Nodal Tangles
Symplectic Geometry
2025-11-07 v3
Abstract
We study piecewise linear knot diagrams in the base of almost toric fibrations of symplectic four-manifolds. These diagrams translate to deformations of the almost toric fibration. We give several applications to symplectic topology, among them a proof of a conjecture by Symington, simpler counterexamples to Lagrangian Poincar\'e recurrence in dimension four, the calculation of the displacement energy for many fibres of toric moment maps, and an elementary recipe for building and distinguishing Lagrangian torus knots.
Cite
@article{arxiv.2506.23754,
title = {Nodal Tangles},
author = {Joel Schmitz},
journal= {arXiv preprint arXiv:2506.23754},
year = {2025}
}
Comments
52 pages, 33 figures; v2: added section 1.2 & 3.3, figure 1 & 4, fixed some typos; v3: slightly simplified proof of Thm. 3.25