Invariants of knotted surfaces from link homology and bridge trisections

TL;DR

This paper develops new invariants for knotted surfaces in four-dimensional space using link homology and bridge trisections, enabling distinction of unknotted and certain knotted spheres.

Contribution

It introduces novel invariants derived from link homology and $A_0$-algebras, connecting these to hyperboxes of chain complexes for knotted surfaces.

Findings

Invariant distinguishes unknotted sphere from some knotted spheres.

Constructs an $A_0$-algebra invariant for bridge-trisected surfaces.

Establishes a new link between $A_0$-algebras and hyperboxes of chain complexes.

Abstract

Meier and Zupan showed that every surface in the four-sphere admits a bridge trisection and can therefore be represented by three simple tangles. This raises the possibility of applying methods from link homology to knotted surfaces. We use link homology to construct an invariant of knotted surfaces (up to isotopy) which distinguishes the unknotted sphere from certain knotted spheres. We also construct an invariant of a bridge-trisected surface in the the form of an $A_\infty$-algebra. Both invariants are defined by a novel connection between $A_\infty$-algebras and Manolescu and Ozsv\'ath's hyperboxes of chain complexes.