluding the proof of the first assertion.

In order to prove the second assertion, notice that our hypothesis implies that the submodule L is trivial. Since L is a metabolizer of X, we deduce that X is itself trivial. The conclusion now follows from the first assertion.

The situation simplifies considerably when $\gamma _{K}$ is abelian (i.e., factors through the abelianization of $\pi _{1}(M_{K})$ ). Let $\mathbb {F}[t_{K}^{\pm 1}]$ be the group algebra of the abelianization of $\pi _{1}(M_{K})$ with $t_{K}$ denoting the generator corresponding to the meridian identified with $\eta $ . In this case, ${\mathbb {F}}[t^{\pm 1}]^{d}_{\gamma _{K}}$ becomes a module over $\mathbb {F}[t_{K}^{\pm 1}]$ via the homomorphism $\gamma _{K} \colon \mathbb {F}[t_K^{\pm 1}] \to {\mathbb {F}}[t^{\pm 1}]^d_{\gamma _{K}} $ (i.e., via the action $t_{K} \cdot v = v\gamma _K(\eta ) $ with $v \in {\mathbb {F}}[t^{\pm 1}]^{d}_{\gamma _{K}}$ ).

In this setting, Theorem 3.11 takes the following form (compare with [Reference Friedl, Leidy, Nagel and Powell21, Theorem 1.3]).

Corollary 3.15. Using the same assumptions as in Theorem 3.11, if $\gamma _{K}$ is an abelian representation, then there exists an isometry of linking forms

$$\begin{align*}\operatorname{\mathrm{Bl}}_{\gamma}(P(K,\eta)) \cong \operatorname{\mathrm{Bl}}_{\gamma_P}(P) \oplus \operatorname{\mathrm{Bl}}_{\gamma_K}(K).\end{align*}$$

If we denote by $\mathbb {F}[\eta ] \subset \mathbb {F}[\pi (M_{P})]$ the subring generated by $\eta $ and we assume that $\det (\gamma _{P}(x)) \neq 0$ , for any $x \in \mathbb {F}[\eta ] \setminus \{0\}$ , then $H_{1}(M_{K};{\mathbb {F}}[t^{\pm 1}]^{d}_{\gamma })$ is isomorphic to $H_{1}(M_{K};\mathbb {F}[t_{K}^{\pm 1}]) \otimes _{\mathbb {F}[t_{K}^{\pm 1}]} {\mathbb {F}}[t^{\pm 1}]^{d}_{\gamma _{K}},$ and there is an isometry

$$ \begin{align*}\operatorname{\mathrm{Bl}}_{\gamma_K}(K) \cong \operatorname{\mathrm{Bl}}(K) \otimes_{\mathbb{F}[t_{K}^{\pm1}]} {\mathbb{F}}[t^{\pm 1}]^{d}_{\gamma_{K}}.\end{align*} $$

Next, we discuss the consequences of Theorem 3.11 on signatures of satellite knots.

Corollary 3.16. Let $P(K,\eta )$ be a satellite knot with pattern P, companion K and infection curve $\eta $ . Let $\gamma \colon \pi _1(M_{P(K,\eta )}) \to GL_d({\mathbb {F}}[t^{\pm 1}])$ be a unitary acyclic $\eta $ -regular representation. If the assumptions of Theorem 3.11 hold, then for any $\omega \in S^1$ , the averaged twisted signatures of $P(K,\eta ),P$ and K satisfy

$$ \begin{align*}\sigma_{P(K,\eta),\gamma}^{\operatorname{av}}(\omega)=\sigma_{P,\gamma_P}^{\operatorname{av}}(\omega)+\sigma_{K,\gamma_K}^{\operatorname{av}}(\omega).\end{align*} $$

In particular, under these assumptions, the averaged twisted signature is additive on connected sums:

$$ \begin{align*}\sigma_{P \# K,\gamma}^{\operatorname{av}}(\omega)=\sigma_{P,\gamma_P}^{\operatorname{av}}(\omega)+\sigma_{K,\gamma_K}^{\operatorname{av}}(\omega).\end{align*} $$

Finally, in the abelian case, we recover results concerning the behavior of the Blanchfield pairing and Levine-Tristram signature under satellite operations (see [Reference Livingston and Melvin39] and [Reference Shinohara43, Reference Litherland37]).

Corollary 3.17. Let $P(K,\eta )$ be a satellite knot with pattern P, companion K, infection curve $\eta $ and winding number $w = \operatorname {\mathrm {lk}}(\eta ,P)$ . The Blanchfield pairing and Levine-Tristram signature sastify

$$ \begin{align*} \operatorname{\mathrm{Bl}}(P(K,\eta))(t)&=\operatorname{\mathrm{Bl}}(P)(t) \oplus \operatorname{\mathrm{Bl}}(K)(t^w), \\ \sigma_{P(K,\eta)}(\omega)&=\sigma_{P}(\omega)+\sigma_{K}(\omega^w). \end{align*} $$

4.1 The metabelian Blanchfield pairing

In this subsection, we review two equivalent definitions of a particular twisted Blanchfield pairing associated to $M_K$ . The first makes use of a metabelian representation $\alpha (n,\chi ) \colon \pi _1(M_K) \to GL_n({\mathbb {C}}[t^{\pm 1}])$ , while the second uses the n-fold cover $M_n \to M_K$ . References for this subsection include [Reference Friedl19, Reference Herald, Kirk and Livingston28, Reference Miller and Powell40].

Use $\phi _K \colon \pi _1(M_K) \to H_1(M_K;\mathbb {Z})\cong \mathbb {Z}=\langle t_K \rangle $ to denote the abelianization homomorphism. Since $\phi _K$ endows $\mathbb {Z}[t_K^{\pm 1}]$ with a right $\pi _1(M_K)$ -module structure, it gives rise to the twisted homology $\mathbb {Z}[t_K^{\pm 1}]$ -module $H_1(M_K;\mathbb {Z}[t_{K}^{\pm 1}])$ . This module is known to coincide with the Alexander module of K. In what follows, we shall frequently identify $H_1(\Sigma _n(K);\mathbb {Z})$ with $H_1(M_K;\mathbb {Z}[t_K^{\pm 1}])/(t_K^n-1)$ as in [Reference Friedl and Powell23, Corollary 2.4]. We now 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, consider the representation

(4.1) $$ \begin{align} \gamma_K(n,\chi) \colon \mathbb{Z} \ltimes H_1(\Sigma_n(K);\mathbb{Z}) &\to ~\operatorname{GL}_n({\mathbb{C}}[t^{\pm 1}]) \nonumber \\ (t_K^j,v) &\mapsto \begin{pmatrix} 0& 1 & \cdots &0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \\ t & 0 & \cdots & 0 \end{pmatrix}^j \begin{pmatrix} \xi_{m}^{\chi(v)} & 0 & \cdots &0 \\ 0 & \xi_{m}^{\chi(t_K \cdot v)} & \cdots &0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \xi_{m}^{\chi(t_K^{n-1} \cdot v)} \end{pmatrix}. \end{align} $$

We will now use $\gamma _K(n,\chi )$ to obtain a representation of $\pi _1(M_K)$ . To achieve this, we identify the Alexander module $H_1(M_K;\mathbb {Z}[t_{K}^{\pm 1}])$ with the derived quotient $\pi _1(M_K)^{(1)}/\pi _1(M_K)^{(2)}$ and consider the following composition of canonical projections:

$$\begin{align*}q_K \colon \pi_{1}(M_{K})^{(1)} \to H_1(M_K;\mathbb{Z}[t_K^{\pm 1}]) \to H_1(\Sigma_n(K);\mathbb{Z}).\end{align*}$$

Fix an element $\mu _{K}$ in $\pi _1(M_K)$ such that $\phi _K(\mu _{K})=t_K$ . Note that for every $g \in \pi _1(M_K)$ , we have $\phi _K(\mu _K^{-\phi _K(g)}g)=1$ . Since $\phi _K$ is the abelianization map, we deduce that $\mu _K^{-\phi _K(g)}g$ belongs to $\pi _1(M_K)^{(1)}$ . As a consequence, we obtain the following map:

(4.2) $$ \begin{align} \widetilde{\rho}_K \colon \pi_1(M_K) &\to \mathbb{Z} \ltimes H_1(\Sigma_n(K);\mathbb{Z}) \\ g &\mapsto (\phi_K(g),q_K(\mu_{K}^{-\phi_K(g)}g)). \nonumber \end{align} $$

Precomposing the representation $\gamma _K(n,\chi )$ with $\widetilde {\rho }_K$ provides the unitary representation

(4.3) $$ \begin{align} \alpha_K(n,\chi) \colon \pi_1(M_K) \stackrel{\widetilde{\rho}}{\to} \mathbb{Z} \ltimes H_1(\Sigma_n(K);\mathbb{Z}) \stackrel{\gamma_K(n,\chi)}{\longrightarrow} \operatorname{GL}_n({\mathbb{C}}[t^{\pm 1}]). \end{align} $$

If m is a prime power and $\chi \colon H_1(\Sigma _n(K);\mathbb {Z}) \to \mathbb {Z}_m$ is nontrivial, it is known that the unitary representation $\alpha (n,\chi )$ is acyclic [Reference Friedl and Powell22] (see also [Reference Miller and Powell40, Lemma 6.6] and [Reference Casson and Gordon6, Corollary following Lemma 4]). When the knot K is clear from the context, we drop it from the notation and simply write $\alpha (n,\chi )$ instead of $\alpha _K(n,\chi )$ .

The next lemma illustrates this notation and will be used to ensure that the hypotheses of Theorem 3.11 hold.

Proof. Let $e_{i} \in \mathbb {C}[t^{\pm 1}]^{n}$ , for $i=1,2,\ldots ,n$ , denote the vectors from the standard basis. By definition, $H_0(M_K;{\mathbb {C}}[t^{\pm 1}]_{\alpha (n,\chi )}^n)={\mathbb {C}}[t^{\pm 1}]^n/V$ , with

$$\begin{align*}V=\langle (\alpha(n,\chi)(g)-\operatorname{id})v \ | \ v \in {\mathbb{C}}[t^{\pm 1}]^n, g \in ~\pi_1(M_K) \rangle.\end{align*}$$

Therefore, it is sufficient to prove that for any $1 \leq i \leq n$ , there exists $g_{i} \in \pi _{1}(M_{K})$ and $w_{i} \in \mathbb {C}[t^{\pm 1}]^{n}$ so that $e_{i} = \left (\alpha (n,\chi )(g_{i}) - \operatorname {id}\right ) \cdot w_{i}$ . Fix $1 \leq i \leq n$ . Choose any $h_{i} \in [\pi _{1}(M_{K}),\pi _{1}(M_{K})]$ so that $\chi (\widetilde {\rho }_K(h_{i})) \neq 1$ ; such an $h_i$ exists because $\chi $ is nontrivial and $\widetilde {\rho }_K$ is surjective. The definition of $\alpha (n,\chi )$ then gives

$$\begin{align*}\left(\alpha(n,\chi)(\mu_K^{-i+1} \cdot h_{i} \cdot \mu_K^{i-1}) - \operatorname{id} \right) \cdot e_{i} = \left(\chi(\widetilde{\rho}_K(h_{i})) - 1\right) e_{i}.\end{align*}$$

Therefore, for $g_{i} := \mu _K^{-i+1} \cdot h_{i} \cdot \mu _K^{i-1}$ and $w_{i} := \frac {1}{\chi (\widetilde {\rho }_K(h_{i}))-1} e_{i}$ , we have $e_{i} = \left (\alpha (n,\chi )(g_{i}) - \operatorname {id}\right ) \cdot w_{i}$ .

We now use $\alpha (n,\chi )$ to endow $\mathbb {C}[t^{\pm 1}]^n$ with a right $\mathbb {Z}[\pi _1(M_K)]$ -module structure and apply Definition 2.2 to define a twisted Blanchfield pairing on $M_K$ .

Definition 4.2. Let K be an oriented knot, n be an integer and m be a prime power, and choose a nontrivial character $\chi \colon H_1(\Sigma _n(K);\mathbb {Z}) \to ~\mathbb {Z}_m$ . The metabelian Blanchfield pairing is the Blanchfield pairing twisted by the unitary acyclic representation $\alpha (n,\chi )$ :

$$ \begin{align*}\operatorname{\mathrm{Bl}}_{\alpha(n,\chi)}(K) \colon H_1(M_K;{\mathbb{C}}[t^{\pm 1}]^n_{\alpha(n,\chi)}) \times H_1(M_K;{\mathbb{C}}[t^{\pm 1}]^n_{\alpha(n,\chi)}) \to \mathbb{C}(t)/ {\mathbb{C}}[t^{\pm 1}].\end{align*} $$

Next, we provide a description of $\operatorname {\mathrm {Bl}}_{\alpha (n,\chi )}(K)$ in terms of the n-fold cyclic cover $p \colon M_n \to M_K$ . The first step is to consider the abelian subgroup $n\mathbb {Z} \times H_1(\Sigma _n(K);\mathbb {Z})$ of $\mathbb {Z} \ltimes H_1(\Sigma _n(K);\mathbb {Z})$ and the following $1$ -dimensional representation:

(4.4) $$ \begin{align} \rho_K(n,\chi) \colon n \mathbb{Z} \times H_1(\Sigma_n(K);\mathbb{Z}) &\to {\mathbb{C}}[t^{\pm 1}] \\ (t_K^{nk},v) &\mapsto t^k \xi_{m}^{\chi(v)}. \nonumber \end{align} $$

We now use $\rho _K(n,\chi )$ to obtain a representation of $\pi _1(M_n)$ . Consider the composition $\pi _1(M_n) \stackrel {p_*}{\rightarrow } \pi _1(M_K) \to H_1(M_K;\mathbb {Z}) \cong \mathbb {Z}$ . Since the image of this map is isomorphic to $n \mathbb {Z}$ , mapping to it produces a surjective map $\alpha \colon \pi _1(M_n) \to n \mathbb {Z}$ . Using the decomposition $H_1(M_n;\mathbb {Z}) \cong H_1(\Sigma _n(K);\mathbb {Z}) \oplus ~\mathbb {Z}$ additionally gives rise to a homomorphism $\rho \colon \pi _1(M_n) \to H_1(\Sigma _n(K);\mathbb {Z})$ . Consider the $\mathbb {Z}[\pi _1(M_n)]$ -module structure on $\mathbb {C}[t^{\pm 1}]$ given by the following composition:

(4.5) $$ \begin{align} \alpha \times \chi \colon \pi_1(M_n) \stackrel{\alpha \times \rho}{\to} n \mathbb{Z} \times H_1(\Sigma_n(K);\mathbb{Z}) \stackrel{\rho_K(n,\chi)}{\to} {\mathbb{C}}[t^{\pm 1}]. \end{align} $$

Under the assumption that m is a prime power and that $\chi $ is a nontrivial character, Casson and Gordon show that $H_*(M_n;\mathbb {C}(t))$ vanishes [Reference Casson and Gordon6, Corollary after Lemma 4]. In other words, under these conditions, the unitary representation $\alpha \times \chi $ is acyclic.

Definition 4.3. Let n be an integer, let m be a prime power and let $\chi \colon H_1(\Sigma _n(K);\mathbb {Z}) \to \mathbb {Z}_m$ be a nontrivial character. The Blanchfield pairing of the cover $\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(M_n)$ is the Blanchfield pairing corresponding to the unitary acyclic representation $\alpha \times \chi :$

(4.6) $$ \begin{align} \operatorname{\mathrm{Bl}}_{\alpha \times \chi}(M_n) \colon H_1(M_n;{\mathbb{C}}[t^{\pm 1}]) \times H_1(M_n;{\mathbb{C}}[t^{\pm 1}]) \to \mathbb{C}(t) / {\mathbb{C}}[t^{\pm 1}]. \end{align} $$

Up to the end of this subsection, we shall use ${\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi }$ to denote the $\mathbb {Z}[\pi _1(M_n)]$ -module structure on ${\mathbb {C}}[t^{\pm 1}]$ given by (4.5) and by ${\mathbb {C}}[t^{\pm 1}]^n_{\alpha (n,\chi )}$ the $\mathbb {Z}[\pi _1(M_K)]$ -module structure on ${\mathbb {C}}[t^{\pm 1}]^n$ given by (4.3). The following result is due to Herald, Kirk and Livingston [Reference Herald, Kirk and Livingston28, Theorem 7.1].

Proposition 4.4. Let n be a positive integer. There is a canonical chain isomorphism of left ${\mathbb {C}}[t^{\pm 1}]$ -modules between $C_*(M_n;\mathbb {C}[t^{\pm 1}]_{\alpha \times \chi })$ and $C_*(M_K;\mathbb {C}[t^{\pm 1}]^n_{\alpha (n,\chi )})$ . In particular, one obtains a canonical isomorphism between the following homology left ${\mathbb {C}}[t^{\pm 1}]$ -modules:

(4.7) $$ \begin{align} H_1(M_n;\mathbb{C}[t^{\pm 1}]_{\alpha \times \chi}) \cong H_1(M_K;\mathbb{C}[t^{\pm 1}]^n_{\alpha(n,\chi)}). \end{align} $$

The proof of Proposition 4.4 relies on the relation between the representation $\gamma _K(n,\chi )$ of (4.1) and the representation $\rho _K(n,\chi )$ of (4.4). Since we shall use this relation in Subsection 4.5, we give some further details.

Remark 4.5. Let H be a subgroup of a group G and let $\rho $ be a representation of H. It is known that there is an induced representation $\operatorname {ind}_H^G\rho $ of G; we refer to [Reference Herald, Kirk and Livingston28] for a construction and a discussion of its properties. Taking $G=\mathbb {Z} \ltimes H_1(\Sigma _n(K);\mathbb {Z})$ and $H=n\mathbb {Z} \times H_1(\Sigma _n(K);\mathbb {Z})$ , the proof of Proposition 4.4 relies heavily on the isomorphism

$$ \begin{align*}\gamma_K(n,\chi)\cong\operatorname{ind}_H^G \rho_K(n,\chi).\end{align*} $$

As the isomorphism described in (4.7) arises from a canonical chain isomorphism (essentially Shapiro’s lemma), we obtain the following result which appears to be implicit in [Reference Miller and Powell40]. However, since no proof has appeared in the literature, we will give a detailed argument in the appendix, and, in particular, this corollary will follow from the more general Proposition A.11.

Summarizing the content of this section, we can choose to work alternatively with $\operatorname {\mathrm {Bl}}_{\alpha (n,\chi )}(K)$ or with $\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(M_n)$ . The former is used in the paper of Miller-Powell [Reference Miller and Powell40] as well as in Subsections 4.2 and 4.5, and in [Reference Borodzik, Conway and Politarczyk2]. The latter is closer to the definition of the Casson-Gordon $\tau $ invariant with which we shall work in Subsections 4.3 and 4.4.

4.2 A sliceness obstruction due to Miller and Powell

The aim of this subsection is to state an obstruction to sliceness which is due to Miller and Powell [Reference Miller and Powell40].

Given an integer n, we use $\lambda _n$ to denote the non-singular $\mathbb {Q}/\mathbb {Z}$ -valued linking form on the finite abelian group $H_1(\Sigma _n(K);\mathbb {Z})$ . Note that $H_1(\Sigma _n(K);\mathbb {Z})$ is also a $\mathbb {Z}[\mathbb {Z}_n]$ -module. A metabolizer of $\lambda _n$ is a $\mathbb {Z}[\mathbb {Z}_n]$ -submodule P of $H_1(\Sigma _n(K);\mathbb {Z})$ such that $P=P^\perp $ , where the orthogonal complement is defined as $P^\perp =\{y\in H_1(\Sigma _n(K);\mathbb {Z})\ | \ \forall x\in P,\ \lambda _n(x,y)=~0\}$ . Finally, given a prime q, integers $b \geq a$ and a character $\chi \colon H_1(\Sigma _n(K);\mathbb {Z}) \to ~\mathbb {Z}_{q^a}$ , we use $\chi _b$ to denote the composition of $\chi $ with the natural inclusion $\mathbb {Z}_{q^a} \to \mathbb {Z}_{q^b}$ .

The following obstruction to sliceness is due to Miller and Powell [Reference Miller and Powell40, Theorem 6.9].

Miller and Powell actually prove Theorem 4.7 under the weaker assumption that K is 2-solvable. We refer to Cochran-Orr-Teichner’s landmark paper [Reference Cochran, Orr and Teichner9] for the definition of n-solvability but do not delve deeper into this issue.

Combining Theorem 4.7 with the earlier sections, we obtain the following result.

The obstruction of Theorem 4.7 is used in [Reference Borodzik, Conway and Politarczyk2] to prove that a certain algebraic knot is not slice, thus recovering a result of Hedden-Kirk-Livingston [Reference Hedden, Kirk and Livingston27].

4.3 The Casson-Gordon $\tau $ -invariant

Given an oriented knot K and a prime power order character $\chi \colon H_1(\Sigma _n(K);\mathbb {Z}) \to \mathbb {Z}_m$ , we recall Casson-Gordon’s construction of the Witt class $\tau (K,\chi )$ [Reference Casson and Gordon6].

Using the isomorphism $ H_1(M_n;\mathbb {Z}) \cong H_1(\Sigma _n(K);\mathbb {Z}) \oplus \mathbb {Z}$ , the character $\chi $ induces a character on $H_1(M_n;\mathbb {Z})$ for which we use the same notation. Recalling that the projection $p \colon M_n \to M_K$ gives rise to a surjection $\alpha \colon \pi _1(M_n) \to n\mathbb {Z} \cong \mathbb {Z}$ , we obtain a homomorphism

(4.8) $$ \begin{align} \alpha \times \chi \colon \pi_1(M_n) \to \mathbb{Z} \times \mathbb{Z}_m. \end{align} $$

Since the bordism group $\Omega _3(\mathbb {Z} \times \mathbb {Z}_m)$ is finite (this can be seen using the Atiyah-Hirzbruch spectral sequence; see, for example, [Reference Conner and Floyd11, Chapter 1, Section 7]), there is an integer r such that r copies of $(M_n,\alpha \times \chi )$ bound $(V_n,\psi )$ for some $4$ -manifold $V_n$ and for some homomorphism $\psi \colon \pi _1(V_n) \to \mathbb {Z} \times \mathbb {Z}_m$ . Consider the map $\mathbb {C}[\mathbb {Z} \times \mathbb {Z}_m] \to \mathbb {C}[t^{\pm 1}]$ which sends the generator of $\mathbb {Z}_m$ to $\xi _{m} = e^{2\pi i/m}$ and the generator of $\mathbb {Z}$ to t. Endow $\mathbb {C}(t)$ with the right $\mathbb {Z}[\pi _1(V_n)]$ -module structure which arises from the composition

(4.9) $$ \begin{align} \mathbb{Z}[\pi_1(V_n)] \xrightarrow{\psi} \mathbb{C}[\mathbb{Z} \times \mathbb{Z}_m] \to {\mathbb{C}}[t^{\pm 1}] \to \mathbb{C}(t). \end{align} $$

We will denote by $\mathbb {C}(t)_{\alpha \times \chi }$ the module $\mathbb {C}(t)$ equipped with the action of the group ring $\mathbb {Z}[\pi _1(V_n)]$ as described above. Observe that (4.9) also provides a right $\mathbb {Z}[\pi _1(V_n)]$ -module structure on ${\mathbb {C}}[t^{\pm 1}]$ (we write ${\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi }$ ) which restricts to the one described in (4.5) on each boundary component of $V_n$ . We will denote the related $\mathbb {C}[\mathbb {Z}\times \mathbb {Z}_{m}]$ -module by ${\mathbb {C}}[t^{\pm 1}]_{\mathbb {Z} \times \mathbb {Z}_m}$ . Next, consider the $\mathbb {C}(t)$ -valued intersection form $\lambda _{\mathbb {C}(t)_{\alpha \times \chi },V_n}$ on $H_2(V_n;\mathbb {C}(t)_{\alpha \times \chi })$ . Casson and Gordon show that if $\chi $ is a character of prime power order, then this Hermitian form is non-singular [Reference Casson and Gordon6, Corollary following Lemma 4]. We record this result for later reference.

Since $\lambda _{\mathbb {C}(t)_{\alpha \times \chi },V_n}$ is non-singular, it gives rise to a Witt class $[\lambda _{\mathbb {C}(t)_{\alpha \times \chi },V_n}]$ in $W(\mathbb {C}(t))$ . However, the standard intersection pairing $\lambda _{\mathbb {Q},V_n}$ may very well be singular. Consequently, we use $\lambda _{\mathbb {Q},V_n}^{\text {nonsing}}$ to denote the form obtained by moding out the radical. The Witt class of $\lambda _{\mathbb {Q},V_n}^{\text {nonsing}}$ provides an element in $W(\mathbb {Q})$ . Since the inclusion map $\mathbb {Q} \to \mathbb {C}(t)$ induces a group homomorphism $i \colon W(\mathbb {Q}) \to W(\mathbb {C}(t))$ , we therefore obtain an element $i([\lambda _{\mathbb {Q},V_n}^{\text {nonsing}}])$ in $W(\mathbb {C}(t))$ .

Definition 4.10. Let K be an oriented knot, let n be an integer, let m be a prime power and let $\chi \colon H_1(\Sigma _n(K);\mathbb {Z}) \to \mathbb {Z}_m$ be a nontrivial character. The Casson-Gordon $\tau $ -invariant is the Witt class

$$ \begin{align*}\tau(K,\chi):=([\lambda_{\mathbb{C}(t)_{\alpha \times \chi},V_n}]-i([\lambda_{\mathbb{Q},V_n}^{\text{nonsing}}]) \otimes \frac{1}{r} \in W(\mathbb{C}(t))\otimes \mathbb{Q}.\end{align*} $$

Casson and Gordon show that $\tau (K,\chi )$ is independent both of the choice of the 4-manifold $V_n$ and of the extension $\psi $ of $\alpha \times \chi $ to $\pi _1(V_n)$ [Reference Casson and Gordon6]. Furthermore, as we shall recall later on, $\tau (K,\chi )$ provides an obstruction to sliceness.

4.4 Casson-Gordon signatures and averaged Blanchfield signatures

In this section, we investigate the relation between the Casson-Gordon invariant $\tau (K,\chi )$ and the twisted Blanchfield pairing $\operatorname {\mathrm {Bl}}_{\alpha (n,\chi )}(K)$ . In order to make both these invariants more tractable, we use averaged signatures.

Let $A(t)$ be a matrix over $\mathbb {C}(t)$ . The function $\operatorname {sign}_\omega (A(t)):=\operatorname {sign}(A(\omega ))$ is a step-function with finitely many discontinuities, and at each discontinuity $\omega $ , we can take the average of the one-sided limits in order to obtain a rational number $\operatorname {sign}_\omega ^{\text {av}}(A(t))$ . As explained in [Reference Casson and Gordon6, discussion preceding Theorem 3], one now obtains a well-defined homomorphism $\operatorname {sign}^{\text {av}}_\omega \colon W(\mathbb {C}(t)) \to \mathbb {Q}$ by setting

$$ \begin{align*}\operatorname{sign}_\omega^{\text{av}}([A(t)]):=\text{sign}^{\text{av}}_\omega(A(t)).\end{align*} $$

This map also gives rise to a well-defined map on $W(\mathbb {C}(t)) \otimes \mathbb {Q}$ . We now apply this discussion to extract signatures from the Casson-Gordon invariant $\tau (K,\chi )$ which is an element of $W(\mathbb {C}(t)) \otimes \mathbb {Q}$ .

Before returning to Blanchfield pairings, let us make Definition 4.12 somewhat more concrete.

Remark 4.13. Suppose $V_n$ is a $4$ -manifold whose boundary consists of a disjoint union of r copies of $M_n$ and over which $\alpha \times \chi $ extends. In other words, we have

$$\begin{align*}\tau(K,\chi)= ([\lambda_{\mathbb{C}(t)_{\alpha\times \chi},V_n}]-i_*([\lambda_{\mathbb{Q},V_n}^{\text{nonsing}}])) \otimes \frac{1}{r}.\end{align*}$$

Since $\operatorname {sign}_\omega ^{\text {av}}$ is a homomorphism, we obtain

$$\begin{align*}\operatorname{sign}_\omega^{\text{av}}(\tau(K,\chi))=\frac{1}{r}(\operatorname{sign}_\omega^{\text{av}}([\lambda_{\mathbb{C}(t)_{\alpha\times \chi},V_n}])-\operatorname{sign}_\omega^{\text{av}}([\lambda_{\mathbb{Q},V_n}^{\text{nonsing}}])).\end{align*}$$

Clearly, the latter term is simply equal to $\operatorname {sign}(V_n)$ , the (untwisted) signature of $V_n$ . Consequently, if $A(t)$ represents $\lambda _{\mathbb {C}(t)_{\alpha \times \chi },V_n}$ , then we deduce that

$$ \begin{align*}\operatorname{sign}_\omega^{\text{av}}(\tau(K,\chi))=\frac{1}{r}(\operatorname{sign}^{\text{av}}_\omega(A(t))-\operatorname{sign}(V_n)).\end{align*} $$

Note that $A(1)$ does not represent the standard intersection form $\lambda _{\mathbb {Q},V_n}$ . It does, however, represent the twisted intersection form $\lambda _{\mathbb {C}_{\xi _{m}},V_n}$ which arises from the coefficient system $\pi _1(V_n) \to \mathbb {Z}_m \to \mathbb {C}$ , where the latter map sends the generator of $\mathbb {Z}_m$ to $\xi _m=\text {exp}(\frac {2\pi i}{m})$ .

Returning to Blanchfield pairings, recall from Definition 4.2 that $\operatorname {\mathrm {Bl}}_{\alpha (n,\chi )}(K)$ denotes the twisted Blanchfield pairing associated to the representation $\alpha (n,\chi ) \colon \pi _1(M_K) \to GL_n({\mathbb {C}}[t^{\pm 1}])$ . Recall furthermore that $\sigma _{K,\alpha (n,\chi )} \colon S^1 \to \mathbb {Z}$ denotes the associated twisted signature function.

The following theorem relates our Blanchfield signatures to the Casson-Gordon signatures.

Before moving toward the proof, we start with a remark and a corollary to Theorem 4.14.

Remark 4.15. Although we stated Theorem 4.14 using the metabelian Blanchfield pairing $\operatorname {\mathrm {Bl}}_{\alpha (n,\chi )}(K)$ , the proof will use the Blanchfield pairing of the cover $\operatorname {Bl}_{\alpha \times \chi }(M_n)$ . Indeed, since Corollary 4.6 states that $\operatorname {Bl}_{\alpha (n,\chi )}(K)$ and $\operatorname {Bl}_{\alpha \times \chi }(M_n)$ are isometric, one linking form is representable if and only if the other one is, and for every $\omega \in S^1$ , the second item of Theorem 4.14 could have been written using the signature function of $\operatorname {Bl}_{\alpha \times \chi }(M_n)$ , since the following equality holds:

(4.10) $$ \begin{align} \sigma^{\text{av}}_{K,\alpha(n,\chi)}(\omega)= \operatorname{sign}_\omega^{\text{av}}(\operatorname{\mathrm{Bl}}_{\alpha \times \chi}(M_n)). \end{align} $$

Since $\tau (K,\chi )$ is known to provide an obstruction to sliceness [Reference Casson and Gordon6], a similar conclusion holds for $\sigma _{K,\alpha (n,\chi )}$ , thus yielding a second proof of (a variation on) Theorem 4.8. Indeed, recalling that $\lambda _n$ denotes the $\mathbb {Q}/\mathbb {Z}$ -valued linking form on $H_1(\Sigma _n(K);\mathbb {Z})$ , the following result is an immediate corollary of Theorem 4.14 and [Reference Casson and Gordon6, Theorem 2].

We now move toward the proof of Theorem 4.14. First of all, as we mentioned in Remark 4.15, we can work with the pairing $\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(M_n)$ instead of $\operatorname {\mathrm {Bl}}_{\alpha (n,\chi )}(K)$ . Next, as in Subsection 4.3, we choose a smooth $4$ -manifold $V_n$ such that the boundary of $V_n$ consists of r copies of $M_n$ and such that the representation extends. Without loss of generality, we can assume that $\pi _1(V_n)=\mathbb {Z}_m \times \mathbb {Z}$ : indeed, since $\mathbb {Z}_m \times \mathbb {Z}$ is finitely normally presented, one can perform finitely many surgeries to obtain the desired fundamental group, while leaving the boundary fixed.

The following lemma collects some algebraic statements which we shall need later on.

Proof. We start by proving (a). Set $G:=\mathbb {Z}_m \times \mathbb {Z}$ for brevity. Recalling the notation introduced below (4.9), we claim that ${\mathbb {C}}[t^{\pm 1}]_{G}$ is a projective $\mathbb {C}[G]$ -module. To see this, first note that $\mathbb {C}[G] = \mathbb {C}[t^{\pm 1}] \otimes _{\mathbb {C}} \mathbb {C}[\mathbb {Z}_m]$ and $\mathbb {C}[\mathbb {Z}_m] = \bigoplus _{j = 0}^{m-1}~\mathbb {C}_{\xi _m^j}$ , where $\mathbb {C}_{\xi _m^j}$ denotes the irreducible complex representation of $\mathbb {Z}_m$ with the action of the cyclic group given by multiplication by the root of unity $\xi _m^j$ , for $\xi _m = \exp \left (\frac {2 \pi i}{m}\right )$ . Therefore, we have $\mathbb {C}[G] = \bigoplus _{j=0}^{m-1} \mathbb {C}[t^{\pm 1}] \otimes _{\mathbb {C}}~\mathbb {C}_{\xi _m^j}$ and ${\mathbb {C}}[t^{\pm 1}]_{G} = \mathbb {C}[t^{\pm 1}] \otimes _{\mathbb {C}}~\mathbb {C}_{\xi _m}$ , concluding the proof of the claim. The claim now implies that $H_1(V_n; {\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ is a summand of $H_1(V_n; \mathbb {C}[G])$ . Since $\pi _1(V_n)=G$ , the corresponding G-cover of $V_n$ is simply-connected, and thus, $H_1(V_n;\mathbb {C}[G])$ vanishes. Consequently, $H_1(V_n; {\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ also vanishes since it is a direct summand of $H_1(V_n; \mathbb {C}[G])$ . This concludes the proof of (a).

We now prove (b) according to which the evaluation map

$$\begin{align*}\operatorname{\mathrm{ev}} \circ \kappa \colon H^2(V_n;{\mathbb{C}}[t^{\pm 1}]_{\alpha \times \chi}) \rightarrow \operatorname{\mathrm{Hom}}_{{\mathbb{C}}[t^{\pm 1}]}(H_2(V_n;{\mathbb{C}}[t^{\pm 1}]_{\alpha \times \chi}),{\mathbb{C}}[t^{\pm 1}])^\#\end{align*}$$

is an isomorphism. Since $H_1(V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })=0$ by the first statement and $H_0(V_n; {\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })=~0$ (recall Lemma 4.1), this follows immediately from the universal coefficient theorem.

We move on to (c) which asserts that $H_2(V_n,\partial V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ is a free ${\mathbb {C}}[t^{\pm 1}]$ -module. Using Poincaré duality and the second statement, we deduce that $H_2(V_n,\partial V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ is isomorphic to $\operatorname {\mathrm {Hom}}_{{\mathbb {C}}[t^{\pm 1}]}(H_2(V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi }),{\mathbb {C}}[t^{\pm 1}])^\#$ which is free because ${\mathbb {C}}[t^{\pm 1}]$ is a PID.

Finally, we deal with (d); that is we show that the inclusion induced map $H^2(V_n,\partial V_n;\mathbb {C}(t)_{\alpha \times \chi }) \to H^2(V_n;\mathbb {C}(t)_{\alpha \times \chi })$ is an isomorphism. Since $H_*(M_n;\mathbb {C}(t)_{\alpha \times \chi })$ vanishes by Lemma 4.9 and $\partial V_n$ consists of r disjoint copies of $M_n$ , we deduce that $H_*(\partial V_n;\mathbb {C}(t)_{\alpha \times \chi })$ also vanishes. The long exact sequence of the pair $(V_n,\partial V_n)$ implies that $H_2(V_n;\mathbb {C}(t)_{\alpha \times \chi }) \rightarrow H_2(V_n,\partial V_n;\mathbb {C}(t)_{\alpha \times \chi })$ is an isomorphism, and the result follows by Poincaré duality. This concludes the proof of (d) and thus the proof of the lemma.

Since ${\mathbb {C}}[t^{\pm 1}]$ is a PID, we can decompose any ${\mathbb {C}}[t^{\pm 1}]$ -module H as the direct sum of its free part $FH$ and its torsion part $TH$ . While $FH$ is typically defined as $H/TH$ , here it is convenient to think of $FH$ as a submodule of H such that $H \cong FH \oplus TH$ . While such a submodule is not unique, different choices will not affect the remainder of the proof. In particular, it follows that

$$\begin{align*}\operatorname{\mathrm{Hom}}_{{\mathbb{C}}[t^{\pm 1}]}(H_2(V_n;{\mathbb{C}}[t^{\pm 1}]_{\alpha \times \chi}),{\mathbb{C}}[t^{\pm 1}])^\#=\operatorname{\mathrm{Hom}}_{{\mathbb{C}}[t^{\pm 1}]}(FH_2(V_n;{\mathbb{C}}[t^{\pm 1}]_{\alpha \times \chi}),{\mathbb{C}}[t^{\pm 1}])^\#.\end{align*}$$

Consequently, point (b) of Lemma 4.17 provides the following well-defined isomorphism:

(4.11) $$ \begin{align} ev \circ \kappa \colon H^2(V_n;{\mathbb{C}}[t^{\pm 1}]_{\alpha \times \chi}) \rightarrow \operatorname{\mathrm{Hom}}_{{\mathbb{C}}[t^{\pm 1}]}(FH_2(V_n;{\mathbb{C}}[t^{\pm 1}]_{\alpha \times \chi}),{\mathbb{C}}[t^{\pm 1}])^\#. \end{align} $$

Composing the inclusion $FH_2(V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi }) \to H_2(V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ with the inclusion induced map $H_2(V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi }) \to H_2(V_n,\partial V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ gives rise to a well-defined map

$$\begin{align*}FH_2(V_n;{\mathbb{C}}[t^{\pm 1}]_{\alpha \times \chi}) \to H_2(V_n,\partial V_n;{\mathbb{C}}[t^{\pm 1}]_{\alpha \times \chi}).\end{align*}$$

Combining these remarks, there is a well-defined intersection form

$$ \begin{align*}\lambda_{{\mathbb{C}}[t^{\pm 1}]_{\alpha \times \chi},V_n} \colon FH_2(V_n;{\mathbb{C}}[t^{\pm 1}]_{\alpha \times \chi}) \times FH_2(V_n;{\mathbb{C}}[t^{\pm 1}]_{\alpha \times \chi}) \to {\mathbb{C}}[t^{\pm 1}].\end{align*} $$

Here, note that the choice of the submodule of $FH$ does not affect the isometry type of $\lambda _{{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi },V_n}$ .

Next, we describe the Blanchfield $\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(\partial V_n)$ on $H_1(\partial V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ . As $\partial V_n$ consists of r copies of $M_n$ , we deduce that $H_1(\partial V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ is isomorphic to the direct sum of r copies of $H_1(M_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ . Since the latter ${\mathbb {C}}[t^{\pm 1}]$ -module is torsion, so is the former. We deduce that $ \operatorname {\mathrm {Bl}}_{\alpha \times \chi }(\partial V_n)$ decomposes as the direct sum $\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(M_n) \oplus \cdots \oplus \operatorname {\mathrm {Bl}}_{\alpha \times \chi }(M_n)$ , where there are r direct summands.

The proof of the next result is postponed until the end of the proof of Theorem 4.14.

Proof of Theorem 4.14 assuming Proposition 4.18

We start by proving the second statement – namely, the equality of signatures. Let $A(t)$ be a matrix representing $\lambda _{{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi },V_n}$ . Since $A(t)$ represents the twisted intersection form $\lambda _{\mathbb {C}(t)_{\alpha \times \chi },V_n}$ , the definition of the Casson-Gordon signature and Remark 4.13 imply that

(4.12) $$ \begin{align} \operatorname{sign}_\omega^{\text{av}}(\tau(K,\chi))=\frac{1}{r}(\operatorname{sign}^{\text{av}}_\omega(A(t))-\operatorname{sign}(V_n)). \end{align} $$

Since Proposition 4.18 tells us that $A(t)$ also represents $-\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(\partial V_n)$ , item (S-5) of Proposition 1.6 implies that $-\operatorname {sign}_\omega ^{\text {av}}(\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(\partial V_n))=\operatorname {sign}^{\text {av}}_\omega (A(t))-\operatorname {sign}^{\text {av}}_1(A(t))$ . However, since $\partial V_n$ consists of r disjoint copies of $M_n$ , we observed previously that $ \operatorname {\mathrm {Bl}}_{\alpha \times \chi }(\partial V_n)=\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(M_n) \oplus \ldots \oplus \operatorname {\mathrm {Bl}}_{\alpha \times \chi }(M_n)$ and, consequently, $\operatorname {sign}_\omega ^{\text {av}}(\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(\partial V_n))=r \cdot \operatorname {sign}_\omega ^{\text {av}}(\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(M_n))$ . Combining these two observations, we obtain the following equality:

(4.13) $$ \begin{align} -\operatorname{sign}^{\text{av}}_\omega(\operatorname{\mathrm{Bl}}_{\alpha \times \chi}( M_n))=\frac{1}{r}(\operatorname{sign}^{\text{av}}_\omega(A(t))-\operatorname{sign}^{\text{av}}_1(A(t))). \end{align} $$

Finally, adding and subtracting a $\text {sign}(V_n)$ term in (4.13) and combining the result with (4.12), we can conclude the proof of the first assertion:

$$ \begin{align*} -\operatorname{sign}^{\text{av}}_\omega(\operatorname{\mathrm{Bl}}_{\alpha \times \chi}( M_n)) &=\frac{1}{r}\left(\operatorname{sign}^{\text{av}}_\omega(A(t))-\operatorname{sign}(V_n)\right)-\frac{1}{r}\left(\operatorname{sign}^{\text{av}}_1(A(t))-\operatorname{sign}(V_n) \right) \\ &=\operatorname{sign}_\omega^{\text{av}}(\tau(K,\chi))-\operatorname{sign}_1^{\text{av}}(\tau(K,\chi)). \end{align*} $$

It remains to prove the first statement – namely, that the twisted Blanchfield pairing $\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(M_n)$ is representable. In the previous paragraph, we showed that $ \operatorname {\mathrm {Bl}}_{\alpha \times \chi }(\partial V_n)=\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(M_n) \oplus \ldots \oplus \operatorname {\mathrm {Bl}}_{\alpha \times \chi }(M_n)$ . Since we know from Proposition 4.18 that $\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(\partial V_n)$ is representable, $\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(M_n) $ is itself representable ( $\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(\partial V_n)$ satisfies the signature condition of item (S-6) of Proposition 1.6, and thus, so does $\operatorname {\mathrm {Bl}}_{\alpha \times \chi }(M_n)$ ), and so the theorem is proved.

We now prove Proposition 4.18; the strategy follows very closely [Reference Conway, Friedl and Toffoli13, Section 5.2] which itself is based on the proof of [Reference Borodzik and Friedl4, Theorem 2.6].

Proof of Proposition 4.18

Consider the following diagram in which all homomorphisms are understood to be homomorphisms of ${\mathbb {C}}[t^{\pm 1}]$ -modules:

(4.14)

To make the diagram more concise, we have used the following shorthands:

• • $\Lambda ={\mathbb {C}}[t^{\pm 1}]$ ;

• • $FH_2=FH_2(V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ ;

• • $H=H_2(V_n,\partial V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ ;

• • $\Omega =\mathbb {C}(t)$ .

The top horizontal line of (4.14) is exact thanks to the long exact sequence of the pair $(V_n,\partial V_n)$ together with Lemma 4.17: exactness at the rightmost end is guaranteed by the first point of Lemma 4.17, while exactness at the middle follows from the third point. In more details, since $H:=H_2(V_n,\partial V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ is free, the image of $H_2(V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ in H is equal to the image of $FH_2(V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ in H.

The commutativity of the top middle square is a consequence of the definition of the Poincaré duality isomorphism. All the squares involving evaluation maps clearly commute, while the upper left (resp. right) square commutes by definition of the intersection (resp. Blanchfield) pairing. Finally, the middle rectangle anti-commutes thanks to [Reference Conway, Friedl and Toffoli13, Lemma 5.4]; see [Reference Conway12, Appendix A] for a proof.

Inspired by (4.14), we define a pairing $\theta $ on $H_2(V_n,\partial V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ by the composition

$$ \begin{align*} H_2(V_n,\partial V_n;{\mathbb{C}}[t^{\pm 1}]_{\alpha \times \chi}) &\xrightarrow{\operatorname{PD}} H^2(V_n;{\mathbb{C}}[t^{\pm 1}]_{\alpha \times \chi}) \rightarrow H^2(V_n;\mathbb{C}(t)_{\alpha\times \chi}) \cong H^2(V_n,\partial V_n;\mathbb{C}(t)_{\alpha\times \chi}) \\ &\rightarrow \operatorname{\mathrm{Hom}}_{{\mathbb{C}}[t^{\pm 1}]}(H_2(V_n,\partial V_n;{\mathbb{C}}[t^{\pm 1}]_{\alpha \times \chi}),\mathbb{C}(t))^\#, \end{align*} $$

where the third map is an isomorphism thanks to the fourth point of Lemma 4.17. The commutativity of the diagram in equation (4.14) immediately implies the commutativity of

(4.15)

We now pick bases in order to obtain matrices for the intersection form. Namely, choose any basis $\mathcal {C}$ of $FH_2(V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ and endow $\operatorname {\mathrm {Hom}}_{{\mathbb {C}}[t^{\pm 1}]}(FH_2(V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi }),{\mathbb {C}}[t^{\pm 1}])^\#$ with the corresponding dual basis $\mathcal {C}^*$ . Let $A(t)$ denote the matrix of the ${\mathbb {C}}[t^{\pm 1}]$ -intersection form $\lambda _{{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi },V_n}$ with respect to these bases.

Next, we use $\mathcal {C}$ and $\mathcal {C}^*$ to base $H_2(V_n;\mathbb {C}(t)_{\alpha \times \chi })$ and $\operatorname {\mathrm {Hom}}_{{\mathbb {C}}[t^{\pm 1}]}(H_2(V_n;\mathbb {C}(t)_{\alpha \times \chi }),\mathbb {C}(t))^\#$ . In other words, as claimed in the first part of Proposition 4.18, there are bases with respect to which both $\lambda _{\mathbb {C}(t)_{\alpha \times \chi },V_n}$ and $\lambda _{{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi },V_n}$ are represented by $A(t)$ . Therefore, to conclude the proof, it only remains to show that $A(t)$ also represents the Blanchfield pairing on $\partial V_n$ .

With this aim in mind, we start by providing a basis for $H_2(V_n,\partial V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ (which is free by Lemma 4.17(b)). This will allow us to represent the inclusion induced map $FH_2(V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi }) \to H_2(V_n,\partial V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ by a matrix. First, consider the following commutative diagram of ${\mathbb {C}}[t^{\pm 1}]$ -homomorphisms:

(4.16)

Here, the bottom-right map is an isomorphism thanks to the second point of Lemma 4.17. We now equip $H_2(V_n,\partial V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ with the basis induced from $\mathcal {C}^*$ and two isomorphisms in the right column of diagram in (4.16). Arguing as in [Reference Conway, Friedl and Toffoli13, Claim in Section 5.3], elementary linear algebra shows that with respect to these bases, the map $FH_2(V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi }) \to H_2(V_n,\partial V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ is represented by the matrix $A(t)^T$ . Rewriting the commutative diagram of equation (4.15) in terms of these bases, we obtain

Here, the middle horizontal map is determined by the top horizontal map, the vertical maps and the commutativity. Since $H_1(V_n;{\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ vanishes by the first point of Lemma 4.17, we deduce that the Blanchfield pairing on $H_1(\partial V_n; {\mathbb {C}}[t^{\pm 1}]_{\alpha \times \chi })$ is represented by $-A(t)$ . This concludes the proof of Proposition 4.18 and thus the proof of Theorem 4.14.

4.5 Satellite formulas for metabelian Blanchfield forms

Given two knots $K,P$ and an unknotted curve $\eta $ in the complement of P, we use $P(K,\eta )$ to denote the resulting satellite knot. As described in Subsection 4.1, for a character $\chi \colon H_1(\Sigma _n(P(K,\eta ));\mathbb {Z}) \to \mathbb {Z}_{m}$ , there is an associated metabelian representation $\alpha (n,\chi ) \colon \pi _1(M_{P(K,\eta )}) \to GL_n(\mathbb {C}[t^{\pm 1}])$ . The goal of this subsection is to apply the satellite formula of Theorem 3.11 to $\alpha (n,\chi )$ . On the level of signatures, the result is reminiscent of Litherland’s description of the behavior of the Casson-Gordon invariants of satellite knots [Reference Litherland38, Theorem 2], but differs in the winding number zero case.