A-polynomial questions for knots in other 3-manifolds

Assume full AC. Let $Y$ be a closed connected oriented smooth $3$-manifold and let $K\subset Y$ be an oriented knot with irreducible exterior $E=Y\setminus \mathrm{Int}\nu K$. Choose a parametrized tubular neighborhood and an oriented integral basis $(\mu ,\lambda )$ on its boundary torus $T$, where $\mu$ bounds a meridian disk in $\nu K$ and $\lambda$ is a chosen framing; no preferred null-homologous longitude is assumed. For each primitive pair $(p,q)$, filling along $p\mu +q\lambda$ means attaching a solid torus whose meridian has that unoriented class.

Define $Z_{E,T}^{\mathrm{i}\mathrm{r}\mathrm{r}}\subset ({\mathbb{C}}^{\ast }{)}^2$ to be the Zariski closure of the pairs $(M,L)$ arising on a common peripheral invariant line of an irreducible representation ${\pi }_1(E)\to S{L}_2(\mathbb{C})$, in the chosen basis. Let $C_{E,T}^{\mathrm{i}\mathrm{r}\mathrm{r}}$ be the union of its one-dimensional irreducible components. Use a square-free Laurent equation over $\mathbb{C}$ of this reduced curve as the enhanced polynomial, up to a Laurent unit, with $1$ for the empty curve. This definition selects curve components; it does not assert that higher-dimensional peripheral-image components are impossible or detected by that equation.

Do the following analogues hold for such closed ambient manifolds: (i) the Newton polygon of this enhanced polynomial is contained in an affine line if and only if $E$ admits a Seifert fibration over a disk with at most two exceptional circle fibers; and (ii) infinitely many distinct primitive filling slopes on $T$ give manifolds whose every $S{L}_2(\mathbb{C})$ representation has abelian image if and only if $E$ admits that Seifert fibration? What additional hypotheses, if any, are required for these equivalences and for the curve-component convention to capture the relevant peripheral representations? An exceptional circle fiber has multiplicity greater than one.

Baldwin and Sivek, Question 1.8

Open generalization question. Baldwin–Sivek, Question 1.8, ask whether their sphere-knot theorems have analogues for knots in other $3$-manifolds, and specifically ask about an irreducible knot complement and the disk-base Seifert condition. Their wording does not explicitly specify that the ambient manifold is closed; closedness is an explicit restriction in the precise version above. The broader assertion for arbitrary compact ambient manifolds with boundary is false: the core of a solid torus has exterior $T^2\times I$, empty irreducible peripheral locus and abelian filling groups at every slope, but two boundary tori preclude the displayed disk-base fibration. Thus that broader branch is a counterexample, not an unresolved equivalence.

The cited source is arXiv:2405.19197v2, revised 28 November 2025 and published in Math. Ann. 394 (2026). The actual author search on 2026-10-19 found the source question and partial results, with no universal resolution identified. This is a bounded literature-search observation, not a claim that no later solution exists. No proof of the closed-manifold analogues or of a general peripheral-image dimension bound is asserted here.

The solid-torus-core counterexample preserves the refuted compact-with-boundary branch of the earlier formulation.

The solid-torus-core counterexample