Weinstein Presentations for High-Dimensional Antisurgery
This talk presents an explicit algorithm for constructing Weinstein presentations of subdomains obtained by carving out regular Lagrangian disks. The core innovation is the boat move, a new higher-dimensional Legendrian isotopy that enables systematic handle slides without creating new obstructions. The method makes previously abstract antisurgery constructions fully diagrammatic, provides explicit front descriptions of P-loose Legendrians, and extends three-dimensional techniques to arbitrary dimensions.Script
Carving out a Lagrangian disk from a symplectic manifold produces a new Weinstein subdomain, but how do you describe its handle structure explicitly? Until now, no general algorithm existed for computing the presentation after antisurgery.
The authors solve this with a new Legendrian isotopy called the boat move. It transforms a critical point of arbitrary Morse index into a local maximum by suspending a lower-dimensional Reidemeister move over a disk.
Here's how the algorithm works. Start with a regular Lagrangian disk presented as the zero section of a cotangent bundle plus critical handles. The relevant Reeb chords connect each attaching Legendrian to the boundary unknot and lie inside a bounded front projection region.
For each relevant Reeb chord with local index k, apply an n comma n minus k boat move to convert its critical point into a maximum. Then slide the attaching Legendrian through a newly added cancelling handle pair. This removes the chord without generating new obstructions, making the process finite and algorithmic.
The method produces explicit front descriptions of P-loose Legendrians for the first time. For a single prime P, the presentation consists of four loose unknots with parallel regions outside a bounded zone, linked according to P inside, and decorated with local boat-move singularities where the critical structure lives.
The carved subdomain has exactly one additional n minus 1 handle and the same number of critical handles as before, with attaching spheres modified by boat moves and cusp gluings. To dive deeper into antisurgery and Weinstein presentations, visit EmergentMind.com where you can explore this paper and create your own research videos.
