Construction of symplectic flexible links

TL;DR

This paper demonstrates that any smooth one-dimensional link in real projective 3-space can be realized as the fixed-point set of a symmetric symplectic surface in complex projective 3-space, with degrees one or two.

Contribution

It establishes the existence of symplectic surfaces invariant under conjugation representing any given flexible link, extending classical topological results.

Findings

Any smooth link can be realized as a fixed-point set of a symmetric symplectic surface.

The degree of the symplectic surface can be chosen as one or two based on the link's homology class.

No obstructions exist beyond classical topology for representing flexible links symplectically.

Abstract

We show that any smooth one-dimensional link in the real projective three-plane is the fixed-point locus of a smooth symplectic surface in the complex projective three-plane which is invariant under complex conjugation. The degree of the surface can be taken to be either one or two, depending on the homology class of the link. In other words, there are no obstructions to finding a symplectic representative of a flexible link beyond the classical topology.