At.algebraic topology – Relationship between Casson-Gordon invariants $ sigma (M, chi) $ and $ sigma_r (M, chi) $

Before I start, I'm sorry if this question should be too basic for mathoverflow. Initially, I wanted to display it on math.SE, but then I chose it on mathoverflow, because the Casson-Gordon invariants are a rather specialized subject and, although my question may be basic, it might also be interesting to 'other.

Setting: I am a student who is currently trying to understand the definition of Casson-Gordon's invariant $ sigma (M, chi) $ (Resp. $ sigma_r (M, chi) $) to see why this particular invariant can be used to show that the only cutting nodes among the twisted doubles of the unknot are the unknot itself and the knot $ 6_1 $ (also known as Stevedore's knot). For that, I read the original papers of Casson and Gordon Cobordism of the classical knots [1] and On the slice kind of knots [2], as well as notes by A. Conways Algebraic and invariant Concordance of Casson-Gordon [3] (The links can be found below).

Now, my question concerns the two different formulations of the Casson-Gordon invariant. $ sigma $, once as $ sigma (M, chi) $ in [1]and once as $ sigma_r (M, chi) $ in [2]. In order to formulate my question, I quickly trace the construction of the invariant $ sigma (M, chi) $, as found in [3] p.14, (resp. [1] in. 183), and $ sigma_r (M, chi) $, as found in [2] p. 41-42 (readers who already know the definitions and / or documents cited might want to skip the next two paragraphs).

Definition of $ sigma (K, chi) $: Given a compact $ 4 $-collecteur $ W $ and a morphism $ psi: pi_1 (W) rightarrow mathbb {Z} _m $, let $ widetilde W rightarrow W $ to be the associates $ mathbb {Z} _m $-couvrant. then $ H_2 ( widetilde W; mathbb {Z}) $ is a $ mathbb {Z}[mathbb{Z}_m]$-module. By mapping the generator $ mathbb {Z} _m $ at $ omega: = e ^ { frac {2 pi i} {m} $we get a map $ mathbb {Z} _m rightarrow mathbb {Q} ( omega) $, which endows $ mathbb {Q} ( omega) $ with a $ ( mathbb {Q} ( omega), mathbb {Z}[mathbb{Z}_m]$-bimodule structure. Together

begin {equation}
H _ * (W; mathbb {Q} ( omega)) = mathbb {Q} ( omega) otimes _ { mathbb {Z}[mathbb{z}_m]} H _ * ( widetilde W; mathbb {Z}).
end {equation}

Then the $ mathbb {Q} ( omega) $-vector space $ H_2 (W; mathbb {Q} ( omega)) $ is endowed with a $ mathbb {Q} ( omega) $Hermitian form valued $ lambda _ { mathbb {Q} ( omega)} $ (whose definition is analogous to the ordinary intersection shape on $ H_2 (W; mathbb {Z}) $). Define the signature of $ W $ twisted by $ psi $ as

begin {equation}
textrm {sign} ^ psi (W): = textrm {sign} ( lambda _ { mathbb {Q} ( omega)}).
end {equation}

Now, given a closed $ 3 $-collecteur $ M $ and a homomorphism $ chi: pi_1 (M) rightarrow mathbb {Z} _m $, the theory of bordism implies that there is a non-negative integer k $, a $ 4 $-hand $ W $ and a homomorphism $ psi: pi_1 (W) rightarrow mathbb {Z} _m $ such as $ partial (W, psi) = k (M, chi) $. Set the invariant $ sigma (M, chi) $ as

begin {equation}
sigma (M, chi): = frac {1} {k} ( textrm {sign} ^ psi (W) – textrm {sign} (W)) in mathbb {Q}.
end {equation}

Definition of $ sigma_r (M, chi) $: Let $ M $ to be a closed $ 3 $and various $ chi: H_1 (M) rightarrow mathbb {Z} _m $ an epimorphism. Again, $ chi $ induces a $ m $cyclic coating $ widetilde M rightarrow M $ with a canonical generator of the group of cover translations, corresponding to $ 1 in mathbb {Z} _m $. Suppose for a positive integer k $ there is a $ mk $cyclic branched fold of $ 4 $-the varieties $ widetilde W rightarrow W $, branched on a surface $ F subset textrm {int} W $, such as $ partial ( widetilde W rightarrow W) = k ( widetilde M rightarrow M) $and such as the cover translation of $ W widetilde which induces a rotation through $ 2 pi / m $ on the fibers of the normal package of $ widetilde F $ limit on each component of $ partial widetilde W $ to the canonical translation covering of $ widetilde M $ determined by $ chi $. Let this cover translation induce $ tau $ sure $ H = H_2 ( widetilde W) otimes mathbb {C} $. In addition, leave $ cdot_H $ denote the form of intersection of $ H_2 ( widetilde W) $ extended to $ H $ (which is Hermitian, but in general not not singular). then $ (H, cdot) $ is broken down into a direct orthogonal sum of clean spaces of $ tau $, corresponding to eigenvalues $ omega ^ r $, $ 0 leq r <m $, or $ omega: = e ^ { frac {2 pi i} {m} $. Let $ varepsilon_r ( widetilde W) $ denote the signature of $ cdot_H $ restricted to the own space corresponding to the eigenvalue $ omega ^ r $. Define, for $ 0 <r <m $, the invariant $ sigma_r (M, chi) $ as

begin {equation}
sigma_r (M, chi): = frac {1} {k} left ( textrm {sign} (W) – varepsilon_r ( widetilde W) – frac {2[F]^ 2r (m-r)} {m ^ 2} right) in mathbb {Q},
end {equation}

or $[F]$ designates the number of self-intersection of the branching surface $ F $.

Problem: Pages 185-186 in [1], Casson and Gordon explain the relationship between $ sigma (M, chi) $ to the signature Atiyah-Singer G-signature. In particular, they conclude on page 186 that (with the notation above),

begin {equation}
k sigma (M, chi ^ r) + textrm {sign} (W) = varepsilon_r ( widetilde W),
end {equation}

which reads for $ r = $ 1 especially as

begin {equation}
k sigma (M, chi) + textrm {sign} (W) = varepsilon_1 ( widetilde W).
end {equation}

In addition, proposition 2.19 at p. 18 in. [3] declares that the formula for the computer $ sigma_r (M, chi) $ in terms of description of the surgery $ M $ (Lemma 3.1 at 42 in [2]) is also valid for $ sigma (K, chi) $. From this, I deduced that we (should) have a relationship like $ sigma (M, chi ^ r) = – sigma_r (M, chi) $ for $ 0 <r <m $. However, there is the summand $ frac {2[F]^ 2r (m-r)} {m ^ 2} $ contained in the definition of $ sigma_r (M, chi) $, and I do not see where and in what form this summary appears in the definition of $ sigma (M, chi) $ (I see however why it appears in the definition of $ sigma_r (M, chi) $, in particular because of the fact that, for a branched cover $ widetilde W rightarrow W $ of closed 4-varieties, there is the formula $ varepsilon_r ( widetilde W) = textrm {sign} (W) – frac {2[F]^ 2r (m-r)} {m ^ 2} $ (Lemma 2.1 at 40 in [2])). So my question is:

Question: Is the relationship $ sigma (M, chi ^ r) = – sigma_r (M, chi) $ correct? If so, where does the summator come from? $ frac {2[F]^ 2r (m-r)} {m ^ 2} $ appears in $ sigma (M, chi) $? If not, is there another relationship that holds for $ sigma (M, chi) $ and $ sigma_r (M, chi) $?

I guess my question has something to do with the connection game $ F $ as I do not see it mentioned in [1] or [3]. However, I am too inexperienced on the subject to come to a conclusion on my own. Thus, any development would be greatly appreciated. Thanks in advance!

References:

[1] A. J. Casson; C. Mc A. Gordon, Cobordism of the classical knotsIn Search of Lost Topology, Progress in Mathematics, Volume 62, p. 181 to 199, Birkhäuser Boston, 1986. With an appendix of P. M. Gilmer (available here)

[2] A. J. Casson; C. Mc A. Gordon, On the nodes in dimension three, Symposium Proceedings in Pure Mathematics, Volume 32, p. 39 – 53, American Mathematical Society, 1978 (available here).

[3] A. Conway, Algebraic and invariant Concordance of Casson-Gordon, notes from a reading group held in Geneva, spring 2017 (available here).