# 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  and On the slice kind of knots , as well as notes by A. Conways Algebraic and invariant Concordance of Casson-Gordon  (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 and once as $$sigma_r (M, chi)$$ in . In order to formulate my question, I quickly trace the construction of the invariant $$sigma (M, chi)$$, as found in  p.14, (resp.  in. 183), and $$sigma_r (M, chi)$$, as found in  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 , 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.  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 ) 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 )). 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  or . 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:

 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)

 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).

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

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