Direct construction of P-loose Legendrian unknots

Construct a P-loose Legendrian unknot directly by pushing a P-Moore space past a region of a Legendrian unknot, not near a cusp, after applying a Legendrian isotopy to the unknot.

Background

The paper constructs P-loose Legendrian unknots indirectly. A neighborhood of a P-Moore space is first used to construct a Lagrangian disk, the disk is carved out of a cotangent bundle, and a flexible handle is then attached. The resulting geometric construction is made explicit through boat moves and cusp connected sums, but it is not a direct analogue of Murphy’s construction of loose Legendrians.

The authors ask whether a P-loose Legendrian unknot can instead be obtained by directly pushing a P-Moore space through a region of the Legendrian unknot away from a cusp. They note that any such operation would need to preserve the expected behavior for disconnected subdomains: when the pushed-through region is disconnected, the resulting Legendrian must be loose.