Contact normalization of divisor graphs using only −1 blow‑ups

Determine whether every plumbing graph associated to a concave symplectic normal crossing (SNC+) divisor can be reduced to Neumann’s topological plumbing calculus (TPC) normal form using only contact‑preserving −1 blow‑ups and −1 blow‑downs, thereby preserving the contactomorphism type of the divisor boundary throughout the normalization process.

Background

In Neumann’s topological plumbing calculus, various graph moves—including ±1 blow‑ups/blow‑downs—are used to reduce a plumbing graph to normal form. However, in the contact setting many of these moves (notably +1 blow‑ups) can change the contactomorphism type or introduce overtwisted structures, so they are not contact‑preserving.

The paper develops a contact‑sensitive framework and proves a modified chain reduction lemma, achieving a "contact normal form" that allows leading 0‑curves and avoids certain non‑contact‑preserving operations. Despite this progress, it remains unclear whether one can reach the full TPC normal form using only −1 blow‑ups/blow‑downs while preserving the boundary contact structure.