Isotopy classification of Z^2-concordances to the Hopf link

Ascertain whether every Z^2-concordance between a two-component link L with multivariable Alexander polynomial 1 and the Hopf link H is isotopic rel. boundary to the concordance obtained by applying a Dehn twist to S^3 × [0,1].

Background

The paper proves that any two Z2-concordances between the Hopf link and an Alexander polynomial one link are equivalent rel. boundary. However, the stronger statement that such concordances are isotopic rel. boundary is not established.

Determining the isotopy classification would refine the uniqueness result for Z2-concordances and align with parallel questions about isotopy versus equivalence in the topological category for 4-manifolds and embedded or immersed surfaces.

This question relates to the structure of the mapping class group of S3 × I and the effect of Dehn twists, which are known to generate nontrivial mapping classes; resolving it would clarify whether these mapping classes can be absorbed by concordance isotopies in the Z2 setting.

References

Our result is stated up to equivalence instead of isotopy because we do not know whether a given $Z2$-concordance from $L$ to $H$ is isotopic rel. boundary to the concordance obtained by applying a Dehn twist to $S3 \times [0,1]$.

— Immersed surfaces with knot group $\mathbb{Z}$ (2410.04635 - Conway et al., 6 Oct 2024) in Introduction, paragraph on Theorem \ref{thm:HopfConcordancesEquivalent}