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Characterizing proper rational tangle replacements between certain 2- and 3-stranded torus knots

Determine for which integers m ≥ 2 there exists a proper rational tangle replacement relating T(3,4) and T(2,2m+1), and for which m ≥ 2 there exists a proper rational tangle replacement relating T(3,5) and T(2,2m+1); equivalently, decide which pairs listed in item (3′) of Proposition thm:pairs_torus_knots are at proper rational Gordian distance 1.

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Background

The authors completely determine when their invariants equal 1 for pairs drawn from 2-stranded and 3-stranded positive torus knots. For proper rational tangle replacements (u_q-distance), they note that while some families are known to be related by a single such replacement, a specific family (item (3′)) remains unresolved.

This question asks to identify precisely which of the pairs {T(3,4), T(2,2m+1)} and {T(3,5), T(2,2m+1)} (with m ≥ 2) are indeed related by a proper rational tangle replacement.

References

On the other hand, the $u_q$-distance has not been completely determined for 2-stranded and 3-stranded torus knots. All knots in \cref{item:pairs_torus_knots:1'} of \cref{thm:pairs_torus_knots} are related by a proper rational tangle replacement $Q_a \leadsto Q_b$, for $a,b\in\mathbb{Z}$. The pairs of knots in \cref{item:pairs_torus_knots:2'} are even related by a crossing change. But we do not know which pairs of knots in \cref{item:pairs_torus_knots:3'} are related by a proper rational tangle replacement.

— Khovanov homology and refined bounds for Gordian distances (2409.05743 - Lewark et al., 9 Sep 2024) in After Proposition thm:pairs_torus_knots, Subsection “λ- and λ^−-distance between torus knots”