Towards a symplectic Khovanov homology for links in fibered $3$-manifolds

TL;DR

This paper introduces a new symplectic Khovanov homology for links in fibered 3-manifolds and proposes conjectural combinatorial algebraic descriptions of associated surface categories.

Contribution

It defines a novel homology theory for transverse links in fibered 3-manifolds and conjectures algebraic models generalizing bordered Floer homology strands algebras.

Findings

Proposes a symplectic Khovanov homology for links in fibered 3-manifolds.

Suggests conjectural combinatorial dga descriptions of surface categories.

Establishes a framework for future algebraic and topological investigations.

Abstract

The goal of this paper is twofold: (i) define a symplectic Khovanov type homology for a transverse link in a fibered closed $3$-manifold $M$ (with an auxiliary choice of a homotopy class of loops that intersect each fiber once) and (ii) give conjectural combinatorial dga descriptions of surface categories that appear in (i). These dgas are higher-dimensional analogs of the strands algebras in bordered Heegaard Floer homology, due to Lipshitz-Ozsv\'ath-Thurston \cite{LOT}.