The Alexander-polynomial expression conjecture for finite type invariants of long odd-dimensional knots

Let $n$ be an odd integer with $n>1$. For each $p$ with $1\leq p\leq \lfloor n/2\rfloor$, let $\Delta_p(t)$ be the suitably normalized generalized Alexander invariant of a long $n$-knot, with $\Delta_p(1)=1$ and $\Delta_p'(1)=0$, and define coefficients $\alpha_j^p$ by

$$\log \Delta_p(t)\big|_{t=e^h}=\alpha_2^p h^2+\alpha_3^p h^3+\cdots\in\mathbb{Q}[[h]].$$

Let $\hat z_j$ denote the associated finite type invariants, and let the subscript $p$ on the polynomial expressions indicate the algebra generated by the coefficients for all such $p$. The Alexander-polynomial expression conjecture. For arbitrary long $n$-knots, one has

$$\hat z_{2k}\in\sum_p\alpha_{2k}^p+\mathbb{R}[\alpha_{2k-1}^p,\ldots,\alpha_3^p,\alpha_2^p]^{(\mathrm{deg}\,\leq 2k-1)}_p$$

and

$$\hat z_{2k+1}\in\mathbb{R}[\alpha_{2k}^p,\ldots,\alpha_3^p,\alpha_2^p]^{(\mathrm{deg}\,\leq 2k)}_p$$

for $k\geq 1$, where $p$ runs over $1\leq p\leq \lfloor n/2\rfloor$. This asserts that these finite type invariants are determined by the generalized Alexander-polynomial coefficients. The source presents this as a conjectural statement for arbitrary long $n$-knots; the preceding theorem establishes an analogous result only for long handle $(p,q)$-knots.