# 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 $$omega ^ r$$, $$0 leq r , or $$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 , 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 $$0 . However, there is the summand $$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, BirkhĂ¤user 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).