Nodal Tangles

TL;DR

This paper explores the use of piecewise linear knot diagrams within almost toric fibrations of symplectic four-manifolds, providing new tools and applications for symplectic topology.

Contribution

It introduces a novel approach to studying symplectic topology through piecewise linear knot diagrams and demonstrates several new applications and results.

Findings

Proof of a conjecture by Symington

Counterexamples to Lagrangian Poincaré recurrence

Calculation of displacement energy for toric fibers

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.