Equivalence of Legendrian band-sum models

Determine to what extent the Type-D and Type-R Legendrian band-sum models are equivalent in general, beyond the connected-sum case of split links.

Background

The paper distinguishes Type-D bands, which directly induce decomposable Lagrangian 1-handles, from Type-R bands, which naturally arise in contact handleslides and regular constructions. Although every Type-D band can be locally induced by a Type-R band, the converse fails in general in the strongest possible sense for one example; the broader relationship between the models remains unclear.

References

For the connected sum of split links, the different models give Legendrian isotopic results , but it is unclear to what extent different models are equivalent in general.

— Derivative links in contact topology  (2609.30182 - Breen et al., 24 Sep 2026) in Section 3, “Legendrian band sums”