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Proper rational tangle replacements for certain torus-knot pairs

Determine, for integers m ≥ 2, which pairs of torus knots among {T(3,4), T(2, 2m+1)} and {T(3,5), T(2, 2m+1)} are related by a proper rational tangle replacement; equivalently, classify the values of m for which a single proper rational tangle replacement exists between these pairs.

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Background

The paper completely determines when pairs of two- and three-stranded torus knots are at Gordian distance 1, and computes the λ and λ− distances for various pairs. However, for certain mixed pairs involving T(3,4) or T(3,5) with T(2, 2m+1), the existence of a proper rational tangle replacement is not resolved.

This question asks for a precise characterization of which m yield a proper rational tangle replacement, i.e., minimal rational tangle operations transforming one knot into the other.

References

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 Section "λ- and Î⁻-distance between torus knots" (after Proposition thm:pairs_torus_knots)