The Alexander polynomial of a slice knot, Reidemeister tosion, Whitehead group

Question

My question is about the Alexander polynomial of a slice knot.

For a slice knot $K$, Fox-Millnor and Terasaka proved that $$ \Delta_{K}(t) \doteq f(t) f(t^{-1})$$ for some polynomial $f(t) \in \mathbb{Z}[t]$, where $\Delta_{K}(t)$ is the Alexander polynomial of $K$ and $\doteq$ means up to units of $\mathbb{Z}[t, t^{-1}]$.

Let $D$ be a slice disk for $K$. Then a folklore result states that $$f(t) \doteq \Delta_{D}(t),$$ where $ \Delta_{D}(t)$ denotes the Alexander polynomial of $D$.

$\textbf{Question 1.} $ Is there a reference of this folklore result ?

Note that this folklore result is important when we calculate the Alexander polynomial of a ribbon knot $R$. Indeed, let $D$ the slice disk in the $4$-ball $B^4$ obtained from a ribbon presentation of $R$. Then we easily obtain the presentation of $\pi_{1}(B^4 \setminus N(D))$ and can determine $ \Delta_{D}(t)$ using Fox calculus, where $N(D)$ is a tubular neighborhood of $D$.

I heard from Akio Kawauchi (who was my adviser) that this folklore result is true, at least, for ribbon knots and he did not know any references. His proof is using the Blanchfield duality (I do not follow the proof fully).

There is another question. Here recall the Reidemeister torsion. Let $K$ be a knot in $S^3$ and $\tau_{\alpha}(S^3 \setminus N(K))$ the Reidemeister torsion associated to the abelian map $\alpha$ of $\pi_{1}(S^3 \setminus N(K) )$. (Precisely, $\alpha : \pi_{1}(S^3 \setminus N(K) ) \to GL(1; \mathbb{Q}(t)).$) Milnor's theorem states that $$ \tau_{\alpha}(S^3 \setminus N(K)) \doteq \dfrac{\Delta_K(t)}{t-1}$$ Let $D$ be a slice disk for some knot and $\tau_{\alpha}((B^4 \setminus N(D)))$ be the Reidemeister torsion of a slice disk $D$ associated to the abelian map $\alpha$ of $\pi_{1}(B^4 \setminus N(D) )$.

$\textbf{Question 2.} $ Is it true that $$ \tau_{\alpha}(B^4 \setminus N(D))\doteq \dfrac{\Delta_D(t)}{t-1} \ ?$$

It seems that, if the Whitehead group of $\pi_{1}(B^4 \setminus N(D))$ is trivial, then Question 2 is true.

$\textbf{Question 3.} $ Is the Whitehead group of $\pi_{1}(B^4 \setminus N(D))$ trivial ?

Finally, note that I am not familiar with the Reidemeister torsion and the Whitehead group, and therefore there might exist some wrong descriptions. If so, please tell me !

Answer 1

It is in general not true, that for a knot $K$ with slice disk $D$ we have $\Delta_K(t)\equiv f(t)\cdot f(t^{-1})$ with $f(t)=\Delta_D(t)$. For example, even if $K$ is the trivial knot, then we can take a slice disk which is given by connect sum of the trivial disk with a knotted $S^2$ in $D^4$. Then $\Delta_D(t)$ will be the Alexander polynomial of the 2-knot, which can be non-trivial. The statement is true though if $D$ is a ribbon disk. The argument is as follows. Let $\Lambda=\Bbb{Z}[t,t^{-1}]$. We denote the zero-framed surgery on $K$ by $N$ and we denote by $X$ the exterior of $D$. We then have a long exact sequence $H_2(X;\Lambda)\to H_2(N,X;\Lambda)\to H_1(N;\Lambda)\to H_1(X;\Lambda)\to 0$. Note that the last map is indeed surjective, since $\pi_1(N)$ surjects onto $\pi_1(X)$. (Here we use that $D$ is a ribbon disk.) By Poincare duality and universal coefficient theorem we have $H_2(N,X;\Lambda)\cong \overline{H_1(X;\Lambda)}$ and $H_2(X;\Lambda)=0$. We thus have a short exact sequence $0\to\overline{H_1(X;\Lambda)}\to H_1(N;\Lambda)\to H_1(X;\Lambda)\to 0$ and the result follows from basis properties of orders of $\Lambda$-modules.

Answer 2

You should check out Kirk and Livingston's paper, Twisted Alexander invariants, Reidemeister torsion and Casson–Gordon invariants, Topology Vol. 38, No. 3, pp. 635-661, 1999, which addresses many of these issues. Milnor's articles on torsion are a great starting point for all of the issues that you raise.

I think your focus in questions 2 and 3 on the Whitehead group of the fundamental group of the disk complement is somewhat misplaced, because the torsions involved are Reidemeister torsions, not Whitehead torsions. In general, to define torsion of a complex X, you want an acyclic complex, and typically one has to tensor the chain complex of the universal cover of X with some $Z[\pi_1(X)]$ module. Whitehead torsion arises in the special case when this module is $Z[\pi_1(X)]$ itself, and typically is not for a complex $X$ but rather a pair $(X,A)$. In that setting, the available torsions are described by the Whitehead group. In the knot theory setting, the torsions typically live in a K-group (more or less the Whitehead group) of some quotient of $Z[\pi_1(X)]$.

As Kim's answer indicates, this was the setting for the published proof (1966) of the Fox-Milnor condition.

A later addition with regard to question 3: I don't believe that the Whitehead group vanishes in general for $\pi_1(B^4 - D)$. As an example, take a twist-spun 2-bridge knot; the fundamental group of the complement of the resulting sphere is a semi direct product $G= Z_p \rtimes Z $ with the generator of $Z$ acting by $-1$ on $Z_p$. (p is odd here.) Connected sum with a trivial disk gives a disk with group $G$. I'm pretty sure this group has a nontrivial Whitehead group for $p \geq 5$ since you can easily construct some nontrivial units in the group ring.